Optimal. Leaf size=182 \[ \frac{a \log \left (c+d x^3\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}-\frac{a \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} \sqrt [3]{b c-a d}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{3 \sqrt{3} c^{5/3} \sqrt [3]{b c-a d}}+\frac{x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )} \]
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Rubi [A] time = 0.210115, antiderivative size = 241, normalized size of antiderivative = 1.32, number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {378, 377, 200, 31, 634, 617, 204, 628} \[ -\frac{2 a \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}+\frac{a \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}+\frac{2 a \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{3 \sqrt{3} c^{5/3} \sqrt [3]{b c-a d}}+\frac{x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )} \]
Antiderivative was successfully verified.
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Rule 378
Rule 377
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^2} \, dx &=\frac{x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac{(2 a) \int \frac{1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{3 c}\\ &=\frac{x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 c}\\ &=\frac{x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{c}-\sqrt [3]{b c-a d} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3}}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{2 \sqrt [3]{c}+\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3}}\\ &=\frac{x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}-\frac{2 a \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{3 c^{4/3}}+\frac{a \operatorname{Subst}\left (\int \frac{\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}\\ &=\frac{x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}-\frac{2 a \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}+\frac{a \log \left (c^{2/3}+\frac{(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{3 c^{5/3} \sqrt [3]{b c-a d}}\\ &=\frac{x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}+\frac{2 a \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3} c^{5/3} \sqrt [3]{b c-a d}}-\frac{2 a \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}+\frac{a \log \left (c^{2/3}+\frac{(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{9 c^{5/3} \sqrt [3]{b c-a d}}\\ \end{align*}
Mathematica [C] time = 0.0295703, size = 78, normalized size = 0.43 \[ \frac{x \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{2}{3},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{c^2 \left (\frac{b x^3}{a}+1\right )^{2/3} \sqrt [3]{\frac{d x^3}{c}+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.401, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{2}} \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{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{\left (c + d x^{3}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{{\left (d x^{3} + c\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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