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