Optimal. Leaf size=123 \[ -\frac{5 a^2 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{8/3}}-\frac{5 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3}}-\frac{5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac{x^5 \sqrt [3]{a+b x^3}}{6 b} \]
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Rubi [A] time = 0.091355, antiderivative size = 176, normalized size of antiderivative = 1.43, number of steps used = 9, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533, Rules used = {321, 331, 292, 31, 634, 617, 204, 628} \[ -\frac{5 a^2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac{5 a^2 \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{54 b^{8/3}}-\frac{5 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3}}-\frac{5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac{x^5 \sqrt [3]{a+b x^3}}{6 b} \]
Antiderivative was successfully verified.
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Rule 321
Rule 331
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^7}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac{x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac{(5 a) \int \frac{x^4}{\left (a+b x^3\right )^{2/3}} \, dx}{6 b}\\ &=-\frac{5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac{x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac{\left (5 a^2\right ) \int \frac{x}{\left (a+b x^3\right )^{2/3}} \, dx}{9 b^2}\\ &=-\frac{5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac{x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-b x^3} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{9 b^2}\\ &=-\frac{5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac{x^5 \sqrt [3]{a+b x^3}}{6 b}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{b} x} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{7/3}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{7/3}}\\ &=-\frac{5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac{x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac{5 a^2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}-\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{a+b x^3}}\right )}{18 b^{7/3}}\\ &=-\frac{5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac{x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac{5 a^2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac{5 a^2 \log \left (1+\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}+\frac{\left (5 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{8/3}}\\ &=-\frac{5 a x^2 \sqrt [3]{a+b x^3}}{18 b^2}+\frac{x^5 \sqrt [3]{a+b x^3}}{6 b}-\frac{5 a^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3}}-\frac{5 a^2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{27 b^{8/3}}+\frac{5 a^2 \log \left (1+\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{54 b^{8/3}}\\ \end{align*}
Mathematica [C] time = 0.0161577, size = 69, normalized size = 0.56 \[ \frac{x^2 \left (5 a^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{b x^3}{b x^3+a}\right )-5 a^2-2 a b x^3+3 b^2 x^6\right )}{18 b^2 \left (a+b x^3\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{{x}^{7} \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(-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 [B] time = 1.54964, size = 521, normalized size = 4.24 \begin{align*} \frac{10 \, \sqrt{3} a^{2} b \sqrt{-\left (-b^{2}\right )^{\frac{1}{3}}} \arctan \left (-\frac{{\left (\sqrt{3} \left (-b^{2}\right )^{\frac{1}{3}} b x - 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{2}{3}}\right )} \sqrt{-\left (-b^{2}\right )^{\frac{1}{3}}}}{3 \, b^{2} x}\right ) - 10 \, \left (-b^{2}\right )^{\frac{2}{3}} a^{2} \log \left (-\frac{\left (-b^{2}\right )^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{1}{3}} b}{x}\right ) + 5 \, \left (-b^{2}\right )^{\frac{2}{3}} a^{2} \log \left (-\frac{\left (-b^{2}\right )^{\frac{1}{3}} b x^{2} -{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-b^{2}\right )^{\frac{2}{3}} x -{\left (b x^{3} + a\right )}^{\frac{2}{3}} b}{x^{2}}\right ) + 3 \,{\left (3 \, b^{3} x^{5} - 5 \, a b^{2} x^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{54 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.13601, size = 37, normalized size = 0.3 \begin{align*} \frac{x^{8} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{11}{3}\right )} \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{x^{7}}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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