Optimal. Leaf size=271 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{4/3} c}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} c}-\frac{\log (x)}{2 a^{4/3} c}+\frac{d^{4/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{4/3}}-\frac{d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{4/3}}-\frac{d^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c (b c-a d)^{4/3}}+\frac{b}{a \sqrt [3]{a+b x^3} (b c-a d)} \]
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Rubi [A] time = 0.316329, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {446, 85, 156, 55, 617, 204, 31, 56} \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{4/3} c}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{4/3} c}-\frac{\log (x)}{2 a^{4/3} c}+\frac{d^{4/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{4/3}}-\frac{d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{4/3}}-\frac{d^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c (b c-a d)^{4/3}}+\frac{b}{a \sqrt [3]{a+b x^3} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 446
Rule 85
Rule 156
Rule 55
Rule 617
Rule 204
Rule 31
Rule 56
Rubi steps
\begin{align*} \int \frac{1}{x \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{4/3} (c+d x)} \, dx,x,x^3\right )\\ &=\frac{b}{a (b c-a d) \sqrt [3]{a+b x^3}}-\frac{\operatorname{Subst}\left (\int \frac{-b c+a d-b d x}{x \sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 a (b c-a d)}\\ &=\frac{b}{a (b c-a d) \sqrt [3]{a+b x^3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )}{3 a c}+\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 c (b c-a d)}\\ &=\frac{b}{a (b c-a d) \sqrt [3]{a+b x^3}}-\frac{\log (x)}{2 a^{4/3} c}+\frac{d^{4/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{4/3}}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 a^{4/3} 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 a c}-\frac{d^{4/3} \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)^{4/3}}+\frac{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 (b c-a d)}\\ &=\frac{b}{a (b c-a d) \sqrt [3]{a+b x^3}}-\frac{\log (x)}{2 a^{4/3} c}+\frac{d^{4/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{4/3}}+\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{4/3} c}-\frac{d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{4/3}}-\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^{4/3} c}+\frac{d^{4/3} \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)^{4/3}}\\ &=\frac{b}{a (b c-a d) \sqrt [3]{a+b x^3}}+\frac{\tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{4/3} c}-\frac{d^{4/3} \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)^{4/3}}-\frac{\log (x)}{2 a^{4/3} c}+\frac{d^{4/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{4/3}}+\frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{4/3} c}-\frac{d^{4/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0296921, size = 86, normalized size = 0.32 \[ \frac{a d \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{d \left (b x^3+a\right )}{a d-b c}\right )+(b c-a d) \, _2F_1\left (-\frac{1}{3},1;\frac{2}{3};\frac{b x^3}{a}+1\right )}{a c \sqrt [3]{a+b x^3} (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{-{\frac{4}{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{4}{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 = 1.66092, size = 2310, normalized size = 8.52 \begin{align*} \text{result too large to display} \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{4}{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 = 2.10361, size = 552, normalized size = 2.04 \begin{align*} -\frac{1}{6} \,{\left (\frac{2 \, d^{2} \left (-\frac{b c - a d}{d}\right )^{\frac{2}{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^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2}} + \frac{6 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{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^{3} c^{3} - 2 \, \sqrt{3} a b^{2} c^{2} d + \sqrt{3} a^{2} b c d^{2}} - \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{2}{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^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2}} - \frac{6}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a b c - a^{2} d\right )}} - \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{4}{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{4}{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{4}{3}} b c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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