Optimal. Leaf size=158 \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2}{9 a b \left (a+b x^3\right )}-\frac{x^2}{6 b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.076943, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {288, 290, 292, 31, 634, 617, 204, 628} \[ \frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3}}+\frac{x^2}{9 a b \left (a+b x^3\right )}-\frac{x^2}{6 b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 288
Rule 290
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+b x^3\right )^3} \, dx &=-\frac{x^2}{6 b \left (a+b x^3\right )^2}+\frac{\int \frac{x}{\left (a+b x^3\right )^2} \, dx}{3 b}\\ &=-\frac{x^2}{6 b \left (a+b x^3\right )^2}+\frac{x^2}{9 a b \left (a+b x^3\right )}+\frac{\int \frac{x}{a+b x^3} \, dx}{9 a b}\\ &=-\frac{x^2}{6 b \left (a+b x^3\right )^2}+\frac{x^2}{9 a b \left (a+b x^3\right )}-\frac{\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{4/3} b^{4/3}}+\frac{\int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{4/3} b^{4/3}}\\ &=-\frac{x^2}{6 b \left (a+b x^3\right )^2}+\frac{x^2}{9 a b \left (a+b x^3\right )}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}+\frac{\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{4/3} b^{5/3}}+\frac{\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a b^{4/3}}\\ &=-\frac{x^2}{6 b \left (a+b x^3\right )^2}+\frac{x^2}{9 a b \left (a+b x^3\right )}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{4/3} b^{5/3}}\\ &=-\frac{x^2}{6 b \left (a+b x^3\right )^2}+\frac{x^2}{9 a b \left (a+b x^3\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{4/3} b^{5/3}}-\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{4/3} b^{5/3}}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{4/3} b^{5/3}}\\ \end{align*}
Mathematica [A] time = 0.0991302, size = 141, normalized size = 0.89 \[ \frac{\frac{6 b^{2/3} x^2}{a^2+a b x^3}+\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}-\frac{2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{a^{4/3}}-\frac{9 b^{2/3} x^2}{\left (a+b x^3\right )^2}}{54 b^{5/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 127, normalized size = 0.8 \begin{align*}{\frac{1}{ \left ( b{x}^{3}+a \right ) ^{2}} \left ({\frac{{x}^{5}}{9\,a}}-{\frac{{x}^{2}}{18\,b}} \right ) }-{\frac{1}{27\,{b}^{2}a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{1}{54\,{b}^{2}a}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{\sqrt{3}}{27\,{b}^{2}a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}} \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 = 1.58234, size = 1152, normalized size = 7.29 \begin{align*} \left [\frac{6 \, a b^{3} x^{5} - 3 \, a^{2} b^{2} x^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{3} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x}{b x^{3} + a}\right ) +{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{54 \,{\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}, \frac{6 \, a b^{3} x^{5} - 3 \, a^{2} b^{2} x^{2} + 6 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{3} + a^{3} b\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) +{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \,{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )} \left (-a b^{2}\right )^{\frac{2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{54 \,{\left (a^{2} b^{5} x^{6} + 2 \, a^{3} b^{4} x^{3} + a^{4} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.619296, size = 70, normalized size = 0.44 \begin{align*} \frac{- a x^{2} + 2 b x^{5}}{18 a^{3} b + 36 a^{2} b^{2} x^{3} + 18 a b^{3} x^{6}} + \operatorname{RootSum}{\left (19683 t^{3} a^{4} b^{5} + 1, \left ( t \mapsto t \log{\left (729 t^{2} a^{3} b^{3} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15351, size = 196, normalized size = 1.24 \begin{align*} -\frac{\left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{27 \, a^{2} b} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{27 \, a^{2} b^{3}} + \frac{2 \, b x^{5} - a x^{2}}{18 \,{\left (b x^{3} + a\right )}^{2} a b} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{54 \, a^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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