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3.176 \(\int \frac{A+B x^3}{x^{3/2} \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=351 \[ -\frac{7 (13 A b-a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{19/6} b^{5/6}}+\frac{7 (13 A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{19/6} b^{5/6}}+\frac{7 (13 A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{216 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B)}{36 a^3 b \sqrt{x}}+\frac{13 A b-a B}{36 a^2 b \sqrt{x} \left (a+b x^3\right )}+\frac{A b-a B}{6 a b \sqrt{x} \left (a+b x^3\right )^2} \]

[Out]

(-7*(13*A*b - a*B))/(36*a^3*b*Sqrt[x]) + (A*b - a*B)/(6*a*b*Sqrt[x]*(a + b*x^3)^
2) + (13*A*b - a*B)/(36*a^2*b*Sqrt[x]*(a + b*x^3)) + (7*(13*A*b - a*B)*ArcTan[Sq
rt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(19/6)*b^(5/6)) - (7*(13*A*b - a*B)
*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(19/6)*b^(5/6)) - (7*(13*
A*b - a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(108*a^(19/6)*b^(5/6)) - (7*(13*A*
b - a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3
]*a^(19/6)*b^(5/6)) + (7*(13*A*b - a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sq
rt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(19/6)*b^(5/6))

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Rubi [A]  time = 1.61154, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{7 (13 A b-a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{19/6} b^{5/6}}+\frac{7 (13 A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{144 \sqrt{3} a^{19/6} b^{5/6}}+\frac{7 (13 A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{216 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{216 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{108 a^{19/6} b^{5/6}}-\frac{7 (13 A b-a B)}{36 a^3 b \sqrt{x}}+\frac{13 A b-a B}{36 a^2 b \sqrt{x} \left (a+b x^3\right )}+\frac{A b-a B}{6 a b \sqrt{x} \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^(3/2)*(a + b*x^3)^3),x]

[Out]

(-7*(13*A*b - a*B))/(36*a^3*b*Sqrt[x]) + (A*b - a*B)/(6*a*b*Sqrt[x]*(a + b*x^3)^
2) + (13*A*b - a*B)/(36*a^2*b*Sqrt[x]*(a + b*x^3)) + (7*(13*A*b - a*B)*ArcTan[Sq
rt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(19/6)*b^(5/6)) - (7*(13*A*b - a*B)
*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(216*a^(19/6)*b^(5/6)) - (7*(13*
A*b - a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(108*a^(19/6)*b^(5/6)) - (7*(13*A*
b - a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(144*Sqrt[3
]*a^(19/6)*b^(5/6)) + (7*(13*A*b - a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sq
rt[x] + b^(1/3)*x])/(144*Sqrt[3]*a^(19/6)*b^(5/6))

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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**3+A)/x**(3/2)/(b*x**3+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.673809, size = 306, normalized size = 0.87 \[ \frac{\frac{72 a^{7/6} x^{5/2} (a B-A b)}{\left (a+b x^3\right )^2}+\frac{7 \sqrt{3} (a B-13 A b) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/6}}+\frac{7 \sqrt{3} (13 A b-a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/6}}+\frac{14 (13 A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{b^{5/6}}-\frac{14 (13 A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{b^{5/6}}+\frac{28 (a B-13 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{b^{5/6}}+\frac{12 \sqrt [6]{a} x^{5/2} (7 a B-19 A b)}{a+b x^3}-\frac{864 \sqrt [6]{a} A}{\sqrt{x}}}{432 a^{19/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^(3/2)*(a + b*x^3)^3),x]

[Out]

((-864*a^(1/6)*A)/Sqrt[x] + (72*a^(7/6)*(-(A*b) + a*B)*x^(5/2))/(a + b*x^3)^2 +
(12*a^(1/6)*(-19*A*b + 7*a*B)*x^(5/2))/(a + b*x^3) + (14*(13*A*b - a*B)*ArcTan[S
qrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/b^(5/6) - (14*(13*A*b - a*B)*ArcTan[Sqrt[
3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/b^(5/6) + (28*(-13*A*b + a*B)*ArcTan[(b^(1/6)
*Sqrt[x])/a^(1/6)])/b^(5/6) + (7*Sqrt[3]*(-13*A*b + a*B)*Log[a^(1/3) - Sqrt[3]*a
^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/b^(5/6) + (7*Sqrt[3]*(13*A*b - a*B)*Log[a^(
1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/b^(5/6))/(432*a^(19/6))

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Maple [A]  time = 0.069, size = 435, normalized size = 1.2 \[ -2\,{\frac{A}{{a}^{3}\sqrt{x}}}-{\frac{19\,{b}^{2}A}{36\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{11}{2}}}}+{\frac{7\,Bb}{36\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{11}{2}}}}-{\frac{25\,Ab}{36\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}+{\frac{13\,B}{36\,a \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{5}{2}}}}-{\frac{91\,A}{108\,{a}^{3}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{91\,Ab\sqrt{3}}{432\,{a}^{4}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{91\,A}{216\,{a}^{3}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{91\,Ab\sqrt{3}}{432\,{a}^{4}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{91\,A}{216\,{a}^{3}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{7\,B}{108\,{a}^{2}b}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+{\frac{7\,B\sqrt{3}}{432\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{7\,B}{216\,{a}^{2}b}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{7\,B\sqrt{3}}{432\,{a}^{3}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{7\,B}{216\,{a}^{2}b}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^(3/2)/(b*x^3+a)^3,x)

[Out]

-2*A/a^3/x^(1/2)-19/36/a^3/(b*x^3+a)^2*x^(11/2)*b^2*A+7/36/a^2/(b*x^3+a)^2*x^(11
/2)*b*B-25/36/a^2/(b*x^3+a)^2*A*x^(5/2)*b+13/36/a/(b*x^3+a)^2*B*x^(5/2)-91/108/a
^3*A/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))-91/432/a^4*A*b*(a/b)^(5/6)*3^(1/2)*
ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-91/216/a^3*A/(a/b)^(1/6)*arctan(-3
^(1/2)+2*x^(1/2)/(a/b)^(1/6))+91/432/a^4*A*b*3^(1/2)*(a/b)^(5/6)*ln(x+3^(1/2)*(a
/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-91/216/a^3*A/(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b)^(
1/6)+3^(1/2))+7/108/a^2*B/b/(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))+7/432/a^3*B*
(a/b)^(5/6)*3^(1/2)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+7/216/a^2*B/b/
(a/b)^(1/6)*arctan(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))-7/432/a^3*B*3^(1/2)*(a/b)^(5/
6)*ln(x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+7/216/a^2*B/b/(a/b)^(1/6)*arcta
n(2*x^(1/2)/(a/b)^(1/6)+3^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.299662, size = 4833, normalized size = 13.77 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/432*(84*(B*a*b - 13*A*b^2)*x^6 + 156*(B*a^2 - 13*A*a*b)*x^3 - 28*sqrt(3)*(a^3*
b^2*x^6 + 2*a^4*b*x^3 + a^5)*sqrt(x)*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*
a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 +
 4826809*A^6*b^6)/(a^19*b^5))^(1/6)*arctan(16807*sqrt(3)*a^16*b^4*(-(B^6*a^6 - 7
8*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^
2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5/6)/(16807*a^16*b^4
*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 42
8415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5/6)
- 33614*(B^5*a^5 - 65*A*B^4*a^4*b + 1690*A^2*B^3*a^3*b^2 - 21970*A^3*B^2*a^2*b^3
 + 142805*A^4*B*a*b^4 - 371293*A^5*b^5)*sqrt(x) + 2*sqrt(-282475249*(B^5*a^21*b^
4 - 65*A*B^4*a^20*b^5 + 1690*A^2*B^3*a^19*b^6 - 21970*A^3*B^2*a^18*b^7 + 142805*
A^4*B*a^17*b^8 - 371293*A^5*a^16*b^9)*sqrt(x)*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535
*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*
B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5/6) + 282475249*(B^10*a^10 - 130*A*B^9*
a^9*b + 7605*A^2*B^8*a^8*b^2 - 263640*A^3*B^7*a^7*b^3 + 5997810*A^4*B^6*a^6*b^4
- 93565836*A^5*B^5*a^5*b^5 + 1013629890*A^6*B^4*a^4*b^6 - 7529822040*A^7*B^3*a^3
*b^7 + 36707882445*A^8*B^2*a^2*b^8 - 106044993730*A^9*B*a*b^9 + 137858491849*A^1
0*b^10)*x - 282475249*(B^6*a^19*b^3 - 78*A*B^5*a^18*b^4 + 2535*A^2*B^4*a^17*b^5
- 43940*A^3*B^3*a^16*b^6 + 428415*A^4*B^2*a^15*b^7 - 2227758*A^5*B*a^14*b^8 + 48
26809*A^6*a^13*b^9)*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A
^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)
/(a^19*b^5))^(2/3)))) - 28*sqrt(3)*(a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)*sqrt(x)*(-(
B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415
*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(1/6)*arct
an(-sqrt(3)*a^16*b^4*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*
A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6
)/(a^19*b^5))^(5/6)/(a^16*b^4*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2
 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 482680
9*A^6*b^6)/(a^19*b^5))^(5/6) + 2*(B^5*a^5 - 65*A*B^4*a^4*b + 1690*A^2*B^3*a^3*b^
2 - 21970*A^3*B^2*a^2*b^3 + 142805*A^4*B*a*b^4 - 371293*A^5*b^5)*sqrt(x) - 2*sqr
t((B^5*a^21*b^4 - 65*A*B^4*a^20*b^5 + 1690*A^2*B^3*a^19*b^6 - 21970*A^3*B^2*a^18
*b^7 + 142805*A^4*B*a^17*b^8 - 371293*A^5*a^16*b^9)*sqrt(x)*(-(B^6*a^6 - 78*A*B^
5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4
- 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5/6) + (B^10*a^10 - 130*A*
B^9*a^9*b + 7605*A^2*B^8*a^8*b^2 - 263640*A^3*B^7*a^7*b^3 + 5997810*A^4*B^6*a^6*
b^4 - 93565836*A^5*B^5*a^5*b^5 + 1013629890*A^6*B^4*a^4*b^6 - 7529822040*A^7*B^3
*a^3*b^7 + 36707882445*A^8*B^2*a^2*b^8 - 106044993730*A^9*B*a*b^9 + 137858491849
*A^10*b^10)*x - (B^6*a^19*b^3 - 78*A*B^5*a^18*b^4 + 2535*A^2*B^4*a^17*b^5 - 4394
0*A^3*B^3*a^16*b^6 + 428415*A^4*B^2*a^15*b^7 - 2227758*A^5*B*a^14*b^8 + 4826809*
A^6*a^13*b^9)*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3
*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19
*b^5))^(2/3)))) - 864*A*a^2 - 14*(a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)*sqrt(x)*(-(B^
6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A
^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(1/6)*log(16
807*a^16*b^4*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*
a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*
b^5))^(5/6) - 16807*(B^5*a^5 - 65*A*B^4*a^4*b + 1690*A^2*B^3*a^3*b^2 - 21970*A^3
*B^2*a^2*b^3 + 142805*A^4*B*a*b^4 - 371293*A^5*b^5)*sqrt(x)) + 14*(a^3*b^2*x^6 +
 2*a^4*b*x^3 + a^5)*sqrt(x)*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 -
 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*
A^6*b^6)/(a^19*b^5))^(1/6)*log(-16807*a^16*b^4*(-(B^6*a^6 - 78*A*B^5*a^5*b + 253
5*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5
*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5/6) - 16807*(B^5*a^5 - 65*A*B^4*a^4*b
+ 1690*A^2*B^3*a^3*b^2 - 21970*A^3*B^2*a^2*b^3 + 142805*A^4*B*a*b^4 - 371293*A^5
*b^5)*sqrt(x)) + 7*(a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)*sqrt(x)*(-(B^6*a^6 - 78*A*B
^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4
 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(1/6)*log(282475249*(B^5*a
^21*b^4 - 65*A*B^4*a^20*b^5 + 1690*A^2*B^3*a^19*b^6 - 21970*A^3*B^2*a^18*b^7 + 1
42805*A^4*B*a^17*b^8 - 371293*A^5*a^16*b^9)*sqrt(x)*(-(B^6*a^6 - 78*A*B^5*a^5*b
+ 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 222775
8*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5/6) + 282475249*(B^10*a^10 - 130*
A*B^9*a^9*b + 7605*A^2*B^8*a^8*b^2 - 263640*A^3*B^7*a^7*b^3 + 5997810*A^4*B^6*a^
6*b^4 - 93565836*A^5*B^5*a^5*b^5 + 1013629890*A^6*B^4*a^4*b^6 - 7529822040*A^7*B
^3*a^3*b^7 + 36707882445*A^8*B^2*a^2*b^8 - 106044993730*A^9*B*a*b^9 + 1378584918
49*A^10*b^10)*x - 282475249*(B^6*a^19*b^3 - 78*A*B^5*a^18*b^4 + 2535*A^2*B^4*a^1
7*b^5 - 43940*A^3*B^3*a^16*b^6 + 428415*A^4*B^2*a^15*b^7 - 2227758*A^5*B*a^14*b^
8 + 4826809*A^6*a^13*b^9)*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 4
3940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^
6*b^6)/(a^19*b^5))^(2/3)) - 7*(a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)*sqrt(x)*(-(B^6*a
^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*
B^2*a^2*b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(1/6)*log(-2824
75249*(B^5*a^21*b^4 - 65*A*B^4*a^20*b^5 + 1690*A^2*B^3*a^19*b^6 - 21970*A^3*B^2*
a^18*b^7 + 142805*A^4*B*a^17*b^8 - 371293*A^5*a^16*b^9)*sqrt(x)*(-(B^6*a^6 - 78*
A*B^5*a^5*b + 2535*A^2*B^4*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*
b^4 - 2227758*A^5*B*a*b^5 + 4826809*A^6*b^6)/(a^19*b^5))^(5/6) + 282475249*(B^10
*a^10 - 130*A*B^9*a^9*b + 7605*A^2*B^8*a^8*b^2 - 263640*A^3*B^7*a^7*b^3 + 599781
0*A^4*B^6*a^6*b^4 - 93565836*A^5*B^5*a^5*b^5 + 1013629890*A^6*B^4*a^4*b^6 - 7529
822040*A^7*B^3*a^3*b^7 + 36707882445*A^8*B^2*a^2*b^8 - 106044993730*A^9*B*a*b^9
+ 137858491849*A^10*b^10)*x - 282475249*(B^6*a^19*b^3 - 78*A*B^5*a^18*b^4 + 2535
*A^2*B^4*a^17*b^5 - 43940*A^3*B^3*a^16*b^6 + 428415*A^4*B^2*a^15*b^7 - 2227758*A
^5*B*a^14*b^8 + 4826809*A^6*a^13*b^9)*(-(B^6*a^6 - 78*A*B^5*a^5*b + 2535*A^2*B^4
*a^4*b^2 - 43940*A^3*B^3*a^3*b^3 + 428415*A^4*B^2*a^2*b^4 - 2227758*A^5*B*a*b^5
+ 4826809*A^6*b^6)/(a^19*b^5))^(2/3)))/((a^3*b^2*x^6 + 2*a^4*b*x^3 + a^5)*sqrt(x
))

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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**3+A)/x**(3/2)/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.307894, size = 444, normalized size = 1.26 \[ -\frac{2 \, A}{a^{3} \sqrt{x}} + \frac{7 \, B a b x^{\frac{11}{2}} - 19 \, A b^{2} x^{\frac{11}{2}} + 13 \, B a^{2} x^{\frac{5}{2}} - 25 \, A a b x^{\frac{5}{2}}}{36 \,{\left (b x^{3} + a\right )}^{2} a^{3}} - \frac{7 \, \sqrt{3}{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 13 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )}{\rm ln}\left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{432 \, a^{4} b^{5}} + \frac{7 \, \sqrt{3}{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 13 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )}{\rm ln}\left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{432 \, a^{4} b^{5}} + \frac{7 \,{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 13 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{216 \, a^{4} b^{5}} + \frac{7 \,{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 13 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{216 \, a^{4} b^{5}} + \frac{7 \,{\left (\left (a b^{5}\right )^{\frac{5}{6}} B a - 13 \, \left (a b^{5}\right )^{\frac{5}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{108 \, a^{4} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(a^3*sqrt(x)) + 1/36*(7*B*a*b*x^(11/2) - 19*A*b^2*x^(11/2) + 13*B*a^2*x^(5/
2) - 25*A*a*b*x^(5/2))/((b*x^3 + a)^2*a^3) - 7/432*sqrt(3)*((a*b^5)^(5/6)*B*a -
13*(a*b^5)^(5/6)*A*b)*ln(sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^4*b^5
) + 7/432*sqrt(3)*((a*b^5)^(5/6)*B*a - 13*(a*b^5)^(5/6)*A*b)*ln(-sqrt(3)*sqrt(x)
*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^4*b^5) + 7/216*((a*b^5)^(5/6)*B*a - 13*(a*b^5
)^(5/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/(a^4*b^5) + 7
/216*((a*b^5)^(5/6)*B*a - 13*(a*b^5)^(5/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6) - 2
*sqrt(x))/(a/b)^(1/6))/(a^4*b^5) + 7/108*((a*b^5)^(5/6)*B*a - 13*(a*b^5)^(5/6)*A
*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a^4*b^5)

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