3.1460 \(\int \frac{(A+B x) (d+e x)^{3/2}}{\left (a-c x^2\right )^3} \, dx\)

Optimal. Leaf size=350 \[ \frac{3 \left (a B e \left (2 \sqrt{c} d-\sqrt{a} e\right )-A \left (-2 \sqrt{a} c d e-a \sqrt{c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{7/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}-\frac{3 \left (a B e \left (\sqrt{a} e+2 \sqrt{c} d\right )-A \left (2 \sqrt{a} c d e-a \sqrt{c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{7/4} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\sqrt{d+e x} (a A e-3 x (2 A c d-a B e))}{16 a^2 c \left (a-c x^2\right )}+\frac{\sqrt{d+e x} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

[Out]

(Sqrt[d + e*x]*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) - (Sqr
t[d + e*x]*(a*A*e - 3*(2*A*c*d - a*B*e)*x))/(16*a^2*c*(a - c*x^2)) + (3*(a*B*e*(
2*Sqrt[c]*d - Sqrt[a]*e) - 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^(7/4
)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - (3*(a*B*e*(2*Sqrt[c]*d + Sqrt[a]*e) - 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^(7/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])

_______________________________________________________________________________________

Rubi [A]  time = 1.53673, antiderivative size = 350, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 \left (a B e \left (2 \sqrt{c} d-\sqrt{a} e\right )-A \left (-2 \sqrt{a} c d e-a \sqrt{c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{7/4} \sqrt{\sqrt{c} d-\sqrt{a} e}}-\frac{3 \left (a B e \left (\sqrt{a} e+2 \sqrt{c} d\right )-A \left (2 \sqrt{a} c d e-a \sqrt{c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{7/4} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\sqrt{d+e x} (a A e-3 x (2 A c d-a B e))}{16 a^2 c \left (a-c x^2\right )}+\frac{\sqrt{d+e x} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(a*(B*d + A*e) + (A*c*d + a*B*e)*x))/(4*a*c*(a - c*x^2)^2) - (Sqr
t[d + e*x]*(a*A*e - 3*(2*A*c*d - a*B*e)*x))/(16*a^2*c*(a - c*x^2)) + (3*(a*B*e*(
2*Sqrt[c]*d - Sqrt[a]*e) - 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^(7/4
)*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) - (3*(a*B*e*(2*Sqrt[c]*d + Sqrt[a]*e) - 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^(7/4)*Sqrt[Sqrt[c]*d + Sqrt[a]*e])

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(3/2)/(-c*x**2+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 1.09211, size = 311, normalized size = 0.89 \[ \frac{-\frac{3 \left (a^{3/2} B \sqrt{c} e^2+A c \left (-2 \sqrt{a} \sqrt{c} d e-a e^2+4 c d^2\right )-2 a B c d e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}-\frac{3 \left (a^{3/2} B \sqrt{c} e^2+A c \left (-2 \sqrt{a} \sqrt{c} d e+a e^2-4 c d^2\right )+2 a B c d e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}+\frac{2 \sqrt{a} c \sqrt{d+e x} \left (a^2 (3 A e+4 B d+B e x)+a c x \left (10 A d+A e x+3 B e x^2\right )-6 A c^2 d x^3\right )}{\left (a-c x^2\right )^2}}{32 a^{5/2} c^2} \]

Antiderivative was successfully verified.

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

[Out]

((2*Sqrt[a]*c*Sqrt[d + e*x]*(-6*A*c^2*d*x^3 + a^2*(4*B*d + 3*A*e + B*e*x) + a*c*
x*(10*A*d + A*e*x + 3*B*e*x^2)))/(a - c*x^2)^2 - (3*(-2*a*B*c*d*e + a^(3/2)*B*Sq
rt[c]*e^2 + A*c*(4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2))*ArcTanh[(Sqrt[c]*Sqrt
[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e] - (3*(2
*a*B*c*d*e + a^(3/2)*B*Sqrt[c]*e^2 + A*c*(-4*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e + a*e
^2))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d +
Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c^2)

_______________________________________________________________________________________

Maple [B]  time = 0.115, size = 1370, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{3}{2}}}{{\left (c x^{2} - a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 1.618, size = 5638, normalized size = 16.11 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/64*(3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c
^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^
3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 + (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^
2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c +
A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 -
a^6*c^3*e^2))*log(-27*(32*A^3*B*c^4*d^5*e^4 - 16*(3*A^2*B^2*a*c^3 + A^4*c^4)*d^
4*e^5 + 8*(3*A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^6 - 4*(B^4*a^3*c - 6*A^2*B^2*a^2
*c^2 - 3*A^4*a*c^3)*d^2*e^7 - 2*(5*A*B^3*a^3*c + 3*A^3*B*a^2*c^2)*d*e^8 + (B^4*a
^4 - A^4*a^2*c^2)*e^9)*sqrt(e*x + d) + 27*(4*A^2*B*a^3*c^4*d^3*e^5 - 2*(2*A*B^2*
a^4*c^3 + A^3*a^3*c^4)*d^2*e^6 + (B^3*a^5*c^2 - A^2*B*a^4*c^3)*d*e^7 + (A*B^2*a^
5*c^2 + A^3*a^4*c^3)*e^8 + (4*A*a^5*c^8*d^5 - 2*B*a^6*c^7*d^4*e - 7*A*a^6*c^7*d^
3*e^2 + 3*B*a^7*c^6*d^2*e^3 + 3*A*a^7*c^6*d*e^4 - B*a^8*c^5*e^5)*sqrt((4*A^2*B^2
*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*
c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))*sqrt((16*A^2*c^3*d^
5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5
*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 + (a^5*c^4*d^2 - a^6*c^3*e
^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2
*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/
(a^5*c^4*d^2 - a^6*c^3*e^2))) - 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16
*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^
2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 + (a^5*c^4*d^2
- a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (
B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*
c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))*log(-27*(32*A^3*B*c^4*d^5*e^4 - 16*(3*A^
2*B^2*a*c^3 + A^4*c^4)*d^4*e^5 + 8*(3*A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^6 - 4*(
B^4*a^3*c - 6*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^7 - 2*(5*A*B^3*a^3*c + 3*A^3*
B*a^2*c^2)*d*e^8 + (B^4*a^4 - A^4*a^2*c^2)*e^9)*sqrt(e*x + d) - 27*(4*A^2*B*a^3*
c^4*d^3*e^5 - 2*(2*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d^2*e^6 + (B^3*a^5*c^2 - A^2*B*a
^4*c^3)*d*e^7 + (A*B^2*a^5*c^2 + A^3*a^4*c^3)*e^8 + (4*A*a^5*c^8*d^5 - 2*B*a^6*c
^7*d^4*e - 7*A*a^6*c^7*d^3*e^2 + 3*B*a^7*c^6*d^2*e^3 + 3*A*a^7*c^6*d*e^4 - B*a^8
*c^5*e^5)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a
^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e
^4)))*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a
^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 +
(a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*
c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*
d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))) + 3*(a^2*c^3*x^4 - 2*a^3*
c^2*x^2 + a^4*c)*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^
3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2
*c)*d*e^4 - (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a
*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 -
2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2))*log(-27*(32*A^3
*B*c^4*d^5*e^4 - 16*(3*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^5 + 8*(3*A*B^3*a^2*c^2 - A
^3*B*a*c^3)*d^3*e^6 - 4*(B^4*a^3*c - 6*A^2*B^2*a^2*c^2 - 3*A^4*a*c^3)*d^2*e^7 -
2*(5*A*B^3*a^3*c + 3*A^3*B*a^2*c^2)*d*e^8 + (B^4*a^4 - A^4*a^2*c^2)*e^9)*sqrt(e*
x + d) + 27*(4*A^2*B*a^3*c^4*d^3*e^5 - 2*(2*A*B^2*a^4*c^3 + A^3*a^3*c^4)*d^2*e^6
+ (B^3*a^5*c^2 - A^2*B*a^4*c^3)*d*e^7 + (A*B^2*a^5*c^2 + A^3*a^4*c^3)*e^8 - (4*
A*a^5*c^8*d^5 - 2*B*a^6*c^7*d^4*e - 7*A*a^6*c^7*d^3*e^2 + 3*B*a^7*c^6*d^2*e^3 +
3*A*a^7*c^6*d*e^4 - B*a^8*c^5*e^5)*sqrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c +
A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^
6*c^8*d^2*e^2 + a^7*c^7*e^4)))*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^4*e + 16*A*
B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2 - (3*B^2*a
^3 - 5*A^2*a^2*c)*d*e^4 - (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2*c^2*d^2*e^
8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*e^10)/
(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*c^3*e^2)))
- 3*(a^2*c^3*x^4 - 2*a^3*c^2*x^2 + a^4*c)*sqrt((16*A^2*c^3*d^5 - 16*A*B*a*c^2*d^
4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*a*c^2)*d^3*e^2
- (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 - (a^5*c^4*d^2 - a^6*c^3*e^2)*sqrt((4*A^2*B^2
*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*
c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*c^4*d^2 - a^6*
c^3*e^2))*log(-27*(32*A^3*B*c^4*d^5*e^4 - 16*(3*A^2*B^2*a*c^3 + A^4*c^4)*d^4*e^5
+ 8*(3*A*B^3*a^2*c^2 - A^3*B*a*c^3)*d^3*e^6 - 4*(B^4*a^3*c - 6*A^2*B^2*a^2*c^2
- 3*A^4*a*c^3)*d^2*e^7 - 2*(5*A*B^3*a^3*c + 3*A^3*B*a^2*c^2)*d*e^8 + (B^4*a^4 -
A^4*a^2*c^2)*e^9)*sqrt(e*x + d) - 27*(4*A^2*B*a^3*c^4*d^3*e^5 - 2*(2*A*B^2*a^4*c
^3 + A^3*a^3*c^4)*d^2*e^6 + (B^3*a^5*c^2 - A^2*B*a^4*c^3)*d*e^7 + (A*B^2*a^5*c^2
+ A^3*a^4*c^3)*e^8 - (4*A*a^5*c^8*d^5 - 2*B*a^6*c^7*d^4*e - 7*A*a^6*c^7*d^3*e^2
+ 3*B*a^7*c^6*d^2*e^3 + 3*A*a^7*c^6*d*e^4 - B*a^8*c^5*e^5)*sqrt((4*A^2*B^2*c^2*
d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*B^2*a*c + A^4*c^2)*
e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))*sqrt((16*A^2*c^3*d^5 - 1
6*A*B*a*c^2*d^4*e + 16*A*B*a^2*c*d^2*e^3 - 2*A*B*a^3*e^5 + 4*(B^2*a^2*c - 5*A^2*
a*c^2)*d^3*e^2 - (3*B^2*a^3 - 5*A^2*a^2*c)*d*e^4 - (a^5*c^4*d^2 - a^6*c^3*e^2)*s
qrt((4*A^2*B^2*c^2*d^2*e^8 - 4*(A*B^3*a*c + A^3*B*c^2)*d*e^9 + (B^4*a^2 + 2*A^2*
B^2*a*c + A^4*c^2)*e^10)/(a^5*c^9*d^4 - 2*a^6*c^8*d^2*e^2 + a^7*c^7*e^4)))/(a^5*
c^4*d^2 - a^6*c^3*e^2))) - 4*(A*a*c*e*x^2 + 4*B*a^2*d + 3*A*a^2*e - 3*(2*A*c^2*d
- B*a*c*e)*x^3 + (10*A*a*c*d + B*a^2*e)*x)*sqrt(e*x + d))/(a^2*c^3*x^4 - 2*a^3*
c^2*x^2 + a^4*c)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out