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

Optimal. Leaf size=269 \[ -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (3 \sqrt {a} A \sqrt {c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (-3 \sqrt {a} A \sqrt {c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac {e \sqrt {d+e x} (5 a B e+A c d)}{2 a c^2}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )} \]

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Rubi [A]  time = 0.54, antiderivative size = 269, 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, 825, 827, 1166, 208} \begin {gather*} -\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (3 \sqrt {a} A \sqrt {c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (-3 \sqrt {a} A \sqrt {c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{9/4}}+\frac {e \sqrt {d+e x} (5 a B e+A c d)}{2 a c^2}+\frac {(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{2 a c \left (a-c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^2,x]

[Out]

(e*(A*c*d + 5*a*B*e)*Sqrt[d + e*x])/(2*a*c^2) + ((d + e*x)^(3/2)*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(2*a*c*(
a - c*x^2)) - ((Sqrt[c]*d - Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt
[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(4*a^(3/2)*c^(9/4)) + ((Sqrt[c]*d + Sqrt[a]*e)^(3/2)*(2*A*c*d - 5*a*B
*e - 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(4*a^(3/2)*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 825

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g*(d + e*x)^m)/
(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x])/(a + c*x^2), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 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)^{5/2}}{\left (a-c x^2\right )^2} \, dx &=\frac {(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}-\frac {\int \frac {\sqrt {d+e x} \left (\frac {1}{2} \left (-2 A c d^2+a e (5 B d+3 A e)\right )+\frac {1}{2} e (A c d+5 a B e) x\right )}{a-c x^2} \, dx}{2 a c}\\ &=\frac {e (A c d+5 a B e) \sqrt {d+e x}}{2 a c^2}+\frac {(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}+\frac {\int \frac {\frac {1}{2} \left (2 A c d \left (c d^2-2 a e^2\right )-5 a B e \left (c d^2+a e^2\right )\right )+\frac {1}{2} c e \left (A c d^2-10 a B d e-3 a A e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )} \, dx}{2 a c^2}\\ &=\frac {e (A c d+5 a B e) \sqrt {d+e x}}{2 a c^2}+\frac {(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} c d e \left (A c d^2-10 a B d e-3 a A e^2\right )+\frac {1}{2} e \left (2 A c d \left (c d^2-2 a e^2\right )-5 a B e \left (c d^2+a e^2\right )\right )+\frac {1}{2} c e \left (A c d^2-10 a B d e-3 a A e^2\right ) x^2}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c^2}\\ &=\frac {e (A c d+5 a B e) \sqrt {d+e x}}{2 a c^2}+\frac {(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}+\frac {\left (\left (\sqrt {c} d+\sqrt {a} e\right )^2 \left (2 A c d-5 a B e-3 \sqrt {a} A \sqrt {c} e\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 )}{4 a^{3/2} c^{3/2}}-\frac {\left (\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (2 A c d-5 a B e+3 \sqrt {a} A \sqrt {c} e\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 )}{4 a^{3/2} c^{3/2}}\\ &=\frac {e (A c d+5 a B e) \sqrt {d+e x}}{2 a c^2}+\frac {(d+e x)^{3/2} (a (B d+A e)+(A c d+a B e) x)}{2 a c \left (a-c x^2\right )}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (2 A c d-5 a B e+3 \sqrt {a} A \sqrt {c} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \left (2 A c d-5 a B e-3 \sqrt {a} A \sqrt {c} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{9/4}}\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 279, normalized size = 1.04 \begin {gather*} \frac {2 \sqrt {a} \sqrt [4]{c} \sqrt {d+e x} \left (5 a^2 B e^2+a c \left (A e (2 d+e x)+B \left (d^2+2 d e x-4 e^2 x^2\right )\right )+A c^2 d^2 x\right )+\left (c x^2-a\right ) \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (3 \sqrt {a} A \sqrt {c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )-\left (c x^2-a\right ) \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \left (-3 \sqrt {a} A \sqrt {c} e-5 a B e+2 A c d\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{4 a^{3/2} c^{9/4} \left (a-c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^2,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 1.64, size = 489, normalized size = 1.82 \begin {gather*} \frac {e \sqrt {d+e x} \left (5 a^2 B e^3+a A c e^2 (d+e x)+a A c d e^2-5 a B c d^2 e+10 a B c d e (d+e x)-4 a B c e (d+e x)^2-A c^2 d^3+A c^2 d^2 (d+e x)\right )}{2 a c^2 \left (a e^2-c d^2+2 c d (d+e x)-c (d+e x)^2\right )}+\frac {\left (-3 a^{3/2} A \sqrt {c} e^3-10 a^{3/2} B \sqrt {c} d e^2-5 a^2 B e^3+\sqrt {a} A c^{3/2} d^2 e-4 a A c d e^2-5 a B c d^2 e+2 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 )}{4 a^{3/2} c^2 \sqrt {-\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right )}}+\frac {\left (-3 a^{3/2} A \sqrt {c} e^3-10 a^{3/2} B \sqrt {c} d e^2+5 a^2 B e^3+\sqrt {a} A c^{3/2} d^2 e+4 a A c d e^2+5 a B c d^2 e-2 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 )}{4 a^{3/2} c^2 \sqrt {-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^2,x]

[Out]

(e*Sqrt[d + e*x]*(-(A*c^2*d^3) - 5*a*B*c*d^2*e + a*A*c*d*e^2 + 5*a^2*B*e^3 + A*c^2*d^2*(d + e*x) + 10*a*B*c*d*
e*(d + e*x) + a*A*c*e^2*(d + e*x) - 4*a*B*c*e*(d + e*x)^2))/(2*a*c^2*(-(c*d^2) + a*e^2 + 2*c*d*(d + e*x) - c*(
d + e*x)^2)) + ((2*A*c^2*d^3 - 5*a*B*c*d^2*e + Sqrt[a]*A*c^(3/2)*d^2*e - 10*a^(3/2)*B*Sqrt[c]*d*e^2 - 4*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)])/(4*a^(3/2)*c^2*Sqrt[-(Sqrt[c]*(Sqrt[c]*d + Sqrt[a]*e))]) + ((-2*A*c^2*d^3 + 5*a*B*c*d^2*e +
Sqrt[a]*A*c^(3/2)*d^2*e - 10*a^(3/2)*B*Sqrt[c]*d*e^2 + 4*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)])/(4*a^(3/2)*c^2*Sqrt[-(Sqrt[
c]*(Sqrt[c]*d - Sqrt[a]*e))])

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fricas [B]  time = 8.66, size = 5611, normalized size = 20.86

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)^2,x, algorithm="fricas")

[Out]

-1/8*((a*c^3*x^2 - a^2*c^2)*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 +
a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*
A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A
^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*
B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2
*c)*d*e^4)/(a^3*c^4))*log(-(120*A^3*B*c^5*d^7*e^2 - 20*(45*A^2*B^2*a*c^4 - A^4*c^5)*d^6*e^3 + 10*(225*A*B^3*a^
2*c^3 - 77*A^3*B*a*c^4)*d^5*e^4 - (1875*B^4*a^3*c^2 - 3000*A^2*B^2*a^2*c^3 + 101*A^4*a*c^4)*d^4*e^5 - 20*(175*
A*B^3*a^3*c^2 - 73*A^3*B*a^2*c^3)*d^3*e^6 + 2*(625*B^4*a^4*c - 1050*A^2*B^2*a^3*c^2 + 81*A^4*a^2*c^3)*d^2*e^7
+ 10*(125*A*B^3*a^4*c - 81*A^3*B*a^3*c^2)*d*e^8 + (625*B^4*a^5 - 81*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) + (30*A^2*
B*a^2*c^5*d^4*e^3 + 5*(15*A*B^2*a^3*c^4 + A^3*a^2*c^5)*d^3*e^4 - 15*(25*B^3*a^4*c^3 + 3*A^2*B*a^3*c^4)*d^2*e^5
- 3*(125*A*B^2*a^4*c^3 + 3*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 - (2*A*a^3*c^8*d^2 -
5*B*a^4*c^7*d*e - 3*A*a^4*c^7*e^2)*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 +
25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7
+ 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^
9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)))*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^
4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 + a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^
3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3
*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A
^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3
*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*sqrt((4*A^2*c^3*d^
5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 + a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(
15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B
^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(2
5*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) +
5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))*log(-(120*A^3*B*c^5*d^7*
e^2 - 20*(45*A^2*B^2*a*c^4 - A^4*c^5)*d^6*e^3 + 10*(225*A*B^3*a^2*c^3 - 77*A^3*B*a*c^4)*d^5*e^4 - (1875*B^4*a^
3*c^2 - 3000*A^2*B^2*a^2*c^3 + 101*A^4*a*c^4)*d^4*e^5 - 20*(175*A*B^3*a^3*c^2 - 73*A^3*B*a^2*c^3)*d^3*e^6 + 2*
(625*B^4*a^4*c - 1050*A^2*B^2*a^3*c^2 + 81*A^4*a^2*c^3)*d^2*e^7 + 10*(125*A*B^3*a^4*c - 81*A^3*B*a^3*c^2)*d*e^
8 + (625*B^4*a^5 - 81*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) - (30*A^2*B*a^2*c^5*d^4*e^3 + 5*(15*A*B^2*a^3*c^4 + A^3*
a^2*c^5)*d^3*e^4 - 15*(25*B^3*a^4*c^3 + 3*A^2*B*a^3*c^4)*d^2*e^5 - 3*(125*A*B^2*a^4*c^3 + 3*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 - (2*A*a^3*c^8*d^2 - 5*B*a^4*c^7*d*e - 3*A*a^4*c^7*e^2)*sqrt((900*
A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4
*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*
A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*
a^2*c^2)*e^10)/(a^3*c^9)))*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 +
a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A
^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^
2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B
^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*
c)*d*e^4)/(a^3*c^4))) + (a*c^3*x^2 - a^2*c^2)*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3
+ 30*A*B*a^3*e^5 - a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*
B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125
*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*
B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5
*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))*log(-(120*A^3*B*c^5*d^7*e^2 - 20*(45*A^2*B^2*a*c^4 - A^4*c^5)*d^6*e^3
+ 10*(225*A*B^3*a^2*c^3 - 77*A^3*B*a*c^4)*d^5*e^4 - (1875*B^4*a^3*c^2 - 3000*A^2*B^2*a^2*c^3 + 101*A^4*a*c^4)*
d^4*e^5 - 20*(175*A*B^3*a^3*c^2 - 73*A^3*B*a^2*c^3)*d^3*e^6 + 2*(625*B^4*a^4*c - 1050*A^2*B^2*a^3*c^2 + 81*A^4
*a^2*c^3)*d^2*e^7 + 10*(125*A*B^3*a^4*c - 81*A^3*B*a^3*c^2)*d*e^8 + (625*B^4*a^5 - 81*A^4*a^3*c^2)*e^9)*sqrt(e
*x + d) + (30*A^2*B*a^2*c^5*d^4*e^3 + 5*(15*A*B^2*a^3*c^4 + A^3*a^2*c^5)*d^3*e^4 - 15*(25*B^3*a^4*c^3 + 3*A^2*
B*a^3*c^4)*d^2*e^5 - 3*(125*A*B^2*a^4*c^3 + 3*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 +
(2*A*a^3*c^8*d^2 - 5*B*a^4*c^7*d*e - 3*A*a^4*c^7*e^2)*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^
3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3
*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A
^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)))*sqrt((4*A^2*c^3*d^5
- 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 - a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(1
5*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^
3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25
*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) +
5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))) - (a*c^3*x^2 - a^2*c^2)*
sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 + 30*A*B*a^3*e^5 - a^3*c^4*sqrt((900*A^2*B^2*c
^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4
*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3
)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*
e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))*log(-(
120*A^3*B*c^5*d^7*e^2 - 20*(45*A^2*B^2*a*c^4 - A^4*c^5)*d^6*e^3 + 10*(225*A*B^3*a^2*c^3 - 77*A^3*B*a*c^4)*d^5*
e^4 - (1875*B^4*a^3*c^2 - 3000*A^2*B^2*a^2*c^3 + 101*A^4*a*c^4)*d^4*e^5 - 20*(175*A*B^3*a^3*c^2 - 73*A^3*B*a^2
*c^3)*d^3*e^6 + 2*(625*B^4*a^4*c - 1050*A^2*B^2*a^3*c^2 + 81*A^4*a^2*c^3)*d^2*e^7 + 10*(125*A*B^3*a^4*c - 81*A
^3*B*a^3*c^2)*d*e^8 + (625*B^4*a^5 - 81*A^4*a^3*c^2)*e^9)*sqrt(e*x + d) - (30*A^2*B*a^2*c^5*d^4*e^3 + 5*(15*A*
B^2*a^3*c^4 + A^3*a^2*c^5)*d^3*e^4 - 15*(25*B^3*a^4*c^3 + 3*A^2*B*a^3*c^4)*d^2*e^5 - 3*(125*A*B^2*a^4*c^3 + 3*
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 + (2*A*a^3*c^8*d^2 - 5*B*a^4*c^7*d*e - 3*A*a^4*c
^7*e^2)*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B^4*a^2*c^2 - 198*A
^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*B^4*a^3*c + 200*A^
2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B^4*a^4 + 450*A^2*B
^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)))*sqrt((4*A^2*c^3*d^5 - 20*A*B*a*c^2*d^4*e + 30*A*B*a^2*c*d^2*e^3 +
30*A*B*a^3*e^5 - a^3*c^4*sqrt((900*A^2*B^2*c^4*d^6*e^4 - 300*(15*A*B^3*a*c^3 - A^3*B*c^4)*d^5*e^5 + 25*(225*B
^4*a^2*c^2 - 198*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^6 + 40*(225*A*B^3*a^2*c^2 - 31*A^3*B*a*c^3)*d^3*e^7 + 30*(125*
B^4*a^3*c + 200*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^8 + 140*(25*A*B^3*a^3*c + 9*A^3*B*a^2*c^2)*d*e^9 + (625*B
^4*a^4 + 450*A^2*B^2*a^3*c + 81*A^4*a^2*c^2)*e^10)/(a^3*c^9)) + 5*(5*B^2*a^2*c - 3*A^2*a*c^2)*d^3*e^2 + 15*(5*
B^2*a^3 + A^2*a^2*c)*d*e^4)/(a^3*c^4))) - 4*(4*B*a*c*e^2*x^2 - B*a*c*d^2 - 2*A*a*c*d*e - 5*B*a^2*e^2 - (A*c^2*
d^2 + 2*B*a*c*d*e + A*a*c*e^2)*x)*sqrt(e*x + d))/(a*c^3*x^2 - a^2*c^2)

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giac [B]  time = 0.78, size = 728, normalized size = 2.71 \begin {gather*} \frac {2 \, \sqrt {x e + d} B e^{2}}{c^{2}} - \frac {{\left (10 \, \sqrt {a c} B a^{3} d {\left | c \right |} e^{3} - {\left (\sqrt {a c} c d^{2} e^{2} - 3 \, \sqrt {a c} a e^{4}\right )} A a^{2} {\left | c \right |} - {\left (a c^{2} d^{3} e - a^{2} c d e^{3}\right )} A {\left | a \right |} {\left | c \right |} - 5 \, {\left (a^{2} c d^{2} e^{2} - a^{3} e^{4}\right )} B {\left | a \right |} {\left | c \right |} + 2 \, {\left (\sqrt {a c} a c^{2} d^{4} - 2 \, \sqrt {a c} a^{2} c d^{2} e^{2}\right )} A {\left | c \right |} - 5 \, {\left (\sqrt {a c} a^{2} c d^{3} e + \sqrt {a c} a^{3} d e^{3}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{3} d + \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} - a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} d - \sqrt {a c} a^{2} c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a \right |}} + \frac {{\left (10 \, B a^{3} c d {\left | c \right |} e^{3} - {\left (c^{2} d^{2} e^{2} - 3 \, a c e^{4}\right )} A a^{2} {\left | c \right |} + {\left (\sqrt {a c} c^{2} d^{3} e - \sqrt {a c} a c d e^{3}\right )} A {\left | a \right |} {\left | c \right |} + 5 \, {\left (\sqrt {a c} a c d^{2} e^{2} - \sqrt {a c} a^{2} e^{4}\right )} B {\left | a \right |} {\left | c \right |} + 2 \, {\left (a c^{3} d^{4} - 2 \, a^{2} c^{2} d^{2} e^{2}\right )} A {\left | c \right |} - 5 \, {\left (a^{2} c^{2} d^{3} e + a^{3} c d e^{3}\right )} B {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{3} d - \sqrt {a^{2} c^{6} d^{2} - {\left (a c^{3} d^{2} - a^{2} c^{2} e^{2}\right )} a c^{3}}}{a c^{3}}}}\right )}{4 \, {\left (a^{2} c^{3} e + \sqrt {a c} a c^{3} d\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a \right |}} - \frac {{\left (x e + d\right )}^{\frac {3}{2}} A c^{2} d^{2} e - \sqrt {x e + d} A c^{2} d^{3} e + 2 \, {\left (x e + d\right )}^{\frac {3}{2}} B a c d e^{2} - \sqrt {x e + d} B a c d^{2} e^{2} + {\left (x e + d\right )}^{\frac {3}{2}} A a c e^{3} + \sqrt {x e + d} A a c d e^{3} + \sqrt {x e + d} B a^{2} e^{4}}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} - a e^{2}\right )} a c^{2}} \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)^2,x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*B*e^2/c^2 - 1/4*(10*sqrt(a*c)*B*a^3*d*abs(c)*e^3 - (sqrt(a*c)*c*d^2*e^2 - 3*sqrt(a*c)*a*e^4)*A
*a^2*abs(c) - (a*c^2*d^3*e - a^2*c*d*e^3)*A*abs(a)*abs(c) - 5*(a^2*c*d^2*e^2 - a^3*e^4)*B*abs(a)*abs(c) + 2*(s
qrt(a*c)*a*c^2*d^4 - 2*sqrt(a*c)*a^2*c*d^2*e^2)*A*abs(c) - 5*(sqrt(a*c)*a^2*c*d^3*e + sqrt(a*c)*a^3*d*e^3)*B*a
bs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^3*d + sqrt(a^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*a*c^3))/(a*c^3)))/((
a^2*c^3*d - sqrt(a*c)*a^2*c^2*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a)) + 1/4*(10*B*a^3*c*d*abs(c)*e^3 - (c^2*d^
2*e^2 - 3*a*c*e^4)*A*a^2*abs(c) + (sqrt(a*c)*c^2*d^3*e - sqrt(a*c)*a*c*d*e^3)*A*abs(a)*abs(c) + 5*(sqrt(a*c)*a
*c*d^2*e^2 - sqrt(a*c)*a^2*e^4)*B*abs(a)*abs(c) + 2*(a*c^3*d^4 - 2*a^2*c^2*d^2*e^2)*A*abs(c) - 5*(a^2*c^2*d^3*
e + a^3*c*d*e^3)*B*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^3*d - sqrt(a^2*c^6*d^2 - (a*c^3*d^2 - a^2*c^2*e^2)*
a*c^3))/(a*c^3)))/((a^2*c^3*e + sqrt(a*c)*a*c^3*d)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(a)) - 1/2*((x*e + d)^(3/2)
*A*c^2*d^2*e - sqrt(x*e + d)*A*c^2*d^3*e + 2*(x*e + d)^(3/2)*B*a*c*d*e^2 - sqrt(x*e + d)*B*a*c*d^2*e^2 + (x*e
+ d)^(3/2)*A*a*c*e^3 + sqrt(x*e + d)*A*a*c*d*e^3 + sqrt(x*e + d)*B*a^2*e^4)/(((x*e + d)^2*c - 2*(x*e + d)*c*d
+ c*d^2 - a*e^2)*a*c^2)

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maple [B]  time = 0.10, size = 1055, normalized size = 3.92

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)^2,x)

[Out]

2*e^2*B/c^2*(e*x+d)^(1/2)-1/2*e^3/c/(c*e^2*x^2-a*e^2)*(e*x+d)^(3/2)*A-1/2*e/(c*e^2*x^2-a*e^2)/a*(e*x+d)^(3/2)*
A*d^2-e^2/c/(c*e^2*x^2-a*e^2)*(e*x+d)^(3/2)*B*d-1/2*e^3/c/(c*e^2*x^2-a*e^2)*(e*x+d)^(1/2)*A*d+1/2*e/(c*e^2*x^2
-a*e^2)/a*(e*x+d)^(1/2)*A*d^3-1/2*e^4/c^2/(c*e^2*x^2-a*e^2)*a*(e*x+d)^(1/2)*B+1/2*e^2/c/(c*e^2*x^2-a*e^2)*(e*x
+d)^(1/2)*B*d^2-e^3/(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+1/2*e*c/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^3-5/4*e^4/c*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-5/4*e^2/(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^2-3/4*e^3/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+1/4*e/a/((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^2-5/2*e^2/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-e^3/(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+1/2*e*c/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^3-5/4*e^4/c*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-5/4*e^2/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*ar
ctan((e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c)*B*d^2+3/4*e^3/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-1/4*e/a/((-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+5/2*e^2/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

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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 )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 3.02, size = 9253, normalized size = 34.40

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)^2,x)

[Out]

atan(((((320*B*a^5*c^4*e^6 + 64*A*a^4*c^5*d*e^5 - 64*A*a^3*c^6*d^3*e^3 - 320*B*a^4*c^5*d^2*e^4)/(8*a^3*c^3) -
64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 +
25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B
^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e -
75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))
^(1/2))*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2
+ 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^
2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^
9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2) + ((d + e*x)^(1/
2)*(25*B^2*a^4*e^8 + 4*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 10*A^2*a^2*c^2*d^2*e^6 + 25*B^2*a^2*c^2*d^4*e^4 - 1
5*A^2*a*c^3*d^4*e^4 + 150*B^2*a^3*c*d^2*e^6 + 100*A*B*a^3*c*d*e^7 - 20*A*B*a*c^3*d^5*e^3))/(a^2*c))*((4*A^2*a^
3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*
e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9
)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*
A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*1i - (((320*B*a^5*c^4*e^6 + 64*A*a
^4*c^5*d*e^5 - 64*A*a^3*c^6*d^3*e^3 - 320*B*a^4*c^5*d^2*e^4)/(8*a^3*c^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*
A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^
6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a
^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2)
+ 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2))*((4*A^2*a^3*c^8*d^5 - 25*B^
2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d
^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*
c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e
^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2) - ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 4*A^2*c^4*d^6*
e^2 + 9*A^2*a^3*c*e^8 + 10*A^2*a^2*c^2*d^2*e^6 + 25*B^2*a^2*c^2*d^4*e^4 - 15*A^2*a*c^3*d^4*e^4 + 150*B^2*a^3*c
*d^2*e^6 + 100*A*B*a^3*c*d*e^7 - 20*A*B*a*c^3*d^5*e^3))/(a^2*c))*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9
)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(
1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c
^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*
e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*1i)/((((320*B*a^5*c^4*e^6 + 64*A*a^4*c^5*d*e^5 - 64*A*a^3*c^6*d^3*e^3
- 320*B*a^4*c^5*d^2*e^4)/(8*a^3*c^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a
^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*
c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*
(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*
a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2))*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*
a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5
*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*
B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/
2))/(64*a^6*c^9))^(1/2) + ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 4*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 10*A^2*a^2*
c^2*d^2*e^6 + 25*B^2*a^2*c^2*d^4*e^4 - 15*A^2*a*c^3*d^4*e^4 + 150*B^2*a^3*c*d^2*e^6 + 100*A*B*a^3*c*d*e^7 - 20
*A*B*a*c^3*d^5*e^3))/(a^2*c))*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 +
25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^
2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e -
75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^
(1/2) + (((320*B*a^5*c^4*e^6 + 64*A*a^4*c^5*d*e^5 - 64*A*a^3*c^6*d^3*e^3 - 320*B*a^4*c^5*d^2*e^4)/(8*a^3*c^3)
+ 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2
+ 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75
*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e
- 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9
))^(1/2))*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e
^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*
A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(
a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2) - ((d + e*x)^(
1/2)*(25*B^2*a^4*e^8 + 4*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 10*A^2*a^2*c^2*d^2*e^6 + 25*B^2*a^2*c^2*d^4*e^4 -
15*A^2*a*c^3*d^4*e^4 + 150*B^2*a^3*c*d^2*e^6 + 100*A*B*a^3*c*d*e^7 - 20*A*B*a*c^3*d^5*e^3))/(a^2*c))*((4*A^2*
a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^
5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c
^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 3
0*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2) + (4*A^3*c^5*d^8*e^3 - 75*A*B^2*
a^5*e^11 + 27*A^3*a^4*c*e^11 - 250*B^3*a^5*d*e^10 + 73*A^3*a^2*c^3*d^4*e^7 - 75*A^3*a^3*c^2*d^2*e^9 - 250*B^3*
a^3*c^2*d^5*e^6 - 29*A^3*a*c^4*d^6*e^5 + 500*B^3*a^4*c*d^3*e^8 + 225*A*B^2*a^2*c^3*d^6*e^5 - 525*A*B^2*a^3*c^2
*d^4*e^7 + 270*A^2*B*a^2*c^3*d^5*e^6 - 360*A^2*B*a^3*c^2*d^3*e^8 + 150*A^2*B*a^4*c*d*e^10 + 375*A*B^2*a^4*c*d^
2*e^9 - 60*A^2*B*a*c^4*d^7*e^4)/(4*a^3*c^3)))*((4*A^2*a^3*c^8*d^5 - 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^
4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 + 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c
^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 - 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) + 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*
a^4*c^7*d^4*e - 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 - 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2)
)/(64*a^6*c^9))^(1/2)*2i + atan(((((320*B*a^5*c^4*e^6 + 64*A*a^4*c^5*d*e^5 - 64*A*a^3*c^6*d^3*e^3 - 320*B*a^4*
c^5*d^2*e^4)/(8*a^3*c^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2)
- 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) +
15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/
2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^
9*c^9)^(1/2))/(64*a^6*c^9))^(1/2))*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e
^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 +
75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4
*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c
^9))^(1/2) + ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 4*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 10*A^2*a^2*c^2*d^2*e^6 +
25*B^2*a^2*c^2*d^4*e^4 - 15*A^2*a*c^3*d^4*e^4 + 150*B^2*a^3*c*d^2*e^6 + 100*A*B*a^3*c*d*e^7 - 20*A*B*a*c^3*d^
5*e^3))/(a^2*c))*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^
6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e
^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^
2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*1i - ((
(320*B*a^5*c^4*e^6 + 64*A*a^4*c^5*d*e^5 - 64*A*a^3*c^6*d^3*e^3 - 320*B*a^4*c^5*d^2*e^4)/(8*a^3*c^3) + 64*a*c^4
*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*
a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c
^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*
a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2))*
((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*
B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^
5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(
1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2) - ((d + e*x)^(1/2)*(25*B
^2*a^4*e^8 + 4*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 10*A^2*a^2*c^2*d^2*e^6 + 25*B^2*a^2*c^2*d^4*e^4 - 15*A^2*a*
c^3*d^4*e^4 + 150*B^2*a^3*c*d^2*e^6 + 100*A*B*a^3*c*d*e^7 - 20*A*B*a*c^3*d^5*e^3))/(a^2*c))*((4*A^2*a^3*c^8*d^
5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*
A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2)
- 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*
c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*1i)/((((320*B*a^5*c^4*e^6 + 64*A*a^4*c^5*d
*e^5 - 64*A*a^3*c^6*d^3*e^3 - 320*B*a^4*c^5*d^2*e^4)/(8*a^3*c^3) - 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*
c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^
5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^
(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*
B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2))*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^
5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(
a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*
e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*
A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2) + ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 4*A^2*c^4*d^6*e^2 + 9*
A^2*a^3*c*e^8 + 10*A^2*a^2*c^2*d^2*e^6 + 25*B^2*a^2*c^2*d^4*e^4 - 15*A^2*a*c^3*d^4*e^4 + 150*B^2*a^3*c*d^2*e^6
+ 100*A*B*a^3*c*d*e^7 - 20*A*B*a*c^3*d^5*e^3))/(a^2*c))*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2)
- 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 1
5*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2
) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9
*c^9)^(1/2))/(64*a^6*c^9))^(1/2) + (((320*B*a^5*c^4*e^6 + 64*A*a^4*c^5*d*e^5 - 64*A*a^3*c^6*d^3*e^3 - 320*B*a^
4*c^5*d^2*e^4)/(8*a^3*c^3) + 64*a*c^4*d*e^2*(d + e*x)^(1/2)*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/
2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2)
+ 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(
1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(
a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2))*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3
*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4
+ 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d
^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6
*c^9))^(1/2) - ((d + e*x)^(1/2)*(25*B^2*a^4*e^8 + 4*A^2*c^4*d^6*e^2 + 9*A^2*a^3*c*e^8 + 10*A^2*a^2*c^2*d^2*e^6
+ 25*B^2*a^2*c^2*d^4*e^4 - 15*A^2*a*c^3*d^4*e^4 + 150*B^2*a^3*c*d^2*e^6 + 100*A*B*a^3*c*d*e^7 - 20*A*B*a*c^3*
d^5*e^3))/(a^2*c))*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*
c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d
*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*
d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2) + (4*
A^3*c^5*d^8*e^3 - 75*A*B^2*a^5*e^11 + 27*A^3*a^4*c*e^11 - 250*B^3*a^5*d*e^10 + 73*A^3*a^2*c^3*d^4*e^7 - 75*A^3
*a^3*c^2*d^2*e^9 - 250*B^3*a^3*c^2*d^5*e^6 - 29*A^3*a*c^4*d^6*e^5 + 500*B^3*a^4*c*d^3*e^8 + 225*A*B^2*a^2*c^3*
d^6*e^5 - 525*A*B^2*a^3*c^2*d^4*e^7 + 270*A^2*B*a^2*c^3*d^5*e^6 - 360*A^2*B*a^3*c^2*d^3*e^8 + 150*A^2*B*a^4*c*
d*e^10 + 375*A*B^2*a^4*c*d^2*e^9 - 60*A^2*B*a*c^4*d^7*e^4)/(4*a^3*c^3)))*((4*A^2*a^3*c^8*d^5 + 25*B^2*a^2*e^5*
(a^9*c^9)^(1/2) - 15*A^2*a^4*c^7*d^3*e^2 + 25*B^2*a^5*c^6*d^3*e^2 + 30*A*B*a^6*c^5*e^5 - 5*A^2*c^2*d^2*e^3*(a^
9*c^9)^(1/2) + 15*A^2*a^5*c^6*d*e^4 + 75*B^2*a^6*c^5*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^9)^(1/2) - 30*A*B*c^2*d^3*e^
2*(a^9*c^9)^(1/2) - 20*A*B*a^4*c^7*d^4*e + 75*B^2*a*c*d^2*e^3*(a^9*c^9)^(1/2) + 30*A*B*a^5*c^6*d^2*e^3 + 70*A*
B*a*c*d*e^4*(a^9*c^9)^(1/2))/(64*a^6*c^9))^(1/2)*2i - (((d + e*x)^(1/2)*(B*a^2*e^4 - A*c^2*d^3*e + A*a*c*d*e^3
- B*a*c*d^2*e^2))/(2*a) + ((d + e*x)^(3/2)*(A*a*c*e^3 + A*c^2*d^2*e + 2*B*a*c*d*e^2))/(2*a))/(c^3*(d + e*x)^2
+ c^3*d^2 - a*c^2*e^2 - 2*c^3*d*(d + e*x)) + (2*B*e^2*(d + e*x)^(1/2))/c^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)**(5/2)/(-c*x**2+a)**2,x)

[Out]

Timed out

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