Optimal. Leaf size=131 \[ \frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}+\frac{\sqrt{c+d x^3} (b c-a d)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.13963, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {446, 98, 156, 63, 208} \[ \frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 b^{3/2}}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}+\frac{\sqrt{c+d x^3} (b c-a d)}{3 a b \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 446
Rule 98
Rule 156
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (c+d x^3\right )^{3/2}}{x \left (a+b x^3\right )^2} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{(c+d x)^{3/2}}{x (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac{(b c-a d) \sqrt{c+d x^3}}{3 a b \left (a+b x^3\right )}+\frac{\operatorname{Subst}\left (\int \frac{b c^2+\frac{1}{2} d (b c+a d) x}{x (a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )}{3 a b}\\ &=\frac{(b c-a d) \sqrt{c+d x^3}}{3 a b \left (a+b x^3\right )}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+d x}} \, dx,x,x^3\right )}{3 a^2}-\frac{((b c-a d) (2 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{(a+b x) \sqrt{c+d x}} \, dx,x,x^3\right )}{6 a^2 b}\\ &=\frac{(b c-a d) \sqrt{c+d x^3}}{3 a b \left (a+b x^3\right )}+\frac{\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{d}+\frac{x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{3 a^2 d}-\frac{((b c-a d) (2 b c+a d)) \operatorname{Subst}\left (\int \frac{1}{a-\frac{b c}{d}+\frac{b x^2}{d}} \, dx,x,\sqrt{c+d x^3}\right )}{3 a^2 b d}\\ &=\frac{(b c-a d) \sqrt{c+d x^3}}{3 a b \left (a+b x^3\right )}-\frac{2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2}+\frac{\sqrt{b c-a d} (2 b c+a d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{3 a^2 b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.163404, size = 122, normalized size = 0.93 \[ \frac{\frac{\sqrt{b c-a d} (a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^3}}{\sqrt{b c-a d}}\right )}{b^{3/2}}+\frac{a \sqrt{c+d x^3} (b c-a d)}{b \left (a+b x^3\right )}-2 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d x^3}}{\sqrt{c}}\right )}{3 a^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.011, size = 1036, normalized size = 7.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x^{3} + c\right )}^{\frac{3}{2}}}{{\left (b x^{3} + a\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81942, size = 1485, normalized size = 11.34 \begin{align*} \left [\frac{{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \,{\left (b^{2} c x^{3} + a b c\right )} \sqrt{c} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + 2 \, \sqrt{d x^{3} + c}{\left (a b c - a^{2} d\right )}}{6 \,{\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac{{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x^{3} + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) +{\left (b^{2} c x^{3} + a b c\right )} \sqrt{c} \log \left (\frac{d x^{3} - 2 \, \sqrt{d x^{3} + c} \sqrt{c} + 2 \, c}{x^{3}}\right ) + \sqrt{d x^{3} + c}{\left (a b c - a^{2} d\right )}}{3 \,{\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac{4 \,{\left (b^{2} c x^{3} + a b c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) +{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b d x^{3} + 2 \, b c - a d + 2 \, \sqrt{d x^{3} + c} b \sqrt{\frac{b c - a d}{b}}}{b x^{3} + a}\right ) + 2 \, \sqrt{d x^{3} + c}{\left (a b c - a^{2} d\right )}}{6 \,{\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac{{\left ({\left (2 \, b^{2} c + a b d\right )} x^{3} + 2 \, a b c + a^{2} d\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (-\frac{\sqrt{d x^{3} + c} b \sqrt{-\frac{b c - a d}{b}}}{b c - a d}\right ) + 2 \,{\left (b^{2} c x^{3} + a b c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{d x^{3} + c} \sqrt{-c}}{c}\right ) + \sqrt{d x^{3} + c}{\left (a b c - a^{2} d\right )}}{3 \,{\left (a^{2} b^{2} x^{3} + a^{3} b\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 = 1.15097, size = 223, normalized size = 1.7 \begin{align*} \frac{1}{3} \, d^{2}{\left (\frac{2 \, c^{2} \arctan \left (\frac{\sqrt{d x^{3} + c}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} d^{2}} + \frac{\sqrt{d x^{3} + c} b c - \sqrt{d x^{3} + c} a d}{{\left ({\left (d x^{3} + c\right )} b - b c + a d\right )} a b d} - \frac{{\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{d x^{3} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a^{2} b d^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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