Optimal. Leaf size=97 \[ -\frac{3 A b-a B}{3 a^2 b x^{3/2}}-\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{5/2} \sqrt{b}}+\frac{A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )} \]
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Rubi [A] time = 0.0571234, 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, Rules used = {457, 325, 329, 275, 205} \[ -\frac{3 A b-a B}{3 a^2 b x^{3/2}}-\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{5/2} \sqrt{b}}+\frac{A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 457
Rule 325
Rule 329
Rule 275
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^{5/2} \left (a+b x^3\right )^2} \, dx &=\frac{A b-a B}{3 a b x^{3/2} \left (a+b x^3\right )}+\frac{\left (\frac{9 A b}{2}-\frac{3 a B}{2}\right ) \int \frac{1}{x^{5/2} \left (a+b x^3\right )} \, dx}{3 a 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 )}-\frac{(3 A b-a B) \int \frac{\sqrt{x}}{a+b x^3} \, dx}{2 a^2}\\ &=-\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 )}-\frac{(3 A b-a B) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^6} \, dx,x,\sqrt{x}\right )}{a^2}\\ &=-\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 )}-\frac{(3 A b-a B) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,x^{3/2}\right )}{3 a^2}\\ &=-\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 )}-\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{3 a^{5/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.116151, size = 79, normalized size = 0.81 \[ \frac{\frac{\sqrt{a} \left (-2 a A+a B x^3-3 A b x^3\right )}{x^{3/2} \left (a+b x^3\right )}+\frac{(a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x^{3/2}}{\sqrt{a}}\right )}{\sqrt{b}}}{3 a^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 93, normalized size = 1. \begin{align*} -{\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}}}}-{\frac{2\,A}{3\,{a}^{2}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92989, size = 498, normalized size = 5.13 \begin{align*} \left [\frac{{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{5} +{\left (B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{3} + 2 \, \sqrt{-a b} x^{\frac{3}{2}} - a}{b x^{3} + a}\right ) - 2 \,{\left (2 \, A a^{2} b -{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{3}\right )} \sqrt{x}}{6 \,{\left (a^{3} b^{2} x^{5} + a^{4} b x^{2}\right )}}, \frac{{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{5} +{\left (B a^{2} - 3 \, A a b\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x^{\frac{3}{2}}}{a}\right ) -{\left (2 \, A a^{2} b -{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{3}\right )} \sqrt{x}}{3 \,{\left (a^{3} b^{2} x^{5} + a^{4} b x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10318, size = 89, normalized size = 0.92 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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