Optimal. Leaf size=245 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} c}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} c}-\frac{\log (x)}{2 a^{2/3} c}+\frac{d^{2/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{2/3}}-\frac{d^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{2/3}}+\frac{d^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c (b c-a d)^{2/3}} \]
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Rubi [A] time = 0.210522, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {446, 86, 57, 617, 204, 31, 58} \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} c}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} c}-\frac{\log (x)}{2 a^{2/3} c}+\frac{d^{2/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{2/3}}-\frac{d^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{2/3}}+\frac{d^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c (b c-a d)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 446
Rule 86
Rule 57
Rule 617
Rule 204
Rule 31
Rule 58
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{3 c}-\frac{d \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{2/3} (c+d x)} \, dx,x,x^3\right )}{3 c}\\ &=-\frac{\log (x)}{2 a^{2/3} c}+\frac{d^{2/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{2/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} c}-\frac{\operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a} c}-\frac{d^{2/3} \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{2/3}}-\frac{\sqrt [3]{d} \operatorname{Subst}\left (\int \frac{1}{\frac{(b c-a d)^{2/3}}{d^{2/3}}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 c \sqrt [3]{b c-a d}}\\ &=-\frac{\log (x)}{2 a^{2/3} c}+\frac{d^{2/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{2/3}}+\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} c}-\frac{d^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{a^{2/3} c}-\frac{d^{2/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{c (b c-a d)^{2/3}}\\ &=-\frac{\tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{2/3} c}+\frac{d^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c (b c-a d)^{2/3}}-\frac{\log (x)}{2 a^{2/3} c}+\frac{d^{2/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{2/3}}+\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} c}-\frac{d^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.278001, size = 308, normalized size = 1.26 \[ -\frac{-\frac{2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{a^{2/3}}+\frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{a^{2/3}}+\frac{2 d^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{2/3}}-\frac{d^{2/3} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{2/3}}+\frac{2 \sqrt{3} d^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt{3}}\right )}{(b c-a d)^{2/3}}}{6 c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \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{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x^{3} + c\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50184, size = 1115, normalized size = 4.55 \begin{align*} -\frac{2 \, \sqrt{3} a^{2} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b c - a d\right )} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{2}{3}} - \sqrt{3} d}{3 \, d}\right ) + a^{2} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} d^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b c d - a d^{2}\right )} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, a^{2} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} d -{\left (b c - a d\right )} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{6}} a \arctan \left (\frac{{\left (a^{2}\right )}^{\frac{1}{6}}{\left (\sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} a + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right )}}{3 \, a^{2}}\right ) +{\left (a^{2}\right )}^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 2 \,{\left (a^{2}\right )}^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right )}{6 \, a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a + b x^{3}\right )^{\frac{2}{3}} \left (c + d x^{3}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.61581, size = 462, normalized size = 1.89 \begin{align*} \frac{1}{6} \,{\left (\frac{2 \, d \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \right |}\right )}{b^{2} c^{2} - a b c d} - \frac{6 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{2} c^{2} - \sqrt{3} a b c d} - \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{2}{3}}\right )}{b^{2} c^{2} - a b c d} - \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{2}{3}} b c} - \frac{\log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{2}{3}} b c} + \frac{2 \, \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}} b c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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