Optimal. Leaf size=217 \[ \frac{5 a^2 \log \left (c+d x^3\right )}{54 c^{8/3} \sqrt [3]{b c-a d}}-\frac{5 a^2 \log \left (\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} \sqrt [3]{b c-a d}}+\frac{5 a^2 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} c^{8/3} \sqrt [3]{b c-a d}}+\frac{5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2} \]
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Rubi [A] time = 0.238557, antiderivative size = 276, normalized size of antiderivative = 1.27, number of steps used = 9, 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} \[ -\frac{5 a^2 \log \left (\sqrt [3]{c}-\frac{x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} \sqrt [3]{b c-a d}}+\frac{5 a^2 \log \left (\frac{x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{54 c^{8/3} \sqrt [3]{b c-a d}}+\frac{5 a^2 \tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt{3} \sqrt [3]{c}}\right )}{9 \sqrt{3} c^{8/3} \sqrt [3]{b c-a d}}+\frac{5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2} \]
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 )^{5/3}}{\left (c+d x^3\right )^3} \, dx &=\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac{(5 a) \int \frac{\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^2} \, dx}{6 c}\\ &=\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac{5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac{\left (5 a^2\right ) \int \frac{1}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx}{9 c^2}\\ &=\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac{5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{c-(b c-a d) x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 c^2}\\ &=\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac{5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac{\left (5 a^2\right ) \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 )}{27 c^{8/3}}+\frac{\left (5 a^2\right ) \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 )}{27 c^{8/3}}\\ &=\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac{5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}-\frac{5 a^2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} \sqrt [3]{b c-a d}}+\frac{\left (5 a^2\right ) \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 )}{18 c^{7/3}}+\frac{\left (5 a^2\right ) \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 )}{54 c^{8/3} \sqrt [3]{b c-a d}}\\ &=\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac{5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}-\frac{5 a^2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} \sqrt [3]{b c-a d}}+\frac{5 a^2 \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 )}{54 c^{8/3} \sqrt [3]{b c-a d}}-\frac{\left (5 a^2\right ) \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 )}{9 c^{8/3} \sqrt [3]{b c-a d}}\\ &=\frac{x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2}+\frac{5 a x \left (a+b x^3\right )^{2/3}}{18 c^2 \left (c+d x^3\right )}+\frac{5 a^2 \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 )}{9 \sqrt{3} c^{8/3} \sqrt [3]{b c-a d}}-\frac{5 a^2 \log \left (\sqrt [3]{c}-\frac{\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{27 c^{8/3} \sqrt [3]{b c-a d}}+\frac{5 a^2 \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 )}{54 c^{8/3} \sqrt [3]{b c-a d}}\\ \end{align*}
Mathematica [C] time = 0.0263331, size = 79, normalized size = 0.36 \[ \frac{a x \left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{5}{3},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{c^3 \left (\frac{b x^3}{a}+1\right )^{2/3} \sqrt [3]{\frac{d x^3}{c}+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d{x}^{3}+c \right ) ^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{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{5}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{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(-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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{3} + a\right )}^{\frac{5}{3}}}{{\left (d x^{3} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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