Optimal. Leaf size=86 \[ -\frac{3 A b-2 a B}{3 a^2 \sqrt{a+b x^3}}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}-\frac{A}{3 a x^3 \sqrt{a+b x^3}} \]
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Rubi [A] time = 0.0650946, antiderivative size = 86, 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 = {446, 78, 51, 63, 208} \[ -\frac{3 A b-2 a B}{3 a^2 \sqrt{a+b x^3}}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}-\frac{A}{3 a x^3 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{A+B x^3}{x^4 \left (a+b x^3\right )^{3/2}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 (a+b x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac{A}{3 a x^3 \sqrt{a+b x^3}}+\frac{\left (-\frac{3 A b}{2}+a B\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{3/2}} \, dx,x,x^3\right )}{3 a}\\ &=-\frac{3 A b-2 a B}{3 a^2 \sqrt{a+b x^3}}-\frac{A}{3 a x^3 \sqrt{a+b x^3}}-\frac{(3 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^3\right )}{6 a^2}\\ &=-\frac{3 A b-2 a B}{3 a^2 \sqrt{a+b x^3}}-\frac{A}{3 a x^3 \sqrt{a+b x^3}}-\frac{(3 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^3}\right )}{3 a^2 b}\\ &=-\frac{3 A b-2 a B}{3 a^2 \sqrt{a+b x^3}}-\frac{A}{3 a x^3 \sqrt{a+b x^3}}+\frac{(3 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b x^3}}{\sqrt{a}}\right )}{3 a^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0180701, size = 57, normalized size = 0.66 \[ \frac{x^3 (2 a B-3 A b) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b x^3}{a}+1\right )-a A}{3 a^2 x^3 \sqrt{a+b x^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 100, normalized size = 1.2 \begin{align*} A \left ( -{\frac{2\,b}{3\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{1}{3\,{a}^{2}{x}^{3}}\sqrt{b{x}^{3}+a}}+{b{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) +B \left ({\frac{2}{3\,a}{\frac{1}{\sqrt{ \left ({x}^{3}+{\frac{a}{b}} \right ) b}}}}-{\frac{2}{3}{\it Artanh} \left ({\sqrt{b{x}^{3}+a}{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{3}{2}}}} \right ) \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 = 1.72835, size = 508, normalized size = 5.91 \begin{align*} \left [-\frac{{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{6} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3}\right )} \sqrt{a} \log \left (\frac{b x^{3} + 2 \, \sqrt{b x^{3} + a} \sqrt{a} + 2 \, a}{x^{3}}\right ) - 2 \,{\left ({\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3} - A a^{2}\right )} \sqrt{b x^{3} + a}}{6 \,{\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}, \frac{{\left ({\left (2 \, B a b - 3 \, A b^{2}\right )} x^{6} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{b x^{3} + a} \sqrt{-a}}{a}\right ) +{\left ({\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3} - A a^{2}\right )} \sqrt{b x^{3} + a}}{3 \,{\left (a^{3} b x^{6} + a^{4} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 78.4772, size = 264, normalized size = 3.07 \begin{align*} A \left (- \frac{1}{3 a \sqrt{b} x^{\frac{9}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} - \frac{\sqrt{b}}{a^{2} x^{\frac{3}{2}} \sqrt{\frac{a}{b x^{3}} + 1}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{\frac{3}{2}}} \right )}}{a^{\frac{5}{2}}}\right ) + B \left (\frac{2 a^{3} \sqrt{1 + \frac{b x^{3}}{a}}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} + \frac{a^{3} \log{\left (\frac{b x^{3}}{a} \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} + \frac{a^{2} b x^{3} \log{\left (\frac{b x^{3}}{a} \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}} - \frac{2 a^{2} b x^{3} \log{\left (\sqrt{1 + \frac{b x^{3}}{a}} + 1 \right )}}{3 a^{\frac{9}{2}} + 3 a^{\frac{7}{2}} b x^{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12342, size = 134, normalized size = 1.56 \begin{align*} \frac{{\left (2 \, B a - 3 \, A b\right )} \arctan \left (\frac{\sqrt{b x^{3} + a}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a} a^{2}} + \frac{2 \,{\left (b x^{3} + a\right )} B a - 2 \, B a^{2} - 3 \,{\left (b x^{3} + a\right )} A b + 2 \, A a b}{3 \,{\left ({\left (b x^{3} + a\right )}^{\frac{3}{2}} - \sqrt{b x^{3} + a} a\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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