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3.13.84 \(\int \frac {(A+B x) (d+e x)^{7/2}}{(a-c x^2)^3} \, dx\)

Optimal. Leaf size=396 \[ \frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (7 a B e \left (3 \sqrt {a} e+2 \sqrt {c} d\right )-A \left (18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (7 a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (-18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{11/4}}+\frac {\sqrt {d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

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Rubi [A]  time = 0.88, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {819, 827, 1166, 208} \begin {gather*} \frac {\sqrt {d+e x} \left (x \left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (a e^2+c d^2\right )\right )+a e \left (-5 a A e^2-14 a B d e+7 A c d^2\right )\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (7 a B e \left (3 \sqrt {a} e+2 \sqrt {c} d\right )-A \left (18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (7 a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (-18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{11/4}}+\frac {(d+e x)^{5/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(7/2))/(a - c*x^2)^3,x]

[Out]

((d + e*x)^(5/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) + (Sqrt[d + e*x]*(a*e*(7*A*c*d^2 -
 14*a*B*d*e - 5*a*A*e^2) + (2*A*c*d*(3*c*d^2 - 2*a*e^2) - 7*a*B*e*(c*d^2 + a*e^2))*x))/(16*a^2*c^2*(a - c*x^2)
) + ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 + 18*Sqrt[a]*c*d*e
 + 5*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(32*a^(5/2)*c^(11/4)) - ((S
qrt[c]*d + Sqrt[a]*e)^(3/2)*(7*a*B*e*(2*Sqrt[c]*d - 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 - 18*Sqrt[a]*c*d*e + 5*a*
Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(11/4))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 819

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x)^(
m - 1)*(a + c*x^2)^(p + 1)*(a*(e*f + d*g) - (c*d*f - a*e*g)*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)),
Int[(d + e*x)^(m - 2)*(a + c*x^2)^(p + 1)*Simp[a*e*(e*f*(m - 1) + d*g*m) - c*d^2*f*(2*p + 3) + e*(a*e*g*m - c*
d*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ
[m, 1] && (EqQ[d, 0] || (EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
 - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
 && NeQ[c*d^2 + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
 q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {(A+B x) (d+e x)^{7/2}}{\left (a-c x^2\right )^3} \, dx &=\frac {(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}-\frac {\int \frac {(d+e x)^{3/2} \left (\frac {1}{2} \left (-6 A c d^2+a e (7 B d+5 A e)\right )-\frac {1}{2} e (A c d-7 a B e) x\right )}{\left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{4} \left (12 A c^2 d^4-14 a B c d^3 e-19 a A c d^2 e^2+28 a^2 B d e^3+5 a^2 A e^4\right )+\frac {1}{4} e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2}\\ &=\frac {(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} e \left (12 A c^2 d^4-14 a B c d^3 e-19 a A c d^2 e^2+28 a^2 B d e^3+5 a^2 A e^4\right )-\frac {1}{4} d e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right )+\frac {1}{4} e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{4 a^2 c^2}\\ &=\frac {(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}-\frac {\left (\left (\sqrt {c} d+\sqrt {a} e\right )^2 \left (7 a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2-18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{c d+\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{32 a^{5/2} c^2}+\frac {\left (\frac {1}{8} e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right )+\frac {-\frac {1}{2} c d e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right )-2 c \left (\frac {1}{4} e \left (12 A c^2 d^4-14 a B c d^3 e-19 a A c d^2 e^2+28 a^2 B d e^3+5 a^2 A e^4\right )-\frac {1}{4} d e \left (2 A c d \left (3 c d^2-4 a e^2\right )-7 a B e \left (c d^2-3 a e^2\right )\right )\right )}{4 \sqrt {a} \sqrt {c} e}\right ) \operatorname {Subst}\left (\int \frac {1}{c d-\sqrt {a} \sqrt {c} e-c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 a^2 c^2}\\ &=\frac {(d+e x)^{5/2} (a (B d+A e)+(A c d+a B e) x)}{4 a c \left (a-c x^2\right )^2}+\frac {\sqrt {d+e x} \left (a e \left (7 A c d^2-14 a B d e-5 a A e^2\right )+\left (2 A c d \left (3 c d^2-2 a e^2\right )-7 a B e \left (c d^2+a e^2\right )\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (7 a B e \left (2 \sqrt {c} d+3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2+18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{11/4}}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \left (7 a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2-18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{11/4}}\\ \end {align*}

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Mathematica [B]  time = 2.96, size = 802, normalized size = 2.03 \begin {gather*} \frac {\frac {2 a c^2 \left (c d^2-a e^2\right ) (-a A e+A c d x+a B (d-e x)) (d+e x)^{9/2}}{\left (a-c x^2\right )^2}+\frac {c^2 \left (6 A c^2 x d^3+a c e (-9 A d-7 B x d+4 A e x) d-a^2 e^2 (-10 B d+A e+3 B e x)\right ) (d+e x)^{9/2}}{2 \left (a-c x^2\right )}-\frac {\left (A \left (54 c^2 d^4+81 a c e^2 d^2+5 a^2 e^4\right )-7 a B d e \left (9 c d^2+11 a e^2\right )\right ) \left (15 \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2}+2 \sqrt {a} \sqrt [4]{c} e \sqrt {d+e x} \left (15 a e^2+c \left (58 d^2+16 e x d+3 e^2 x^2\right )\right )-15 \left (\sqrt {c} d+\sqrt {a} e\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )\right )}{60 \sqrt {a} \sqrt [4]{c}}-\frac {\left (2 A c d \left (3 c d^2+2 a e^2\right )-a B e \left (7 c d^2+3 a e^2\right )\right ) \left (\left (\sqrt {c} d-\sqrt {a} e\right ) \left (15 c^{7/4} (d+e x)^{7/2}+7 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (3 c^{5/4} (d+e x)^{5/2}+5 \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt [4]{c} \sqrt {d+e x} \left (\sqrt {c} (4 d+e x)-3 \sqrt {a} e\right )-3 \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )\right )\right )\right )-\left (\sqrt {c} d+\sqrt {a} e\right ) \left (15 c^{7/4} (d+e x)^{7/2}+7 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (3 c^{5/4} (d+e x)^{5/2}+5 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt [4]{c} \sqrt {d+e x} \left (3 \sqrt {a} e+\sqrt {c} (4 d+e x)\right )-3 \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )\right )\right )\right )\right )}{60 \sqrt {a} c^{3/4}}}{8 a^2 c^2 \left (c d^2-a e^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(7/2))/(a - c*x^2)^3,x]

[Out]

((2*a*c^2*(c*d^2 - a*e^2)*(d + e*x)^(9/2)*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(a - c*x^2)^2 + (c^2*(d + e*x)
^(9/2)*(6*A*c^2*d^3*x + a*c*d*e*(-9*A*d - 7*B*d*x + 4*A*e*x) - a^2*e^2*(-10*B*d + A*e + 3*B*e*x)))/(2*(a - c*x
^2)) - ((-7*a*B*d*e*(9*c*d^2 + 11*a*e^2) + A*(54*c^2*d^4 + 81*a*c*d^2*e^2 + 5*a^2*e^4))*(2*Sqrt[a]*c^(1/4)*e*S
qrt[d + e*x]*(15*a*e^2 + c*(58*d^2 + 16*d*e*x + 3*e^2*x^2)) + 15*(Sqrt[c]*d - Sqrt[a]*e)^(7/2)*ArcTanh[(c^(1/4
)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] - 15*(Sqrt[c]*d + Sqrt[a]*e)^(7/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x
])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(60*Sqrt[a]*c^(1/4)) - ((2*A*c*d*(3*c*d^2 + 2*a*e^2) - a*B*e*(7*c*d^2 + 3*a*
e^2))*((Sqrt[c]*d - Sqrt[a]*e)*(15*c^(7/4)*(d + e*x)^(7/2) + 7*(Sqrt[c]*d - Sqrt[a]*e)*(3*c^(5/4)*(d + e*x)^(5
/2) + 5*(Sqrt[c]*d - Sqrt[a]*e)*(c^(1/4)*Sqrt[d + e*x]*(-3*Sqrt[a]*e + Sqrt[c]*(4*d + e*x)) - 3*(Sqrt[c]*d - S
qrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]))) - (Sqrt[c]*d + Sqrt[a]*e)*(15*
c^(7/4)*(d + e*x)^(7/2) + 7*(Sqrt[c]*d + Sqrt[a]*e)*(3*c^(5/4)*(d + e*x)^(5/2) + 5*(Sqrt[c]*d + Sqrt[a]*e)*(c^
(1/4)*Sqrt[d + e*x]*(3*Sqrt[a]*e + Sqrt[c]*(4*d + e*x)) - 3*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(c^(1/4)*Sqr
t[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])))))/(60*Sqrt[a]*c^(3/4)))/(8*a^2*c^2*(c*d^2 - a*e^2)^2)

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IntegrateAlgebraic [A]  time = 4.78, size = 717, normalized size = 1.81 \begin {gather*} -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (-21 a^{3/2} B e^2+18 \sqrt {a} A c d e+5 a A \sqrt {c} e^2-14 a B \sqrt {c} d e+12 A c^{3/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {\sqrt {a} \sqrt {c} e-c d}}{\sqrt {c} d-\sqrt {a} e}\right )}{32 a^{5/2} c^{5/2} \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (21 a^{3/2} B e^2-18 \sqrt {a} A c d e+5 a A \sqrt {c} e^2-14 a B \sqrt {c} d e+12 A c^{3/2} d^2\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-\sqrt {a} \sqrt {c} e-c d}}{\sqrt {a} e+\sqrt {c} d}\right )}{32 a^{5/2} c^{5/2} \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}-\frac {e \sqrt {d+e x} \left (5 a^3 A e^6+7 a^3 B e^5 (d+e x)+7 a^3 B d e^5-16 a^2 A c d^2 e^4+14 a^2 A c d e^4 (d+e x)-9 a^2 A c e^4 (d+e x)^2-14 a^2 B c d^3 e^3+14 a^2 B c d^2 e^3 (d+e x)+7 a^2 B c d e^3 (d+e x)^2-11 a^2 B c e^3 (d+e x)^3+17 a A c^2 d^4 e^2-32 a A c^2 d^3 e^2 (d+e x)+23 a A c^2 d^2 e^2 (d+e x)^2-8 a A c^2 d e^2 (d+e x)^3+7 a B c^2 d^5 e-21 a B c^2 d^4 e (d+e x)+21 a B c^2 d^3 e (d+e x)^2-7 a B c^2 d^2 e (d+e x)^3-6 A c^3 d^6+18 A c^3 d^5 (d+e x)-18 A c^3 d^4 (d+e x)^2+6 A c^3 d^3 (d+e x)^3\right )}{16 a^2 c^2 \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(7/2))/(a - c*x^2)^3,x]

[Out]

-1/16*(e*Sqrt[d + e*x]*(-6*A*c^3*d^6 + 7*a*B*c^2*d^5*e + 17*a*A*c^2*d^4*e^2 - 14*a^2*B*c*d^3*e^3 - 16*a^2*A*c*
d^2*e^4 + 7*a^3*B*d*e^5 + 5*a^3*A*e^6 + 18*A*c^3*d^5*(d + e*x) - 21*a*B*c^2*d^4*e*(d + e*x) - 32*a*A*c^2*d^3*e
^2*(d + e*x) + 14*a^2*B*c*d^2*e^3*(d + e*x) + 14*a^2*A*c*d*e^4*(d + e*x) + 7*a^3*B*e^5*(d + e*x) - 18*A*c^3*d^
4*(d + e*x)^2 + 21*a*B*c^2*d^3*e*(d + e*x)^2 + 23*a*A*c^2*d^2*e^2*(d + e*x)^2 + 7*a^2*B*c*d*e^3*(d + e*x)^2 -
9*a^2*A*c*e^4*(d + e*x)^2 + 6*A*c^3*d^3*(d + e*x)^3 - 7*a*B*c^2*d^2*e*(d + e*x)^3 - 8*a*A*c^2*d*e^2*(d + e*x)^
3 - 11*a^2*B*c*e^3*(d + e*x)^3))/(a^2*c^2*(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(d + e*x)^2)^2) + ((Sqrt[c]*
d + Sqrt[a]*e)^2*(12*A*c^(3/2)*d^2 - 14*a*B*Sqrt[c]*d*e - 18*Sqrt[a]*A*c*d*e + 21*a^(3/2)*B*e^2 + 5*a*A*Sqrt[c
]*e^2)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(32*a^(5/2)*c^(5/2)*S
qrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) - ((Sqrt[c]*d - Sqrt[a]*e)^2*(12*A*c^(3/2)*d^2 - 14*a*B*Sqrt[c]*d*e +
 18*Sqrt[a]*A*c*d*e - 21*a^(3/2)*B*e^2 + 5*a*A*Sqrt[c]*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d +
e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(32*a^(5/2)*c^(5/2)*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e))])

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fricas [B]  time = 11.79, size = 6669, normalized size = 16.84

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="fricas")

[Out]

-1/64*((a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*
d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*
a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^
3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 53
2*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)
/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7
*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(30240*A^3*B*c^6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*
e^5 + 504*(245*A*B^3*a^2*c^4 - 207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*c^4 - 2727
*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^5*e^8 + (242501*B^4*a^4*c^2 - 573888*A^2
*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^9 + 28*(29743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481
*B^4*a^5*c - 122892*A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c + 625*A^3*B*a^4*c^2)*
d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x + d) + (1260*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a
^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^6 + 49*(35*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*
A^3*a^4*c^5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*B^2*a^6*c^3 + 25*A^3*a^5*c^4)*
e^10 + (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 13*A*a^6*c^9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^
4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^
4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 +
25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3
*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^
3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3
+ 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*
c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441
*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*
c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a
^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 - 336*A*B*a
*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2
*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4
*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2
 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*
a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11
*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(30240*A^3*B*c^6*d^9*e^4 - 3024*(35
*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3*a^2*c^4 - 207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 -
 108486*A^2*B^2*a^2*c^4 - 2727*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^5*e^8 + (2
42501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^9 + 28*(29743*A*B^3*a^4*c^2 - 1051*A^3*B
*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c - 122892*A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B
^3*a^5*c + 625*A^3*B*a^4*c^2)*d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x + d) - (1260*A^2*B*a^
3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^6 + 49*(35*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7
 + 3*(1911*A*B^2*a^5*c^4 + 85*A^3*a^4*c^5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*
B^2*a^6*c^3 + 25*A^3*a^5*c^4)*e^10 + (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 13*A*a^6*c^9*d*e^2 + 21*B*a^7*c
^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 -
 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*
a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (1944
81*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d
^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 + a^5*c^5*sqrt((44100*A^2*B^2*c^4*d
^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*
d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*
A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c
+ 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a
^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) + (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqr
t((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7
 - a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^
2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B
^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (1
94481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d
^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))*log(-(3024
0*A^3*B*c^6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3*a^2*c^4 - 207*A^3*B*a*c^5)*d^
7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*c^4 - 2727*A^4*a*c^5)*d^6*e^7 - 14*(40523*A*B^3*a^3*c^3 - 80
19*A^3*B*a^2*c^4)*d^5*e^8 + (242501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509*A^4*a^2*c^4)*d^4*e^9 + 28*(29
743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c - 122892*A^2*B^2*a^4*c^2 - 3125*A^4*a^3
*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c + 625*A^3*B*a^4*c^2)*d*e^12 + (194481*B^4*a^6 - 625*A^4*a^4*c^2)*e^13)*
sqrt(e*x + d) + (1260*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c^6)*d^4*e^6 + 49*(35*B^3*a^5*c
^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*A^3*a^4*c^5)*d^2*e^8 - 21*(147*B^3*a^6*c^3 - 55*A^
2*B*a^5*c^4)*d*e^9 - 5*(441*A*B^2*a^6*c^3 + 25*A^3*a^5*c^4)*e^10 - (12*A*a^5*c^10*d^3 - 14*B*a^6*c^9*d^2*e - 1
3*A*a^6*c^9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^
5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a
*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25
*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)))*sqrt((144
*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d^2*e^5 + 210*A*B*a^4*e^7 - a^5
*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 20
70*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3
*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*
B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2
 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^6)/(a^5*c^5))) - (a^2*c^4*x^4
- 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a
^3*c*d^2*e^5 + 210*A*B*a^4*e^7 - a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)
*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*
B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c +
 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7
*B^2*a^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c
)*d*e^6)/(a^5*c^5))*log(-(30240*A^3*B*c^6*d^9*e^4 - 3024*(35*A^2*B^2*a*c^5 + A^4*c^6)*d^8*e^5 + 504*(245*A*B^3
*a^2*c^4 - 207*A^3*B*a*c^5)*d^7*e^6 - 4*(12005*B^4*a^3*c^3 - 108486*A^2*B^2*a^2*c^4 - 2727*A^4*a*c^5)*d^6*e^7
- 14*(40523*A*B^3*a^3*c^3 - 8019*A^3*B*a^2*c^4)*d^5*e^8 + (242501*B^4*a^4*c^2 - 573888*A^2*B^2*a^3*c^3 - 13509
*A^4*a^2*c^4)*d^4*e^9 + 28*(29743*A*B^3*a^4*c^2 - 1051*A^3*B*a^3*c^3)*d^3*e^10 - 2*(194481*B^4*a^5*c - 122892*
A^2*B^2*a^4*c^2 - 3125*A^4*a^3*c^3)*d^2*e^11 - 14*(27783*A*B^3*a^5*c + 625*A^3*B*a^4*c^2)*d*e^12 + (194481*B^4
*a^6 - 625*A^4*a^4*c^2)*e^13)*sqrt(e*x + d) - (1260*A^2*B*a^3*c^6*d^5*e^5 - 42*(70*A*B^2*a^4*c^5 + 3*A^3*a^3*c
^6)*d^4*e^6 + 49*(35*B^3*a^5*c^4 - 51*A^2*B*a^4*c^5)*d^3*e^7 + 3*(1911*A*B^2*a^5*c^4 + 85*A^3*a^4*c^5)*d^2*e^8
 - 21*(147*B^3*a^6*c^3 - 55*A^2*B*a^5*c^4)*d*e^9 - 5*(441*A*B^2*a^6*c^3 + 25*A^3*a^5*c^4)*e^10 - (12*A*a^5*c^1
0*d^3 - 14*B*a^6*c^9*d^2*e - 13*A*a^6*c^9*d*e^2 + 21*B*a^7*c^8*e^3)*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35
*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(56
35*A*B^3*a^2*c^2 + 387*A^3*B*a*c^3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^
12 - 532*(441*A*B^3*a^3*c + 25*A^3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2
)*e^14)/(a^5*c^11)))*sqrt((144*A^2*c^4*d^7 - 336*A*B*a*c^3*d^6*e + 1120*A*B*a^2*c^2*d^4*e^3 - 1050*A*B*a^3*c*d
^2*e^5 + 210*A*B*a^4*e^7 - a^5*c^5*sqrt((44100*A^2*B^2*c^4*d^6*e^8 - 2940*(35*A*B^3*a*c^3 + 3*A^3*B*c^4)*d^5*e
^9 + 49*(1225*B^4*a^2*c^2 - 2070*A^2*B^2*a*c^3 + 9*A^4*c^4)*d^4*e^10 + 56*(5635*A*B^3*a^2*c^2 + 387*A^3*B*a*c^
3)*d^3*e^11 - 42*(5145*B^4*a^3*c - 952*A^2*B^2*a^2*c^2 + 25*A^4*a*c^3)*d^2*e^12 - 532*(441*A*B^3*a^3*c + 25*A^
3*B*a^2*c^2)*d*e^13 + (194481*B^4*a^4 + 22050*A^2*B^2*a^3*c + 625*A^4*a^2*c^2)*e^14)/(a^5*c^11)) + 28*(7*B^2*a
^2*c^2 - 15*A^2*a*c^3)*d^5*e^2 - 35*(21*B^2*a^3*c - 11*A^2*a^2*c^2)*d^3*e^4 + 105*(7*B^2*a^4 - A^2*a^3*c)*d*e^
6)/(a^5*c^5))) - 4*(4*B*a^2*c*d^3 + 11*A*a^2*c*d^2*e - 14*B*a^3*d*e^2 - 5*A*a^3*e^3 - (6*A*c^3*d^3 - 7*B*a*c^2
*d^2*e - 8*A*a*c^2*d*e^2 - 11*B*a^2*c*e^3)*x^3 + (A*a*c^2*d^2*e + 26*B*a^2*c*d*e^2 + 9*A*a^2*c*e^3)*x^2 + (10*
A*a*c^2*d^3 + 5*B*a^2*c*d^2*e + 4*A*a^2*c*d*e^2 - 7*B*a^3*e^3)*x)*sqrt(e*x + d))/(a^2*c^4*x^4 - 2*a^3*c^3*x^2
+ a^4*c^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="giac")

[Out]

sage2

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maple [B]  time = 0.11, size = 1828, normalized size = 4.62

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(7/2)/(-c*x^2+a)^3,x)

[Out]

3/8*e/a^2*c/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(
1/2)*c)*A*d^4+3/8*e/a^2*c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2
)^(1/2))*c)^(1/2)*c)*A*d^4-21/32*e^4/c^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2
)^(1/2))*c)^(1/2)*c)*B+11/16*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(7/2)*B+9/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^
(5/2)*A+21/32*e^4/c^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)
*B+7/32*e^2/a/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d^
2-1/4*e^3/a/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d-7/3
2*e^2/a/c/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d^2+7/8*e
^4/c/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)
*B*d-7/16*e^2/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*
c)^(1/2)*c)*B*d^3-19/32*e^3/a/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*
c*e^2)^(1/2))*c)^(1/2)*c)*A*d^2+3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(1/2)*A*d^6+21/16*e^2/(c*e^2*x^2-a*e^2
)^2/a*(e*x+d)^(3/2)*B*d^4-5/16*e^7/(c*e^2*x^2-a*e^2)^2*a/c^2*(e*x+d)^(1/2)*A+e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)
^(1/2)*A*d^2-17/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*A*d^4-7/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*
B*d^5-7/16*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(5/2)*B*d-7/8*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*B*d^2+7/8*e
^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*B*d^3+5/32*e^5/c/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arcta
n((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A-3/16*e/a^2/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctan((e*x+
d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d^3+5/32*e^5/c/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*
arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A+3/16*e/a^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(
(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d^3+1/2*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(7/2)*A*d+7/16*e^
2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(7/2)*B*d^2-23/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*A*d^2-21/16*e^2/(c*e
^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*B*d^3-7/8*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*A*d+2*e^3/(c*e^2*x^2-a*e^2)^
2/a*(e*x+d)^(3/2)*A*d^3-7/16*e^6/(c*e^2*x^2-a*e^2)^2/c^2*a*(e*x+d)^(3/2)*B-3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*
x+d)^(7/2)*A*d^3+9/8*e/(c*e^2*x^2-a*e^2)^2*c/a^2*(e*x+d)^(5/2)*A*d^4+1/4*e^3/a/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1
/2)*arctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d-7/16*e^2/a/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1
/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d^3-19/32*e^3/a/(a*c*e^2)^(1/2)/((c*d
+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A*d^2+7/8*e^4/c/(a*c*e^2)^
(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d-7/16*e^6/(c
*e^2*x^2-a*e^2)^2*a/c^2*(e*x+d)^(1/2)*B*d-9/8*e/(c*e^2*x^2-a*e^2)^2*c/a^2*(e*x+d)^(3/2)*A*d^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {7}{2}}}{{\left (c x^{2} - a\right )}^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(7/2)/(-c*x^2+a)^3,x, algorithm="maxima")

[Out]

-integrate((B*x + A)*(e*x + d)^(7/2)/(c*x^2 - a)^3, x)

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mupad [B]  time = 3.69, size = 11687, normalized size = 29.51

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(7/2))/(a - c*x^2)^3,x)

[Out]

- atan(((((20480*A*a^7*c^6*e^7 + 28672*B*a^7*c^6*d*e^6 + 24576*A*a^5*c^8*d^4*e^3 - 45056*A*a^6*c^7*d^2*e^5 - 2
8672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c^5) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e
^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a
^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*
a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6
*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c
*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2))*((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2)
- 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*
A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^
2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*
e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1
/2))/(4096*a^10*c^11))^(1/2) + ((d + e*x)^(1/2)*(441*B^2*a^5*e^10 + 144*A^2*c^5*d^8*e^2 + 25*A^2*a^4*c*e^10 +
385*A^2*a^2*c^3*d^4*e^6 - 126*A^2*a^3*c^2*d^2*e^8 + 196*B^2*a^2*c^3*d^6*e^4 - 735*B^2*a^3*c^2*d^4*e^6 - 420*A^
2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2*e^8 - 56*A*B*a^4*c*d*e^9 - 336*A*B*a*c^4*d^7*e^3 + 1120*A*B*a^2*c^3*d^5*e^
5 - 840*A*B*a^3*c^2*d^3*e^7))/(64*a^4*c^2))*((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A
^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9
*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e
^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^
15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4
096*a^10*c^11))^(1/2)*1i - (((20480*A*a^7*c^6*e^7 + 28672*B*a^7*c^6*d*e^6 + 24576*A*a^5*c^8*d^4*e^3 - 45056*A*
a^6*c^7*d^2*e^5 - 28672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c^5) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^10*
d^7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*
d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c
^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) -
336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^
2*e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2))*((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7
*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8
*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^
9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e
 + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d
*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) - ((d + e*x)^(1/2)*(441*B^2*a^5*e^10 + 144*A^2*c^5*d^8*e^2 + 2
5*A^2*a^4*c*e^10 + 385*A^2*a^2*c^3*d^4*e^6 - 126*A^2*a^3*c^2*d^2*e^8 + 196*B^2*a^2*c^3*d^6*e^4 - 735*B^2*a^3*c
^2*d^4*e^6 - 420*A^2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2*e^8 - 56*A*B*a^4*c*d*e^9 - 336*A*B*a*c^4*d^7*e^3 + 1120
*A*B*a^2*c^3*d^5*e^5 - 840*A*B*a^3*c^2*d^3*e^7))/(64*a^4*c^2))*((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7*(a^15*
c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^
3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d
*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*
B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6*(a
^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2)*1i)/((((20480*A*a^7*c^6*e^7 + 28672*B*a^7*c^6*d*e^6 + 24576*A*a^5*c^8
*d^4*e^3 - 45056*A*a^6*c^7*d^2*e^5 - 28672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c^5) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*
((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4
 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1
/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(
a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 -
1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2))*((144*A^2*a^5*c^10*d^
7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^
5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7
*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 33
6*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*
e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) + ((d + e*x)^(1/2)*(441*B^2*a^5*e^10 + 144*
A^2*c^5*d^8*e^2 + 25*A^2*a^4*c*e^10 + 385*A^2*a^2*c^3*d^4*e^6 - 126*A^2*a^3*c^2*d^2*e^8 + 196*B^2*a^2*c^3*d^6*
e^4 - 735*B^2*a^3*c^2*d^4*e^6 - 420*A^2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2*e^8 - 56*A*B*a^4*c*d*e^9 - 336*A*B*a
*c^4*d^7*e^3 + 1120*A*B*a^2*c^3*d^5*e^5 - 840*A*B*a^3*c^2*d^3*e^7))/(64*a^4*c^2))*((144*A^2*a^5*c^10*d^7 - 441
*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 -
 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6
+ 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a
^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 2
66*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) + (((20480*A*a^7*c^6*e^7 + 28672*B*a^7*c^6*d*e^6 +
 24576*A*a^5*c^8*d^4*e^3 - 45056*A*a^6*c^7*d^2*e^5 - 28672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c^5) + 64*a*c^4*d*e^2*
(d + e*x)^(1/2)*((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2
*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^
5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A
*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^
7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2))*((144
*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 19
6*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) -
 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*
c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*
A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) - ((d + e*x)^(1/2)*(441*B^2
*a^5*e^10 + 144*A^2*c^5*d^8*e^2 + 25*A^2*a^4*c*e^10 + 385*A^2*a^2*c^3*d^4*e^6 - 126*A^2*a^3*c^2*d^2*e^8 + 196*
B^2*a^2*c^3*d^6*e^4 - 735*B^2*a^3*c^2*d^4*e^6 - 420*A^2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2*e^8 - 56*A*B*a^4*c*d
*e^9 - 336*A*B*a*c^4*d^7*e^3 + 1120*A*B*a^2*c^3*d^5*e^5 - 840*A*B*a^3*c^2*d^3*e^7))/(64*a^4*c^2))*((144*A^2*a^
5*c^10*d^7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a
^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^
2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(
1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8
*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) + (9261*B^3*a^7*e^14 - 864*A^3*c^7
*d^11*e^3 - 7398*A^3*a^2*c^5*d^7*e^7 + 6140*A^3*a^3*c^4*d^5*e^9 - 2182*A^3*a^4*c^3*d^3*e^11 + 1372*B^3*a^3*c^4
*d^8*e^6 - 9947*B^3*a^4*c^3*d^6*e^8 + 25039*B^3*a^5*c^2*d^4*e^10 - 525*A^2*B*a^6*c*e^14 + 4104*A^3*a*c^6*d^9*e
^5 + 200*A^3*a^5*c^2*d*e^13 - 25725*B^3*a^6*c*d^2*e^12 - 3528*A*B^2*a^2*c^5*d^9*e^5 + 22638*A*B^2*a^3*c^4*d^7*
e^7 - 51156*A*B^2*a^4*c^3*d^5*e^9 + 48510*A*B^2*a^5*c^2*d^3*e^11 - 16884*A^2*B*a^2*c^5*d^8*e^6 + 34083*A^2*B*a
^3*c^4*d^6*e^8 - 30135*A^2*B*a^4*c^3*d^4*e^10 + 10437*A^2*B*a^5*c^2*d^2*e^12 - 16464*A*B^2*a^6*c*d*e^13 + 3024
*A^2*B*a*c^6*d^10*e^4)/(2048*a^6*c^5)))*((144*A^2*a^5*c^10*d^7 - 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a
^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6
*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 - 25*A^2*a*c*e^7*(
a^15*c^11)^(1/2) - 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e + 245*B^2*a*c*d^2*e^5*(a^15*c
^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 + 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*
a^10*c^11))^(1/2)*2i - atan(((((20480*A*a^7*c^6*e^7 + 28672*B*a^7*c^6*d*e^6 + 24576*A*a^5*c^8*d^4*e^3 - 45056*
A*a^6*c^7*d^2*e^5 - 28672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c^5) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^1
0*d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^
8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8
*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2)
- 336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*
d^2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2))*((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e
^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a
^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*
a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6
*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c
*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) + ((d + e*x)^(1/2)*(441*B^2*a^5*e^10 + 144*A^2*c^5*d^8*e^2 +
 25*A^2*a^4*c*e^10 + 385*A^2*a^2*c^3*d^4*e^6 - 126*A^2*a^3*c^2*d^2*e^8 + 196*B^2*a^2*c^3*d^6*e^4 - 735*B^2*a^3
*c^2*d^4*e^6 - 420*A^2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2*e^8 - 56*A*B*a^4*c*d*e^9 - 336*A*B*a*c^4*d^7*e^3 + 11
20*A*B*a^2*c^3*d^5*e^5 - 840*A*B*a^3*c^2*d^3*e^7))/(64*a^4*c^2))*((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^1
5*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*
d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6
*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 24
5*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*
(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2)*1i - (((20480*A*a^7*c^6*e^7 + 28672*B*a^7*c^6*d*e^6 + 24576*A*a^5*c
^8*d^4*e^3 - 45056*A*a^6*c^7*d^2*e^5 - 28672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c^5) + 64*a*c^4*d*e^2*(d + e*x)^(1/2
)*((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e
^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^
(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4
*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3
- 1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2))*((144*A^2*a^5*c^10*
d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*
d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c
^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) -
336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^
2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) - ((d + e*x)^(1/2)*(441*B^2*a^5*e^10 + 14
4*A^2*c^5*d^8*e^2 + 25*A^2*a^4*c*e^10 + 385*A^2*a^2*c^3*d^4*e^6 - 126*A^2*a^3*c^2*d^2*e^8 + 196*B^2*a^2*c^3*d^
6*e^4 - 735*B^2*a^3*c^2*d^4*e^6 - 420*A^2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2*e^8 - 56*A*B*a^4*c*d*e^9 - 336*A*B
*a*c^4*d^7*e^3 + 1120*A*B*a^2*c^3*d^5*e^5 - 840*A*B*a^3*c^2*d^3*e^7))/(64*a^4*c^2))*((144*A^2*a^5*c^10*d^7 + 4
41*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2
 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^
6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B
*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 -
 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2)*1i)/((((20480*A*a^7*c^6*e^7 + 28672*B*a^7*c^6*d*
e^6 + 24576*A*a^5*c^8*d^4*e^3 - 45056*A*a^6*c^7*d^2*e^5 - 28672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c^5) - 64*a*c^4*d
*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 38
5*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d
^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) +
210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A
*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2))*
((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4
 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1
/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(
a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 -
1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) + ((d + e*x)^(1/2)*(44
1*B^2*a^5*e^10 + 144*A^2*c^5*d^8*e^2 + 25*A^2*a^4*c*e^10 + 385*A^2*a^2*c^3*d^4*e^6 - 126*A^2*a^3*c^2*d^2*e^8 +
 196*B^2*a^2*c^3*d^6*e^4 - 735*B^2*a^3*c^2*d^4*e^6 - 420*A^2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2*e^8 - 56*A*B*a^
4*c*d*e^9 - 336*A*B*a*c^4*d^7*e^3 + 1120*A*B*a^2*c^3*d^5*e^5 - 840*A*B*a^3*c^2*d^3*e^7))/(64*a^4*c^2))*((144*A
^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*
B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 1
05*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(a^15*c^
11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*
B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) + (((20480*A*a^7*c^6*e^7 + 28
672*B*a^7*c^6*d*e^6 + 24576*A*a^5*c^8*d^4*e^3 - 45056*A*a^6*c^7*d^2*e^5 - 28672*B*a^6*c^7*d^3*e^4)/(4096*a^6*c
^5) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*
c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^
7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^1
5*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11
)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^1
0*c^11))^(1/2))*((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2
*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^
5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A
*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^
7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) - ((d
+ e*x)^(1/2)*(441*B^2*a^5*e^10 + 144*A^2*c^5*d^8*e^2 + 25*A^2*a^4*c*e^10 + 385*A^2*a^2*c^3*d^4*e^6 - 126*A^2*a
^3*c^2*d^2*e^8 + 196*B^2*a^2*c^3*d^6*e^4 - 735*B^2*a^3*c^2*d^4*e^6 - 420*A^2*a*c^4*d^6*e^4 + 490*B^2*a^4*c*d^2
*e^8 - 56*A*B*a^4*c*d*e^9 - 336*A*B*a*c^4*d^7*e^3 + 1120*A*B*a^2*c^3*d^5*e^5 - 840*A*B*a^3*c^2*d^3*e^7))/(64*a
^4*c^2))*((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^15*c^11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^
8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15
*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*
d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 245*B^2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d
^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*(a^15*c^11)^(1/2))/(4096*a^10*c^11))^(1/2) + (9261*B^3*a
^7*e^14 - 864*A^3*c^7*d^11*e^3 - 7398*A^3*a^2*c^5*d^7*e^7 + 6140*A^3*a^3*c^4*d^5*e^9 - 2182*A^3*a^4*c^3*d^3*e^
11 + 1372*B^3*a^3*c^4*d^8*e^6 - 9947*B^3*a^4*c^3*d^6*e^8 + 25039*B^3*a^5*c^2*d^4*e^10 - 525*A^2*B*a^6*c*e^14 +
 4104*A^3*a*c^6*d^9*e^5 + 200*A^3*a^5*c^2*d*e^13 - 25725*B^3*a^6*c*d^2*e^12 - 3528*A*B^2*a^2*c^5*d^9*e^5 + 226
38*A*B^2*a^3*c^4*d^7*e^7 - 51156*A*B^2*a^4*c^3*d^5*e^9 + 48510*A*B^2*a^5*c^2*d^3*e^11 - 16884*A^2*B*a^2*c^5*d^
8*e^6 + 34083*A^2*B*a^3*c^4*d^6*e^8 - 30135*A^2*B*a^4*c^3*d^4*e^10 + 10437*A^2*B*a^5*c^2*d^2*e^12 - 16464*A*B^
2*a^6*c*d*e^13 + 3024*A^2*B*a*c^6*d^10*e^4)/(2048*a^6*c^5)))*((144*A^2*a^5*c^10*d^7 + 441*B^2*a^2*e^7*(a^15*c^
11)^(1/2) - 420*A^2*a^6*c^9*d^5*e^2 + 385*A^2*a^7*c^8*d^3*e^4 + 196*B^2*a^7*c^8*d^5*e^2 - 735*B^2*a^8*c^7*d^3*
e^4 + 210*A*B*a^9*c^6*e^7 - 21*A^2*c^2*d^2*e^5*(a^15*c^11)^(1/2) - 105*A^2*a^8*c^7*d*e^6 + 735*B^2*a^9*c^6*d*e
^6 + 25*A^2*a*c*e^7*(a^15*c^11)^(1/2) + 210*A*B*c^2*d^3*e^4*(a^15*c^11)^(1/2) - 336*A*B*a^6*c^9*d^6*e - 245*B^
2*a*c*d^2*e^5*(a^15*c^11)^(1/2) + 1120*A*B*a^7*c^8*d^4*e^3 - 1050*A*B*a^8*c^7*d^2*e^5 - 266*A*B*a*c*d*e^6*(a^1
5*c^11)^(1/2))/(4096*a^10*c^11))^(1/2)*2i - (((d + e*x)^(1/2)*(5*A*a^3*e^7 + 7*B*a^3*d*e^6 - 6*A*c^3*d^6*e + 1
7*A*a*c^2*d^4*e^3 - 16*A*a^2*c*d^2*e^5 + 7*B*a*c^2*d^5*e^2 - 14*B*a^2*c*d^3*e^4))/(16*a^2*c^2) + ((d + e*x)^(3
/2)*(7*B*a^3*e^6 + 18*A*c^3*d^5*e - 32*A*a*c^2*d^3*e^3 - 21*B*a*c^2*d^4*e^2 + 14*B*a^2*c*d^2*e^4 + 14*A*a^2*c*
d*e^5))/(16*a^2*c^2) + ((d + e*x)^(5/2)*(7*B*a^2*d*e^4 - 9*A*a^2*e^5 - 18*A*c^2*d^4*e + 23*A*a*c*d^2*e^3 + 21*
B*a*c*d^3*e^2))/(16*a^2*c) - ((d + e*x)^(7/2)*(11*B*a^2*e^4 - 6*A*c^2*d^3*e + 8*A*a*c*d*e^3 + 7*B*a*c*d^2*e^2)
)/(16*a^2*c))/(c^2*(d + e*x)^4 + a^2*e^4 + c^2*d^4 + (6*c^2*d^2 - 2*a*c*e^2)*(d + e*x)^2 - (4*c^2*d^3 - 4*a*c*
d*e^2)*(d + e*x) - 4*c^2*d*(d + e*x)^3 - 2*a*c*d^2*e^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(7/2)/(-c*x**2+a)**3,x)

[Out]

Timed out

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