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3.13.85 \(\int \frac {(A+B x) (d+e x)^{5/2}}{(a-c x^2)^3} \, dx\)

Optimal. Leaf size=372 \[ \frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (5 a B e \left (\sqrt {a} e+2 \sqrt {c} d\right )-3 A \left (2 \sqrt {a} c d e-a \sqrt {c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{9/4}}-\frac {\sqrt {\sqrt {a} e+\sqrt {c} d} \left (5 a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (-6 \sqrt {a} c d e-3 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{32 a^{5/2} c^{9/4}}+\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

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Rubi [A]  time = 0.69, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {819, 821, 827, 1166, 208} \begin {gather*} \frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (5 a B e \left (\sqrt {a} e+2 \sqrt {c} d\right )-3 A \left (2 \sqrt {a} c d e-a \sqrt {c} e^2+4 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^{9/4}}-\frac {\sqrt {\sqrt {a} e+\sqrt {c} d} \left (5 a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-A \left (-6 \sqrt {a} c d e-3 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^{9/4}}+\frac {\sqrt {d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {(d+e x)^{3/2} (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)^(5/2))/(a - c*x^2)^3,x]

[Out]

((d + e*x)^(3/2)*(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*(3*A*c*d - 5
*a*B*e) + c*(6*A*c*d^2 - a*e*(5*B*d + 3*A*e))*x))/(16*a^2*c^2*(a - c*x^2)) + (Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(5*a
*B*e*(2*Sqrt[c]*d + Sqrt[a]*e) - 3*A*(4*c^(3/2)*d^2 + 2*Sqrt[a]*c*d*e - 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^(9/4)) - (Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(5*a*B*e*(2*Sqrt[c
]*d - Sqrt[a]*e) - A*(12*c^(3/2)*d^2 - 6*Sqrt[a]*c*d*e - 3*a*Sqrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqr
t[Sqrt[c]*d + Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))

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 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*
(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*
x^2)^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*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, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

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)^{5/2}}{\left (a-c x^2\right )^3} \, dx &=\frac {(d+e x)^{3/2} (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 {\sqrt {d+e x} \left (\frac {1}{2} \left (-6 A c d^2+a e (5 B d+3 A e)\right )-\frac {1}{2} e (3 A c d-5 a B e) x\right )}{\left (a-c x^2\right )^2} \, dx}{4 a c}\\ &=\frac {(d+e x)^{3/2} (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 (3 A c d-5 a B e)+c \left (6 A c d^2-a e (5 B d+3 A e)\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{4} \left (3 A c d \left (4 c d^2-3 a e^2\right )-5 a B e \left (2 c d^2-a e^2\right )\right )+\frac {1}{4} c e \left (6 A c d^2-a e (5 B d+3 A e)\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{8 a^2 c^2}\\ &=\frac {(d+e x)^{3/2} (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 (3 A c d-5 a B e)+c \left (6 A c d^2-a e (5 B d+3 A e)\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{4} c d e \left (6 A c d^2-a e (5 B d+3 A e)\right )+\frac {1}{4} e \left (3 A c d \left (4 c d^2-3 a e^2\right )-5 a B e \left (2 c d^2-a e^2\right )\right )+\frac {1}{4} c e \left (6 A c d^2-a e (5 B d+3 A e)\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)^{3/2} (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 (3 A c d-5 a B e)+c \left (6 A c d^2-a e (5 B d+3 A e)\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right ) \left (5 a B e \left (2 \sqrt {c} d+\sqrt {a} e\right )-3 A \left (4 c^{3/2} d^2+2 \sqrt {a} c d e-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^{3/2}}+\frac {\left (\frac {1}{8} c e \left (6 A c d^2-a e (5 B d+3 A e)\right )-\frac {-\frac {1}{2} c^2 d e \left (6 A c d^2-a e (5 B d+3 A e)\right )-2 c \left (-\frac {1}{4} c d e \left (6 A c d^2-a e (5 B d+3 A e)\right )+\frac {1}{4} e \left (3 A c d \left (4 c d^2-3 a e^2\right )-5 a B e \left (2 c d^2-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)^{3/2} (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 (3 A c d-5 a B e)+c \left (6 A c d^2-a e (5 B d+3 A e)\right ) x\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (5 a B e \left (2 \sqrt {c} d+\sqrt {a} e\right )-3 A \left (4 c^{3/2} d^2+2 \sqrt {a} c d e-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^{9/4}}-\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \left (5 a B e \left (2 \sqrt {c} d-\sqrt {a} e\right )-3 A \left (4 c^{3/2} d^2-2 \sqrt {a} c d e-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^{9/4}}\\ \end {align*}

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Mathematica [A]  time = 1.56, size = 569, normalized size = 1.53 \begin {gather*} \frac {\frac {c^2 (d+e x)^{7/2} \left (a^2 e^2 (A e+6 B d-B e x)-a c d^2 e (7 A+5 B x)+6 A c^2 d^3 x\right )}{2 \left (a-c x^2\right )}+\frac {\sqrt [4]{c} \left (5 a B d e \left (5 a e^2+7 c d^2\right )-3 A \left (-a^2 e^4+7 a c d^2 e^2+14 c^2 d^4\right )\right ) \left (2 \sqrt {a} c^{3/4} e \sqrt {d+e x} (7 d+e x)+3 \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-3 \left (\sqrt {a} e+\sqrt {c} d\right )^{5/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\right )}{12 \sqrt {a}}+\frac {2 a c^2 (d+e x)^{7/2} \left (c d^2-a e^2\right ) (-a A e+a B (d-e x)+A c d x)}{\left (a-c x^2\right )^2}+\frac {\left (6 A c^2 d^3-a B e \left (a e^2+5 c d^2\right )\right ) \left (2 \sqrt {a} \sqrt [4]{c} e \sqrt {d+e x} \left (15 a e^2+c \left (58 d^2+16 d e x+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 )-15 \left (\sqrt {a} e+\sqrt {c} d\right )^{7/2} \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )\right )}{12 \sqrt {a} \sqrt [4]{c}}}{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)^(5/2))/(a - c*x^2)^3,x]

[Out]

((2*a*c^2*(c*d^2 - a*e^2)*(d + e*x)^(7/2)*(-(a*A*e) + A*c*d*x + a*B*(d - e*x)))/(a - c*x^2)^2 + (c^2*(d + e*x)
^(7/2)*(6*A*c^2*d^3*x - a*c*d^2*e*(7*A + 5*B*x) + a^2*e^2*(6*B*d + A*e - B*e*x)))/(2*(a - c*x^2)) + (c^(1/4)*(
5*a*B*d*e*(7*c*d^2 + 5*a*e^2) - 3*A*(14*c^2*d^4 + 7*a*c*d^2*e^2 - a^2*e^4))*(2*Sqrt[a]*c^(3/4)*e*Sqrt[d + e*x]
*(7*d + e*x) + 3*(Sqrt[c]*d - Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]] -
3*(Sqrt[c]*d + Sqrt[a]*e)^(5/2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]))/(12*Sqrt[a]) +
((6*A*c^2*d^3 - a*B*e*(5*c*d^2 + a*e^2))*(2*Sqrt[a]*c^(1/4)*e*Sqrt[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]]))/(12*Sqrt[a]*c
^(1/4)))/(8*a^2*c^2*(c*d^2 - a*e^2)^2)

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IntegrateAlgebraic [A]  time = 4.54, size = 669, normalized size = 1.80 \begin {gather*} \frac {\left (-3 a^{3/2} A \sqrt {c} e^3-5 a^{3/2} B \sqrt {c} d e^2+5 a^2 B e^3+6 \sqrt {a} A c^{3/2} d^2 e-9 a A c d e^2-10 a B c d^2 e+12 A c^2 d^3\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^2 \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (-3 a^{3/2} A \sqrt {c} e^3-5 a^{3/2} B \sqrt {c} d e^2-5 a^2 B e^3+6 \sqrt {a} A c^{3/2} d^2 e+9 a A c d e^2+10 a B c d^2 e-12 A c^2 d^3\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^2 \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}}-\frac {e \sqrt {d+e x} \left (5 a^3 B e^5-a^2 A c e^4 (d+e x)-6 a^2 A c d e^4-10 a^2 B c d^2 e^3+15 a^2 B c d e^3 (d+e x)-9 a^2 B c e^3 (d+e x)^2+12 a A c^2 d^3 e^2-17 a A c^2 d^2 e^2 (d+e x)+8 a A c^2 d e^2 (d+e x)^2-3 a A c^2 e^2 (d+e x)^3+5 a B c^2 d^4 e-15 a B c^2 d^3 e (d+e x)+15 a B c^2 d^2 e (d+e x)^2-5 a B c^2 d e (d+e x)^3-6 A c^3 d^5+18 A c^3 d^4 (d+e x)-18 A c^3 d^3 (d+e x)^2+6 A c^3 d^2 (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)^(5/2))/(a - c*x^2)^3,x]

[Out]

-1/16*(e*Sqrt[d + e*x]*(-6*A*c^3*d^5 + 5*a*B*c^2*d^4*e + 12*a*A*c^2*d^3*e^2 - 10*a^2*B*c*d^2*e^3 - 6*a^2*A*c*d
*e^4 + 5*a^3*B*e^5 + 18*A*c^3*d^4*(d + e*x) - 15*a*B*c^2*d^3*e*(d + e*x) - 17*a*A*c^2*d^2*e^2*(d + e*x) + 15*a
^2*B*c*d*e^3*(d + e*x) - a^2*A*c*e^4*(d + e*x) - 18*A*c^3*d^3*(d + e*x)^2 + 15*a*B*c^2*d^2*e*(d + e*x)^2 + 8*a
*A*c^2*d*e^2*(d + e*x)^2 - 9*a^2*B*c*e^3*(d + e*x)^2 + 6*A*c^3*d^2*(d + e*x)^3 - 5*a*B*c^2*d*e*(d + e*x)^3 - 3
*a*A*c^2*e^2*(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) + ((12*A*c^2*d^3 -
 10*a*B*c*d^2*e + 6*Sqrt[a]*A*c^(3/2)*d^2*e - 5*a^(3/2)*B*Sqrt[c]*d*e^2 - 9*a*A*c*d*e^2 + 5*a^2*B*e^3 - 3*a^(3
/2)*A*Sqrt[c]*e^3)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(32*a^(5/
2)*c^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) + ((-12*A*c^2*d^3 + 10*a*B*c*d^2*e + 6*Sqrt[a]*A*c^(3/2)*d^2*
e - 5*a^(3/2)*B*Sqrt[c]*d*e^2 + 9*a*A*c*d*e^2 - 5*a^2*B*e^3 - 3*a^(3/2)*A*Sqrt[c]*e^3)*ArcTan[(Sqrt[-(c*d) + S
qrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(32*a^(5/2)*c^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]
*e))])

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fricas [B]  time = 1.22, size = 3382, normalized size = 9.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(-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^3*d^5 - 240*A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2
*e^3 - 30*A*B*a^3*e^5 + a^5*c^4*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B
^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)) + 20*(5*B^2*a^2*c - 9*A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a
^3 - 3*A^2*a^2*c)*d*e^4)/(a^5*c^4))*log(-(4320*A^3*B*c^4*d^5*e^4 - 432*(25*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^5
+ 360*(25*A*B^3*a^2*c^2 - 3*A^3*B*a*c^3)*d^3*e^6 - 4*(625*B^4*a^3*c - 1350*A^2*B^2*a^2*c^2 - 243*A^4*a*c^3)*d^
2*e^7 - 30*(125*A*B^3*a^3*c + 27*A^3*B*a^2*c^2)*d*e^8 + (625*B^4*a^4 - 81*A^4*a^2*c^2)*e^9)*sqrt(e*x + d) + (1
80*A^2*B*a^3*c^4*d^2*e^5 - 6*(50*A*B^2*a^4*c^3 + 9*A^3*a^3*c^4)*d*e^6 + 5*(25*B^3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e
^7 - (12*A*a^5*c^8*d^2 - 10*B*a^6*c^7*d*e - 3*A*a^6*c^7*e^2)*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c
+ 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)))*sqrt((144*A^2*c^3*d^5 -
240*A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 - 30*A*B*a^3*e^5 + a^5*c^4*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*
A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)) + 20*(5*B^2*a^2
*c - 9*A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a^3 - 3*A^2*a^2*c)*d*e^4)/(a^5*c^4))) - (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a
^4*c^2)*sqrt((144*A^2*c^3*d^5 - 240*A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 - 30*A*B*a^3*e^5 + a^5*c^4*sqrt((9
00*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*
e^10)/(a^5*c^9)) + 20*(5*B^2*a^2*c - 9*A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a^3 - 3*A^2*a^2*c)*d*e^4)/(a^5*c^4))*log
(-(4320*A^3*B*c^4*d^5*e^4 - 432*(25*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^5 + 360*(25*A*B^3*a^2*c^2 - 3*A^3*B*a*c^3
)*d^3*e^6 - 4*(625*B^4*a^3*c - 1350*A^2*B^2*a^2*c^2 - 243*A^4*a*c^3)*d^2*e^7 - 30*(125*A*B^3*a^3*c + 27*A^3*B*
a^2*c^2)*d*e^8 + (625*B^4*a^4 - 81*A^4*a^2*c^2)*e^9)*sqrt(e*x + d) - (180*A^2*B*a^3*c^4*d^2*e^5 - 6*(50*A*B^2*
a^4*c^3 + 9*A^3*a^3*c^4)*d*e^6 + 5*(25*B^3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e^7 - (12*A*a^5*c^8*d^2 - 10*B*a^6*c^7*d
*e - 3*A*a^6*c^7*e^2)*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 4
50*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)))*sqrt((144*A^2*c^3*d^5 - 240*A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2
*e^3 - 30*A*B*a^3*e^5 + a^5*c^4*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B
^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)) + 20*(5*B^2*a^2*c - 9*A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a
^3 - 3*A^2*a^2*c)*d*e^4)/(a^5*c^4))) + (a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^3*d^5 - 240*A*B
*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 - 30*A*B*a^3*e^5 - a^5*c^4*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a
*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)) + 20*(5*B^2*a^2*c - 9*
A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a^3 - 3*A^2*a^2*c)*d*e^4)/(a^5*c^4))*log(-(4320*A^3*B*c^4*d^5*e^4 - 432*(25*A^2
*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^5 + 360*(25*A*B^3*a^2*c^2 - 3*A^3*B*a*c^3)*d^3*e^6 - 4*(625*B^4*a^3*c - 1350*A^2
*B^2*a^2*c^2 - 243*A^4*a*c^3)*d^2*e^7 - 30*(125*A*B^3*a^3*c + 27*A^3*B*a^2*c^2)*d*e^8 + (625*B^4*a^4 - 81*A^4*
a^2*c^2)*e^9)*sqrt(e*x + d) + (180*A^2*B*a^3*c^4*d^2*e^5 - 6*(50*A*B^2*a^4*c^3 + 9*A^3*a^3*c^4)*d*e^6 + 5*(25*
B^3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e^7 + (12*A*a^5*c^8*d^2 - 10*B*a^6*c^7*d*e - 3*A*a^6*c^7*e^2)*sqrt((900*A^2*B^2
*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5
*c^9)))*sqrt((144*A^2*c^3*d^5 - 240*A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 - 30*A*B*a^3*e^5 - a^5*c^4*sqrt((9
00*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*
e^10)/(a^5*c^9)) + 20*(5*B^2*a^2*c - 9*A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a^3 - 3*A^2*a^2*c)*d*e^4)/(a^5*c^4))) -
(a^2*c^4*x^4 - 2*a^3*c^3*x^2 + a^4*c^2)*sqrt((144*A^2*c^3*d^5 - 240*A*B*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 -
30*A*B*a^3*e^5 - a^5*c^4*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2
+ 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)) + 20*(5*B^2*a^2*c - 9*A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a^3 - 3*
A^2*a^2*c)*d*e^4)/(a^5*c^4))*log(-(4320*A^3*B*c^4*d^5*e^4 - 432*(25*A^2*B^2*a*c^3 + 3*A^4*c^4)*d^4*e^5 + 360*(
25*A*B^3*a^2*c^2 - 3*A^3*B*a*c^3)*d^3*e^6 - 4*(625*B^4*a^3*c - 1350*A^2*B^2*a^2*c^2 - 243*A^4*a*c^3)*d^2*e^7 -
 30*(125*A*B^3*a^3*c + 27*A^3*B*a^2*c^2)*d*e^8 + (625*B^4*a^4 - 81*A^4*a^2*c^2)*e^9)*sqrt(e*x + d) - (180*A^2*
B*a^3*c^4*d^2*e^5 - 6*(50*A*B^2*a^4*c^3 + 9*A^3*a^3*c^4)*d*e^6 + 5*(25*B^3*a^5*c^2 + 9*A^2*B*a^4*c^3)*e^7 + (1
2*A*a^5*c^8*d^2 - 10*B*a^6*c^7*d*e - 3*A*a^6*c^7*e^2)*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a*c + 9*A^3
*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)))*sqrt((144*A^2*c^3*d^5 - 240*A*B
*a*c^2*d^4*e + 240*A*B*a^2*c*d^2*e^3 - 30*A*B*a^3*e^5 - a^5*c^4*sqrt((900*A^2*B^2*c^2*d^2*e^8 - 60*(25*A*B^3*a
*c + 9*A^3*B*c^2)*d*e^9 + (625*B^4*a^2 + 450*A^2*B^2*a*c + 81*A^4*c^2)*e^10)/(a^5*c^9)) + 20*(5*B^2*a^2*c - 9*
A^2*a*c^2)*d^3*e^2 - 15*(5*B^2*a^3 - 3*A^2*a^2*c)*d*e^4)/(a^5*c^4))) - 4*(4*B*a^2*c*d^2 + 7*A*a^2*c*d*e - 5*B*
a^3*e^2 - (6*A*c^3*d^2 - 5*B*a*c^2*d*e - 3*A*a*c^2*e^2)*x^3 + (A*a*c^2*d*e + 9*B*a^2*c*e^2)*x^2 + (10*A*a*c^2*
d^2 + 3*B*a^2*c*d*e + A*a^2*c*e^2)*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 [B]  time = 0.60, size = 728, normalized size = 1.96 \begin {gather*} -\frac {6 \, {\left (x e + d\right )}^{\frac {7}{2}} A c^{3} d^{2} e - 18 \, {\left (x e + d\right )}^{\frac {5}{2}} A c^{3} d^{3} e + 18 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{3} d^{4} e - 6 \, \sqrt {x e + d} A c^{3} d^{5} e - 5 \, {\left (x e + d\right )}^{\frac {7}{2}} B a c^{2} d e^{2} + 15 \, {\left (x e + d\right )}^{\frac {5}{2}} B a c^{2} d^{2} e^{2} - 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a c^{2} d^{3} e^{2} + 5 \, \sqrt {x e + d} B a c^{2} d^{4} e^{2} - 3 \, {\left (x e + d\right )}^{\frac {7}{2}} A a c^{2} e^{3} + 8 \, {\left (x e + d\right )}^{\frac {5}{2}} A a c^{2} d e^{3} - 17 \, {\left (x e + d\right )}^{\frac {3}{2}} A a c^{2} d^{2} e^{3} + 12 \, \sqrt {x e + d} A a c^{2} d^{3} e^{3} - 9 \, {\left (x e + d\right )}^{\frac {5}{2}} B a^{2} c e^{4} + 15 \, {\left (x e + d\right )}^{\frac {3}{2}} B a^{2} c d e^{4} - 10 \, \sqrt {x e + d} B a^{2} c d^{2} e^{4} - {\left (x e + d\right )}^{\frac {3}{2}} A a^{2} c e^{5} - 6 \, \sqrt {x e + d} A a^{2} c d e^{5} + 5 \, \sqrt {x e + d} B a^{3} e^{6}}{16 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )}^{2} a^{2} c^{2}} - \frac {{\left (3 \, {\left (2 \, a c d e - 4 \, \sqrt {a c} c d^{2} + \sqrt {a c} a e^{2}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} A {\left | c \right |} + 5 \, {\left (2 \, \sqrt {a c} a d e - a^{2} e^{2}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d + \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, a^{3} c^{4}} - \frac {{\left (3 \, {\left (2 \, a c d e + 4 \, \sqrt {a c} c d^{2} - \sqrt {a c} a e^{2}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} A {\left | c \right |} - 5 \, {\left (2 \, \sqrt {a c} a d e + a^{2} e^{2}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a^{2} c^{3} d - \sqrt {a^{4} c^{6} d^{2} - {\left (a^{2} c^{3} d^{2} - a^{3} c^{2} e^{2}\right )} a^{2} c^{3}}}{a^{2} c^{3}}}}\right )}{32 \, a^{3} c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(5/2)/(-c*x^2+a)^3,x, algorithm="giac")

[Out]

-1/16*(6*(x*e + d)^(7/2)*A*c^3*d^2*e - 18*(x*e + d)^(5/2)*A*c^3*d^3*e + 18*(x*e + d)^(3/2)*A*c^3*d^4*e - 6*sqr
t(x*e + d)*A*c^3*d^5*e - 5*(x*e + d)^(7/2)*B*a*c^2*d*e^2 + 15*(x*e + d)^(5/2)*B*a*c^2*d^2*e^2 - 15*(x*e + d)^(
3/2)*B*a*c^2*d^3*e^2 + 5*sqrt(x*e + d)*B*a*c^2*d^4*e^2 - 3*(x*e + d)^(7/2)*A*a*c^2*e^3 + 8*(x*e + d)^(5/2)*A*a
*c^2*d*e^3 - 17*(x*e + d)^(3/2)*A*a*c^2*d^2*e^3 + 12*sqrt(x*e + d)*A*a*c^2*d^3*e^3 - 9*(x*e + d)^(5/2)*B*a^2*c
*e^4 + 15*(x*e + d)^(3/2)*B*a^2*c*d*e^4 - 10*sqrt(x*e + d)*B*a^2*c*d^2*e^4 - (x*e + d)^(3/2)*A*a^2*c*e^5 - 6*s
qrt(x*e + d)*A*a^2*c*d*e^5 + 5*sqrt(x*e + d)*B*a^3*e^6)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 - a*e^2)^2*a
^2*c^2) - 1/32*(3*(2*a*c*d*e - 4*sqrt(a*c)*c*d^2 + sqrt(a*c)*a*e^2)*sqrt(-c^2*d - sqrt(a*c)*c*e)*A*abs(c) + 5*
(2*sqrt(a*c)*a*d*e - a^2*e^2)*sqrt(-c^2*d - sqrt(a*c)*c*e)*B*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d + s
qrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2*e^2)*a^2*c^3))/(a^2*c^3)))/(a^3*c^4) - 1/32*(3*(2*a*c*d*e + 4*sqrt(a*
c)*c*d^2 - sqrt(a*c)*a*e^2)*sqrt(-c^2*d + sqrt(a*c)*c*e)*A*abs(c) - 5*(2*sqrt(a*c)*a*d*e + a^2*e^2)*sqrt(-c^2*
d + sqrt(a*c)*c*e)*B*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a^2*c^3*d - sqrt(a^4*c^6*d^2 - (a^2*c^3*d^2 - a^3*c^2
*e^2)*a^2*c^3))/(a^2*c^3)))/(a^3*c^4)

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maple [B]  time = 0.11, size = 1447, normalized size = 3.89

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(5/2)/(-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)*arctanh((e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1
/2)*c)*A*d^3+3/8*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*A*d+5/8*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(1/2)*B*d^2
+5/32*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+5/32*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-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^2+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^2-15/16*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*B*d+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^3+17/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*
x+d)^(3/2)*A*d^2+15/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(3/2)*B*d^3-3/32*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+3/32*e^3/a/c/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*a
rctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*A-15/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*B*d^2-
1/2*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(5/2)*A*d-3/4*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(1/2)*A*d^3-5/16*e^6/(c*
e^2*x^2-a*e^2)^2*a/c^2*(e*x+d)^(1/2)*B+9/16*e^4/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(5/2)*B-5/16*e^2/(c*e^2*x^2-a*e^
2)^2/a*(e*x+d)^(1/2)*B*d^4+5/16*e^2/(c*e^2*x^2-a*e^2)^2/a*(e*x+d)^(7/2)*B*d+3/16*e^3/(c*e^2*x^2-a*e^2)^2/a*(e*
x+d)^(7/2)*A+1/16*e^5/(c*e^2*x^2-a*e^2)^2/c*(e*x+d)^(3/2)*A-5/16*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^2-9/32*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+9/8*e/(c*e^2*x^2-a*e^2)^2/a
^2*c*(e*x+d)^(5/2)*A*d^3+3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(1/2)*A*d^5-3/8*e/(c*e^2*x^2-a*e^2)^2/a^2*(e*
x+d)^(7/2)*A*c*d^2-9/32*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-5/16*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^2-5/32*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+5/32*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-9/8*e/(c*e^2*x^2-a*e^2)^2/a^2*c*(e*x+d)^(3/2)*A*d^4

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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 {5}{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)^(5/2)/(-c*x^2+a)^3,x, algorithm="maxima")

[Out]

-integrate((B*x + A)*(e*x + d)^(5/2)/(c*x^2 - a)^3, x)

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mupad [B]  time = 3.00, size = 7702, normalized size = 20.70

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^3,x)

[Out]

((e*(d + e*x)^(7/2)*(3*A*a*e^2 - 6*A*c*d^2 + 5*B*a*d*e))/(16*a^2) - ((d + e*x)^(1/2)*(5*B*a^3*e^6 - 6*A*c^3*d^
5*e + 12*A*a*c^2*d^3*e^3 + 5*B*a*c^2*d^4*e^2 - 10*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5))/(16*a^2*c^2) + ((d + e*x
)^(3/2)*(A*a^2*e^5 - 15*B*a^2*d*e^4 - 18*A*c^2*d^4*e + 17*A*a*c*d^2*e^3 + 15*B*a*c*d^3*e^2))/(16*a^2*c) + (e*(
d + e*x)^(5/2)*(18*A*c^2*d^3 + 9*B*a^2*e^3 - 8*A*a*c*d*e^2 - 15*B*a*c*d^2*e))/(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) - atan(((((20480*B*a^7*c^4*e^6 - 24576*A*a^6*c^5*d*e^5 + 24576*A*a^5*c^6*d^3*e^3 - 20480
*B*a^6*c^5*d^2*e^4)/(4096*a^6*c^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2
*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c
^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2)
 - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2))*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9
*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*
a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c
^6*d^2*e^3)/(4096*a^10*c^9))^(1/2) + ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 144*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8
+ 45*A^2*a^2*c^2*d^2*e^6 + 100*B^2*a^2*c^2*d^4*e^4 - 180*A^2*a*c^3*d^4*e^4 - 75*B^2*a^3*c*d^2*e^6 - 60*A*B*a^3
*c*d*e^7 - 240*A*B*a*c^3*d^5*e^3 + 240*A*B*a^2*c^2*d^3*e^5))/(64*a^4*c))*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 14
4*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*
a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^
(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2)*1i - (((20480*B*a^7*c^4*e^6 - 24576*A*a^6*c^5*d*e^5 +
24576*A*a^5*c^6*d^3*e^3 - 20480*B*a^6*c^5*d^2*e^4)/(4096*a^6*c^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(25*B^2*a
*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2
*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e -
30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2))*(-(25*B^2*a*e^5*(a^15*c^9)^
(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2
 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(
a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2) - ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 144*A^2
*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 45*A^2*a^2*c^2*d^2*e^6 + 100*B^2*a^2*c^2*d^4*e^4 - 180*A^2*a*c^3*d^4*e^4 - 75
*B^2*a^3*c*d^2*e^6 - 60*A*B*a^3*c*d*e^7 - 240*A*B*a*c^3*d^5*e^3 + 240*A*B*a^2*c^2*d^3*e^5))/(64*a^4*c))*(-(25*
B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 10
0*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4
*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2)*1i)/((((20480*B*a^7*c^4
*e^6 - 24576*A*a^6*c^5*d*e^5 + 24576*A*a^5*c^6*d^3*e^3 - 20480*B*a^6*c^5*d^2*e^4)/(4096*a^6*c^3) - 64*a*c^4*d*
e^2*(d + e*x)^(1/2)*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 18
0*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d
*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/
2))*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^
3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a
^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2) + ((d + e*x)^
(1/2)*(25*B^2*a^4*e^8 + 144*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 45*A^2*a^2*c^2*d^2*e^6 + 100*B^2*a^2*c^2*d^4*e
^4 - 180*A^2*a*c^3*d^4*e^4 - 75*B^2*a^3*c*d^2*e^6 - 60*A*B*a^3*c*d*e^7 - 240*A*B*a*c^3*d^5*e^3 + 240*A*B*a^2*c
^2*d^3*e^5))/(64*a^4*c))*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2)
 + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*
c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9)
)^(1/2) + (((20480*B*a^7*c^4*e^6 - 24576*A*a^6*c^5*d*e^5 + 24576*A*a^5*c^6*d^3*e^3 - 20480*B*a^6*c^5*d^2*e^4)/
(4096*a^6*c^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2
*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*
c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d
^2*e^3)/(4096*a^10*c^9))^(1/2))*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9
)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B
^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^
10*c^9))^(1/2) - ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 144*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 45*A^2*a^2*c^2*d^2
*e^6 + 100*B^2*a^2*c^2*d^4*e^4 - 180*A^2*a*c^3*d^4*e^4 - 75*B^2*a^3*c*d^2*e^6 - 60*A*B*a^3*c*d*e^7 - 240*A*B*a
*c^3*d^5*e^3 + 240*A*B*a^2*c^2*d^3*e^5))/(64*a^4*c))*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 +
9*A^2*c*e^5*(a^15*c^9)^(1/2) + 180*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2
*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*
c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2) - (864*A^3*c^5*d^8*e^3 - 75*A*B^2*a^5*e^11 + 27*A^3*a^4*c*e^11 - 125*B^3*a
^5*d*e^10 + 1458*A^3*a^2*c^3*d^4*e^7 - 405*A^3*a^3*c^2*d^2*e^9 - 500*B^3*a^3*c^2*d^5*e^6 - 1944*A^3*a*c^4*d^6*
e^5 + 625*B^3*a^4*c*d^3*e^8 + 1800*A*B^2*a^2*c^3*d^6*e^5 - 2850*A*B^2*a^3*c^2*d^4*e^7 + 4140*A^2*B*a^2*c^3*d^5
*e^6 - 2385*A^2*B*a^3*c^2*d^3*e^8 + 405*A^2*B*a^4*c*d*e^10 + 1125*A*B^2*a^4*c*d^2*e^9 - 2160*A^2*B*a*c^4*d^7*e
^4)/(2048*a^6*c^3)))*(-(25*B^2*a*e^5*(a^15*c^9)^(1/2) - 144*A^2*a^5*c^8*d^5 + 9*A^2*c*e^5*(a^15*c^9)^(1/2) + 1
80*A^2*a^6*c^7*d^3*e^2 - 100*B^2*a^7*c^6*d^3*e^2 + 30*A*B*a^8*c^5*e^5 - 45*A^2*a^7*c^6*d*e^4 + 75*B^2*a^8*c^5*
d*e^4 + 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) - 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1
/2)*2i - atan(((((20480*B*a^7*c^4*e^6 - 24576*A*a^6*c^5*d*e^5 + 24576*A*a^5*c^6*d^3*e^3 - 20480*B*a^6*c^5*d^2*
e^4)/(4096*a^6*c^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9
*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*
a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c
^6*d^2*e^3)/(4096*a^10*c^9))^(1/2))*((144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*
c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 7
5*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096
*a^10*c^9))^(1/2) + ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 144*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 45*A^2*a^2*c^2*
d^2*e^6 + 100*B^2*a^2*c^2*d^4*e^4 - 180*A^2*a*c^3*d^4*e^4 - 75*B^2*a^3*c*d^2*e^6 - 60*A*B*a^3*c*d*e^7 - 240*A*
B*a*c^3*d^5*e^3 + 240*A*B*a^2*c^2*d^3*e^5))/(64*a^4*c))*((144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2)
+ 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A
^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^
7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2)*1i - (((20480*B*a^7*c^4*e^6 - 24576*A*a^6*c^5*d*e^5 + 24576*A*a^5*c^6*d^
3*e^3 - 20480*B*a^6*c^5*d^2*e^4)/(4096*a^6*c^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^8*d^5 + 25*B^
2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2 -
30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^1
5*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2))*((144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9
)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^
5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 24
0*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2) - ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 144*A^2*c^4*d^6*e^2 + 9*A^2
*a^3*c*e^8 + 45*A^2*a^2*c^2*d^2*e^6 + 100*B^2*a^2*c^2*d^4*e^4 - 180*A^2*a*c^3*d^4*e^4 - 75*B^2*a^3*c*d^2*e^6 -
 60*A*B*a^3*c*d*e^7 - 240*A*B*a*c^3*d^5*e^3 + 240*A*B*a^2*c^2*d^3*e^5))/(64*a^4*c))*((144*A^2*a^5*c^8*d^5 + 25
*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2
 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(
a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2)*1i)/((((20480*B*a^7*c^4*e^6 - 24576*A*a^6*c^
5*d*e^5 + 24576*A*a^5*c^6*d^3*e^3 - 20480*B*a^6*c^5*d^2*e^4)/(4096*a^6*c^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*(
(144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2
+ 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7
*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2))*((144*A^2*a^5*c^8*
d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6
*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c
*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2) + ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 +
 144*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 45*A^2*a^2*c^2*d^2*e^6 + 100*B^2*a^2*c^2*d^4*e^4 - 180*A^2*a*c^3*d^4*
e^4 - 75*B^2*a^3*c*d^2*e^6 - 60*A*B*a^3*c*d*e^7 - 240*A*B*a*c^3*d^5*e^3 + 240*A*B*a^2*c^2*d^3*e^5))/(64*a^4*c)
)*((144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e
^2 + 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*
c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2) + (((20480*B*a^7
*c^4*e^6 - 24576*A*a^6*c^5*d*e^5 + 24576*A*a^5*c^6*d^3*e^3 - 20480*B*a^6*c^5*d^2*e^4)/(4096*a^6*c^3) + 64*a*c^
4*d*e^2*(d + e*x)^(1/2)*((144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) -
 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^
5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^
(1/2))*((144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*
d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B
*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2) - ((d + e*x
)^(1/2)*(25*B^2*a^4*e^8 + 144*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 45*A^2*a^2*c^2*d^2*e^6 + 100*B^2*a^2*c^2*d^4
*e^4 - 180*A^2*a*c^3*d^4*e^4 - 75*B^2*a^3*c*d^2*e^6 - 60*A*B*a^3*c*d*e^7 - 240*A*B*a*c^3*d^5*e^3 + 240*A*B*a^2
*c^2*d^3*e^5))/(64*a^4*c))*((144*A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2
) - 180*A^2*a^6*c^7*d^3*e^2 + 100*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8
*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9
))^(1/2) - (864*A^3*c^5*d^8*e^3 - 75*A*B^2*a^5*e^11 + 27*A^3*a^4*c*e^11 - 125*B^3*a^5*d*e^10 + 1458*A^3*a^2*c^
3*d^4*e^7 - 405*A^3*a^3*c^2*d^2*e^9 - 500*B^3*a^3*c^2*d^5*e^6 - 1944*A^3*a*c^4*d^6*e^5 + 625*B^3*a^4*c*d^3*e^8
 + 1800*A*B^2*a^2*c^3*d^6*e^5 - 2850*A*B^2*a^3*c^2*d^4*e^7 + 4140*A^2*B*a^2*c^3*d^5*e^6 - 2385*A^2*B*a^3*c^2*d
^3*e^8 + 405*A^2*B*a^4*c*d*e^10 + 1125*A*B^2*a^4*c*d^2*e^9 - 2160*A^2*B*a*c^4*d^7*e^4)/(2048*a^6*c^3)))*((144*
A^2*a^5*c^8*d^5 + 25*B^2*a*e^5*(a^15*c^9)^(1/2) + 9*A^2*c*e^5*(a^15*c^9)^(1/2) - 180*A^2*a^6*c^7*d^3*e^2 + 100
*B^2*a^7*c^6*d^3*e^2 - 30*A*B*a^8*c^5*e^5 + 45*A^2*a^7*c^6*d*e^4 - 75*B^2*a^8*c^5*d*e^4 - 240*A*B*a^6*c^7*d^4*
e - 30*A*B*c*d*e^4*(a^15*c^9)^(1/2) + 240*A*B*a^7*c^6*d^2*e^3)/(4096*a^10*c^9))^(1/2)*2i

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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)**(5/2)/(-c*x**2+a)**3,x)

[Out]

Timed out

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