3.170 \(\int \frac{A+B x^3}{x^{7/2} \left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=318 \[ \frac{(11 A b-5 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{17/6} \sqrt [6]{b}}-\frac{(11 A b-5 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{17/6} \sqrt [6]{b}}+\frac{(11 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac{(11 A b-5 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac{(11 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}-\frac{11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac{A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )} \]

[Out]

-(11*A*b - 5*a*B)/(15*a^2*b*x^(5/2)) + (A*b - a*B)/(3*a*b*x^(5/2)*(a + b*x^3)) +
((11*A*b - 5*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(17/6)*b
^(1/6)) - ((11*A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a
^(17/6)*b^(1/6)) - ((11*A*b - 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(17
/6)*b^(1/6)) + ((11*A*b - 5*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] +
b^(1/3)*x])/(12*Sqrt[3]*a^(17/6)*b^(1/6)) - ((11*A*b - 5*a*B)*Log[a^(1/3) + Sqr
t[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(17/6)*b^(1/6))

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Rubi [A]  time = 1.12795, antiderivative size = 318, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ \frac{(11 A b-5 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{17/6} \sqrt [6]{b}}-\frac{(11 A b-5 a B) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{12 \sqrt{3} a^{17/6} \sqrt [6]{b}}+\frac{(11 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac{(11 A b-5 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{18 a^{17/6} \sqrt [6]{b}}-\frac{(11 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{9 a^{17/6} \sqrt [6]{b}}-\frac{11 A b-5 a B}{15 a^2 b x^{5/2}}+\frac{A b-a B}{3 a b x^{5/2} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^2),x]

[Out]

-(11*A*b - 5*a*B)/(15*a^2*b*x^(5/2)) + (A*b - a*B)/(3*a*b*x^(5/2)*(a + b*x^3)) +
((11*A*b - 5*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a^(17/6)*b
^(1/6)) - ((11*A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/(18*a
^(17/6)*b^(1/6)) - ((11*A*b - 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/(9*a^(17
/6)*b^(1/6)) + ((11*A*b - 5*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] +
b^(1/3)*x])/(12*Sqrt[3]*a^(17/6)*b^(1/6)) - ((11*A*b - 5*a*B)*Log[a^(1/3) + Sqr
t[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/(12*Sqrt[3]*a^(17/6)*b^(1/6))

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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**(7/2)/(b*x**3+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.540126, size = 278, normalized size = 0.87 \[ \frac{\frac{60 a^{5/6} \sqrt{x} (a B-A b)}{a+b x^3}-\frac{72 a^{5/6} A}{x^{5/2}}+\frac{5 \sqrt{3} (11 A b-5 a B) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [6]{b}}+\frac{5 \sqrt{3} (5 a B-11 A b) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} \sqrt{x}+\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [6]{b}}+\frac{10 (11 A b-5 a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac{10 (11 A b-5 a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{\sqrt [6]{b}}+\frac{20 (5 a B-11 A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}}{180 a^{17/6}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^(7/2)*(a + b*x^3)^2),x]

[Out]

((-72*a^(5/6)*A)/x^(5/2) + (60*a^(5/6)*(-(A*b) + a*B)*Sqrt[x])/(a + b*x^3) + (10
*(11*A*b - 5*a*B)*ArcTan[Sqrt[3] - (2*b^(1/6)*Sqrt[x])/a^(1/6)])/b^(1/6) - (10*(
11*A*b - 5*a*B)*ArcTan[Sqrt[3] + (2*b^(1/6)*Sqrt[x])/a^(1/6)])/b^(1/6) + (20*(-1
1*A*b + 5*a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1/6)])/b^(1/6) + (5*Sqrt[3]*(11*A*b -
5*a*B)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/3)*x])/b^(1/6) + (5
*Sqrt[3]*(-11*A*b + 5*a*B)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*Sqrt[x] + b^(1/
3)*x])/b^(1/6))/(180*a^(17/6))

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Maple [A]  time = 0.063, size = 389, normalized size = 1.2 \[ -{\frac{2\,A}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{Ab}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }\sqrt{x}}+{\frac{B}{3\,a \left ( b{x}^{3}+a \right ) }\sqrt{x}}-{\frac{11\,Ab}{9\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{11\,Ab\sqrt{3}}{36\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{11\,Ab}{18\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{11\,Ab\sqrt{3}}{36\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{11\,Ab}{18\,{a}^{3}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{5\,B}{9\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{5\,B\sqrt{3}}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5\,B}{18\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( -\sqrt{3}+2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }+{\frac{5\,B\sqrt{3}}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ( x+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}\sqrt{x}+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5\,B}{18\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{\sqrt{x}{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^(7/2)/(b*x^3+a)^2,x)

[Out]

-2/5*A/a^2/x^(5/2)-1/3/a^2*x^(1/2)/(b*x^3+a)*A*b+1/3/a*x^(1/2)/(b*x^3+a)*B-11/9/
a^3*A*b*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))+11/36/a^3*A*b*3^(1/2)*(a/b)^(1/6
)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-11/18/a^3*A*b*(a/b)^(1/6)*arctan
(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))-11/36/a^3*A*b*3^(1/2)*(a/b)^(1/6)*ln(x+3^(1/2)*
(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))-11/18/a^3*A*b*(a/b)^(1/6)*arctan(2*x^(1/2)/(a/b
)^(1/6)+3^(1/2))+5/9/a^2*B*(a/b)^(1/6)*arctan(x^(1/2)/(a/b)^(1/6))-5/36/a^2*B*3^
(1/2)*(a/b)^(1/6)*ln(x-3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+5/18/a^2*B*(a/b)
^(1/6)*arctan(-3^(1/2)+2*x^(1/2)/(a/b)^(1/6))+5/36/a^2*B*3^(1/2)*(a/b)^(1/6)*ln(
x+3^(1/2)*(a/b)^(1/6)*x^(1/2)+(a/b)^(1/3))+5/18/a^2*B*(a/b)^(1/6)*arctan(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)^2*x^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276996, size = 3048, normalized size = 9.58 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^2*x^(7/2)),x, algorithm="fricas")

[Out]

1/180*(12*(5*B*a - 11*A*b)*x^3 + 20*sqrt(3)*(a^2*b*x^5 + a^3*x^2)*sqrt(x)*(-(156
25*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*
b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))
^(1/6)*arctan(-sqrt(3)*a^3*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B
^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a
*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)/(a^3*(-(15625*B^6*a^6 - 206250*A*B^5*a^5
*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4
- 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6) + 2*(5*B*a - 11*A*b)*s
qrt(x) - 2*sqrt(a^6*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*
b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 +
1771561*A^6*b^6)/(a^17*b))^(1/3) + (25*B^2*a^2 - 110*A*B*a*b + 121*A^2*b^2)*x +
(5*B*a^4 - 11*A*a^3*b)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A
^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5
*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)))) + 20*sqrt(3)*(a^2*b*x^5 + a^3*x^2
)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327
500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^
6*b^6)/(a^17*b))^(1/6)*arctan(sqrt(3)*a^3*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b
+ 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 -
4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)/(a^3*(-(15625*B^6*a^6 - 2
06250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*
A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6) - 2*(5*
B*a - 11*A*b)*sqrt(x) + 2*sqrt(a^6*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 11343
75*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530
*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/3) + (25*B^2*a^2 - 110*A*B*a*b + 12
1*A^2*b^2)*x - (5*B*a^4 - 11*A*a^3*b)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^
5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^
4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)))) + 5*(a^2*b*x^5 + a
^3*x^2)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2
- 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771
561*A^6*b^6)/(a^17*b))^(1/6)*log(a^6*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 113
4375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 48315
30*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/3) + (25*B^2*a^2 - 110*A*B*a*b +
121*A^2*b^2)*x + (5*B*a^4 - 11*A*a^3*b)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*
a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*
b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)) - 5*(a^2*b*x^5 + a
^3*x^2)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2
- 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771
561*A^6*b^6)/(a^17*b))^(1/6)*log(a^6*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 113
4375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 48315
30*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/3) + (25*B^2*a^2 - 110*A*B*a*b +
121*A^2*b^2)*x - (5*B*a^4 - 11*A*a^3*b)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*
a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*
b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)) - 10*(a^2*b*x^5 +
a^3*x^2)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2
- 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 177
1561*A^6*b^6)/(a^17*b))^(1/6)*log(a^3*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 11
34375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831
530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6) - (5*B*a - 11*A*b)*sqrt(x)) +
10*(a^2*b*x^5 + a^3*x^2)*sqrt(x)*(-(15625*B^6*a^6 - 206250*A*B^5*a^5*b + 113437
5*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*B^2*a^2*b^4 - 4831530*
A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6)*log(-a^3*(-(15625*B^6*a^6 - 20625
0*A*B^5*a^5*b + 1134375*A^2*B^4*a^4*b^2 - 3327500*A^3*B^3*a^3*b^3 + 5490375*A^4*
B^2*a^2*b^4 - 4831530*A^5*B*a*b^5 + 1771561*A^6*b^6)/(a^17*b))^(1/6) - (5*B*a -
11*A*b)*sqrt(x)) - 72*A*a)/((a^2*b*x^5 + a^3*x^2)*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**(7/2)/(b*x**3+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.235231, size = 423, normalized size = 1.33 \[ \frac{\sqrt{3}{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 11 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a^{3} b} - \frac{\sqrt{3}{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 11 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )}{\rm ln}\left (-\sqrt{3} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{6}} + x + \left (\frac{a}{b}\right )^{\frac{1}{3}}\right )}{36 \, a^{3} b} + \frac{B a \sqrt{x} - A b \sqrt{x}}{3 \,{\left (b x^{3} + a\right )} a^{2}} + \frac{{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 11 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} + 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{3} b} + \frac{{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 11 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (-\frac{\sqrt{3} \left (\frac{a}{b}\right )^{\frac{1}{6}} - 2 \, \sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a^{3} b} + \frac{{\left (5 \, \left (a b^{5}\right )^{\frac{1}{6}} B a - 11 \, \left (a b^{5}\right )^{\frac{1}{6}} A b\right )} \arctan \left (\frac{\sqrt{x}}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{9 \, a^{3} b} - \frac{2 \, A}{5 \, a^{2} x^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^2*x^(7/2)),x, algorithm="giac")

[Out]

1/36*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 11*(a*b^5)^(1/6)*A*b)*ln(sqrt(3)*sqrt(x)*(a/
b)^(1/6) + x + (a/b)^(1/3))/(a^3*b) - 1/36*sqrt(3)*(5*(a*b^5)^(1/6)*B*a - 11*(a*
b^5)^(1/6)*A*b)*ln(-sqrt(3)*sqrt(x)*(a/b)^(1/6) + x + (a/b)^(1/3))/(a^3*b) + 1/3
*(B*a*sqrt(x) - A*b*sqrt(x))/((b*x^3 + a)*a^2) + 1/18*(5*(a*b^5)^(1/6)*B*a - 11*
(a*b^5)^(1/6)*A*b)*arctan((sqrt(3)*(a/b)^(1/6) + 2*sqrt(x))/(a/b)^(1/6))/(a^3*b)
+ 1/18*(5*(a*b^5)^(1/6)*B*a - 11*(a*b^5)^(1/6)*A*b)*arctan(-(sqrt(3)*(a/b)^(1/6
) - 2*sqrt(x))/(a/b)^(1/6))/(a^3*b) + 1/9*(5*(a*b^5)^(1/6)*B*a - 11*(a*b^5)^(1/6
)*A*b)*arctan(sqrt(x)/(a/b)^(1/6))/(a^3*b) - 2/5*A/(a^2*x^(5/2))