3.1459 \(\int \frac{(A+B x) (d+e x)^{5/2}}{\left (a-c x^2\right )^3} \, dx\)

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

[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) + (S
qrt[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[S
qrt[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])/Sqrt[Sqrt[c]*
d + Sqrt[a]*e]])/(32*a^(5/2)*c^(9/4))

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Rubi [A]  time = 1.42744, antiderivative size = 372, 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 \[ \frac{\sqrt{\sqrt{c} d-\sqrt{a} e} \left (5 a B e \left (\sqrt{a} e+2 \sqrt{c} d\right )-3 A \left (2 \sqrt{a} c d e-a \sqrt{c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{32 a^{5/2} c^{9/4}}-\frac{\sqrt{\sqrt{a} e+\sqrt{c} d} \left (5 a B e \left (2 \sqrt{c} d-\sqrt{a} e\right )-3 A \left (-2 \sqrt{a} c d e-a \sqrt{c} e^2+4 c^{3/2} d^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{32 a^{5/2} c^{9/4}}+\frac{\sqrt{d+e x} \left (c x \left (6 A c d^2-a e (3 A e+5 B d)\right )+a e (3 A c d-5 a B e)\right )}{16 a^2 c^2 \left (a-c x^2\right )}+\frac{(d+e x)^{3/2} (x (a B e+A c d)+a (A e+B d))}{4 a c \left (a-c x^2\right )^2} \]

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) + (S
qrt[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[S
qrt[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))

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

[Out]

Timed out

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Mathematica [A]  time = 1.37, size = 395, normalized size = 1.06 \[ \frac{\frac{2 \sqrt{a} \sqrt{d+e x} \left (-5 a^3 B e^2+a^2 c \left (A e (7 d+e x)+B \left (4 d^2+3 d e x+9 e^2 x^2\right )\right )+a c^2 x \left (A \left (10 d^2+d e x+3 e^2 x^2\right )+5 B d e x^2\right )-6 A c^3 d^2 x^3\right )}{\left (a-c x^2\right )^2}-\frac{\left (\sqrt{c} d-\sqrt{a} e\right ) \left (3 A \left (2 \sqrt{a} c d e-a \sqrt{c} e^2+4 c^{3/2} d^2\right )-5 a B e \left (\sqrt{a} e+2 \sqrt{c} d\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}\right )}{\sqrt{c d-\sqrt{a} \sqrt{c} e}}+\frac{\left (\sqrt{a} e+\sqrt{c} d\right ) \left (3 A \left (-2 \sqrt{a} c d e-a \sqrt{c} e^2+4 c^{3/2} d^2\right )+5 a B e \left (\sqrt{a} e-2 \sqrt{c} d\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}\right )}{\sqrt{\sqrt{a} \sqrt{c} e+c d}}}{32 a^{5/2} c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^(5/2))/(a - c*x^2)^3,x]

[Out]

((2*Sqrt[a]*Sqrt[d + e*x]*(-5*a^3*B*e^2 - 6*A*c^3*d^2*x^3 + a*c^2*x*(5*B*d*e*x^2
+ A*(10*d^2 + d*e*x + 3*e^2*x^2)) + a^2*c*(A*e*(7*d + e*x) + B*(4*d^2 + 3*d*e*x
+ 9*e^2*x^2))))/(a - c*x^2)^2 - ((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[(
Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d - Sqrt[a]*Sqrt[c
]*e] + ((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[(Sqrt[c]*Sqrt[d + e*x])/Sqr
t[c*d + Sqrt[a]*Sqrt[c]*e]])/Sqrt[c*d + Sqrt[a]*Sqrt[c]*e])/(32*a^(5/2)*c^2)

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Maple [B]  time = 0.083, size = 1931, normalized size = 5.2 \[ \text{result too large to display} \]

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]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (B x + A\right )}{\left (e x + d\right )}^{\frac{5}{2}}}{{\left (c x^{2} - a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(B*x + A)*(e*x + d)^(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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Fricas [A]  time = 1.05683, size = 4566, normalized size = 12.27 \[ \text{result too large to display} \]

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 - 1
5*(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((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 - 43
2*(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 - 3
0*(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((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 - 4
32*(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((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)) + 2
0*(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 - 3
0*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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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)**(5/2)/(-c*x**2+a)**3,x)

[Out]

Timed out

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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)^(5/2)/(c*x^2 - a)^3,x, algorithm="giac")

[Out]

Timed out