3.177 \(\int \frac{A+B x^3}{x^{5/2} \left (a+b x^3\right )^3} \, dx\)

Optimal. Leaf size=130 \[ -\frac{(5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{4 a^{7/2} \sqrt{b}}-\frac{5 A b-a B}{4 a^3 b x^{3/2}}+\frac{5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2} \]

[Out]

-(5*A*b - a*B)/(4*a^3*b*x^(3/2)) + (A*b - a*B)/(6*a*b*x^(3/2)*(a + b*x^3)^2) + (
5*A*b - a*B)/(12*a^2*b*x^(3/2)*(a + b*x^3)) - ((5*A*b - a*B)*ArcTan[(Sqrt[b]*x^(
3/2))/Sqrt[a]])/(4*a^(7/2)*Sqrt[b])

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Rubi [A]  time = 0.211944, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{(5 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{4 a^{7/2} \sqrt{b}}-\frac{5 A b-a B}{4 a^3 b x^{3/2}}+\frac{5 A b-a B}{12 a^2 b x^{3/2} \left (a+b x^3\right )}+\frac{A b-a B}{6 a b x^{3/2} \left (a+b x^3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^3)/(x^(5/2)*(a + b*x^3)^3),x]

[Out]

-(5*A*b - a*B)/(4*a^3*b*x^(3/2)) + (A*b - a*B)/(6*a*b*x^(3/2)*(a + b*x^3)^2) + (
5*A*b - a*B)/(12*a^2*b*x^(3/2)*(a + b*x^3)) - ((5*A*b - a*B)*ArcTan[(Sqrt[b]*x^(
3/2))/Sqrt[a]])/(4*a^(7/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 23.6294, size = 109, normalized size = 0.84 \[ \frac{A b - B a}{6 a b x^{\frac{3}{2}} \left (a + b x^{3}\right )^{2}} + \frac{5 A b - B a}{12 a^{2} b x^{\frac{3}{2}} \left (a + b x^{3}\right )} - \frac{5 A b - B a}{4 a^{3} b x^{\frac{3}{2}}} - \frac{\left (5 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{4 a^{\frac{7}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**3+A)/x**(5/2)/(b*x**3+a)**3,x)

[Out]

(A*b - B*a)/(6*a*b*x**(3/2)*(a + b*x**3)**2) + (5*A*b - B*a)/(12*a**2*b*x**(3/2)
*(a + b*x**3)) - (5*A*b - B*a)/(4*a**3*b*x**(3/2)) - (5*A*b - B*a)*atan(sqrt(b)*
x**(3/2)/sqrt(a))/(4*a**(7/2)*sqrt(b))

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Mathematica [A]  time = 0.337097, size = 194, normalized size = 1.49 \[ \frac{\frac{2 a^{3/2} x^{3/2} (a B-A b)}{\left (a+b x^3\right )^2}+\frac{\sqrt{a} x^{3/2} (3 a B-7 A b)}{a+b x^3}+\frac{3 (5 A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt{b}}-\frac{3 (5 A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{\sqrt{b}}+\frac{3 (5 A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt{b}}-\frac{8 \sqrt{a} A}{x^{3/2}}}{12 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^3)/(x^(5/2)*(a + b*x^3)^3),x]

[Out]

((-8*Sqrt[a]*A)/x^(3/2) + (2*a^(3/2)*(-(A*b) + a*B)*x^(3/2))/(a + b*x^3)^2 + (Sq
rt[a]*(-7*A*b + 3*a*B)*x^(3/2))/(a + b*x^3) + (3*(5*A*b - a*B)*ArcTan[Sqrt[3] -
(2*b^(1/6)*Sqrt[x])/a^(1/6)])/Sqrt[b] - (3*(5*A*b - a*B)*ArcTan[Sqrt[3] + (2*b^(
1/6)*Sqrt[x])/a^(1/6)])/Sqrt[b] + (3*(5*A*b - a*B)*ArcTan[(b^(1/6)*Sqrt[x])/a^(1
/6)])/Sqrt[b])/(12*a^(7/2))

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Maple [A]  time = 0.028, size = 133, normalized size = 1. \[ -{\frac{2\,A}{3\,{a}^{3}}{x}^{-{\frac{3}{2}}}}-{\frac{7\,{b}^{2}A}{12\,{a}^{3} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{9}{2}}}}+{\frac{Bb}{4\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{9}{2}}}}-{\frac{3\,Ab}{4\,{a}^{2} \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}+{\frac{5\,B}{12\,a \left ( b{x}^{3}+a \right ) ^{2}}{x}^{{\frac{3}{2}}}}-{\frac{5\,Ab}{4\,{a}^{3}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{4\,{a}^{2}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^3+A)/x^(5/2)/(b*x^3+a)^3,x)

[Out]

-2/3*A/a^3/x^(3/2)-7/12/a^3/(b*x^3+a)^2*x^(9/2)*b^2*A+1/4/a^2/(b*x^3+a)^2*x^(9/2
)*b*B-3/4/a^2/(b*x^3+a)^2*A*x^(3/2)*b+5/12/a/(b*x^3+a)^2*B*x^(3/2)-5/4/a^3/(a*b)
^(1/2)*arctan(x^(3/2)*b/(a*b)^(1/2))*A*b+1/4/a^2/(a*b)^(1/2)*arctan(x^(3/2)*b/(a
*b)^(1/2))*B

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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^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.292869, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{7} + 2 \,{\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt{x} \log \left (-\frac{2 \, a b x^{\frac{3}{2}} -{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right ) - 2 \,{\left (3 \,{\left (B a b - 5 \, A b^{2}\right )} x^{6} + 5 \,{\left (B a^{2} - 5 \, A a b\right )} x^{3} - 8 \, A a^{2}\right )} \sqrt{-a b}}{24 \,{\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )} \sqrt{-a b} \sqrt{x}}, \frac{3 \,{\left ({\left (B a b^{2} - 5 \, A b^{3}\right )} x^{7} + 2 \,{\left (B a^{2} b - 5 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 5 \, A a^{2} b\right )} x\right )} \sqrt{x} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right ) +{\left (3 \,{\left (B a b - 5 \, A b^{2}\right )} x^{6} + 5 \,{\left (B a^{2} - 5 \, A a b\right )} x^{3} - 8 \, A a^{2}\right )} \sqrt{a b}}{12 \,{\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )} \sqrt{a b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^3*x^(5/2)),x, algorithm="fricas")

[Out]

[-1/24*(3*((B*a*b^2 - 5*A*b^3)*x^7 + 2*(B*a^2*b - 5*A*a*b^2)*x^4 + (B*a^3 - 5*A*
a^2*b)*x)*sqrt(x)*log(-(2*a*b*x^(3/2) - (b*x^3 - a)*sqrt(-a*b))/(b*x^3 + a)) - 2
*(3*(B*a*b - 5*A*b^2)*x^6 + 5*(B*a^2 - 5*A*a*b)*x^3 - 8*A*a^2)*sqrt(-a*b))/((a^3
*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*sqrt(-a*b)*sqrt(x)), 1/12*(3*((B*a*b^2 - 5*A*b^3
)*x^7 + 2*(B*a^2*b - 5*A*a*b^2)*x^4 + (B*a^3 - 5*A*a^2*b)*x)*sqrt(x)*arctan(sqrt
(a*b)*x^(3/2)/a) + (3*(B*a*b - 5*A*b^2)*x^6 + 5*(B*a^2 - 5*A*a*b)*x^3 - 8*A*a^2)
*sqrt(a*b))/((a^3*b^2*x^7 + 2*a^4*b*x^4 + a^5*x)*sqrt(a*b)*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**(5/2)/(b*x**3+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220152, size = 119, normalized size = 0.92 \[ \frac{{\left (B a - 5 \, A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{4 \, \sqrt{a b} a^{3}} - \frac{2 \, A}{3 \, a^{3} x^{\frac{3}{2}}} + \frac{3 \, B a b x^{\frac{9}{2}} - 7 \, A b^{2} x^{\frac{9}{2}} + 5 \, B a^{2} x^{\frac{3}{2}} - 9 \, A a b x^{\frac{3}{2}}}{12 \,{\left (b x^{3} + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)/((b*x^3 + a)^3*x^(5/2)),x, algorithm="giac")

[Out]

1/4*(B*a - 5*A*b)*arctan(b*x^(3/2)/sqrt(a*b))/(sqrt(a*b)*a^3) - 2/3*A/(a^3*x^(3/
2)) + 1/12*(3*B*a*b*x^(9/2) - 7*A*b^2*x^(9/2) + 5*B*a^2*x^(3/2) - 9*A*a*b*x^(3/2
))/((b*x^3 + a)^2*a^3)