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

Optimal. Leaf size=97 \[ -\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{5/2} \sqrt{b}}-\frac{3 A b-a B}{3 a^2 b x^{3/2}}+\frac{A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.173487, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{5/2} \sqrt{b}}-\frac{3 A b-a B}{3 a^2 b x^{3/2}}+\frac{A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.0133, size = 80, normalized size = 0.82 \[ \frac{A b - B a}{3 a b x^{\frac{3}{2}} \left (a + b x^{3}\right )} - \frac{3 A b - B a}{3 a^{2} b x^{\frac{3}{2}}} - \frac{\left (3 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 a^{\frac{5}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.389144, size = 162, normalized size = 1.67 \[ \frac{\frac{\sqrt{a} x^{3/2} (a B-A b)}{a+b x^3}+\frac{(3 A b-a B) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt{b}}-\frac{(3 A b-a B) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}+\sqrt{3}\right )}{\sqrt{b}}+\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{\sqrt{b}}-\frac{2 \sqrt{a} A}{x^{3/2}}}{3 a^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 93, normalized size = 1. \[ -{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}}-{\frac{Ab}{3\,{a}^{2} \left ( b{x}^{3}+a \right ) }{x}^{{\frac{3}{2}}}}+{\frac{B}{3\,a \left ( b{x}^{3}+a \right ) }{x}^{{\frac{3}{2}}}}-{\frac{Ab}{{a}^{2}}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{3\,a}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*A/a^2/x^(3/2)-1/3/a^2*x^(3/2)/(b*x^3+a)*A*b+1/3/a*x^(3/2)/(b*x^3+a)*B-1/a^2
/(a*b)^(1/2)*arctan(x^(3/2)*b/(a*b)^(1/2))*A*b+1/3/a/(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)^2*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253117, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{4} +{\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt{x} \log \left (-\frac{2 \, a b x^{\frac{3}{2}} -{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right ) - 2 \,{\left ({\left (B a - 3 \, A b\right )} x^{3} - 2 \, A a\right )} \sqrt{-a b}}{6 \,{\left (a^{2} b x^{4} + a^{3} x\right )} \sqrt{-a b} \sqrt{x}}, \frac{{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{4} +{\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt{x} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right ) +{\left ({\left (B a - 3 \, A b\right )} x^{3} - 2 \, A a\right )} \sqrt{a b}}{3 \,{\left (a^{2} b x^{4} + a^{3} x\right )} \sqrt{a b} \sqrt{x}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222052, size = 89, normalized size = 0.92 \[ \frac{{\left (B a - 3 \, A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} a^{2}} + \frac{B a x^{3} - 3 \, A b x^{3} - 2 \, A a}{3 \,{\left (b x^{\frac{9}{2}} + a x^{\frac{3}{2}}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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