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

Optimal. Leaf size=53 \[ -\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{3/2} \sqrt{b}}-\frac{2 A}{3 a x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.113014, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{3/2} \sqrt{b}}-\frac{2 A}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.6981, size = 49, normalized size = 0.92 \[ - \frac{2 A}{3 a x^{\frac{3}{2}}} - \frac{2 \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x^{\frac{3}{2}}}{\sqrt{a}} \right )}}{3 a^{\frac{3}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A/(3*a*x**(3/2)) - 2*(A*b - B*a)*atan(sqrt(b)*x**(3/2)/sqrt(a))/(3*a**(3/2)*s
qrt(b))

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Mathematica [B]  time = 0.178389, size = 160, normalized size = 3.02 \[ \frac{2 (a B-A b) \tan ^{-1}\left (\frac{2 \sqrt [6]{b} \sqrt{x}-\sqrt{3} \sqrt [6]{a}}{\sqrt [6]{a}}\right )}{3 a^{3/2} \sqrt{b}}+\frac{2 (a B-A b) \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 a^{3/2} \sqrt{b}}-\frac{2 (a B-A b) \tan ^{-1}\left (\frac{\sqrt [6]{b} \sqrt{x}}{\sqrt [6]{a}}\right )}{3 a^{3/2} \sqrt{b}}-\frac{2 A}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 53, normalized size = 1. \[ -{\frac{2\,Ab}{3\,a}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{2\,B}{3}\arctan \left ({b{x}^{{\frac{3}{2}}}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{2\,A}{3\,a}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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)*x^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245175, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (B a - A b\right )} x^{\frac{3}{2}} \log \left (-\frac{2 \, a b x^{\frac{3}{2}} -{\left (b x^{3} - a\right )} \sqrt{-a b}}{b x^{3} + a}\right ) + 2 \, \sqrt{-a b} A}{3 \, \sqrt{-a b} a x^{\frac{3}{2}}}, \frac{2 \,{\left ({\left (B a - A b\right )} x^{\frac{3}{2}} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right ) - \sqrt{a b} A\right )}}{3 \, \sqrt{a b} a x^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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**3+A)/x**(5/2)/(b*x**3+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212065, size = 53, normalized size = 1. \[ \frac{2 \,{\left (B a - A b\right )} \arctan \left (\frac{b x^{\frac{3}{2}}}{\sqrt{a b}}\right )}{3 \, \sqrt{a b} a} - \frac{2 \, A}{3 \, a x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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