Optimal. Leaf size=139 \[ -\frac{5 a^2 \log \left (\sqrt [3]{b} \sqrt{x}-\sqrt [3]{a+b x^{3/2}}\right )}{9 b^{8/3}}-\frac{10 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3}}-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b} \]
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Rubi [A] time = 0.162139, antiderivative size = 198, normalized size of antiderivative = 1.42, number of steps used = 10, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529, Rules used = {341, 321, 331, 292, 31, 634, 617, 204, 628} \[ -\frac{10 a^2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{8/3}}+\frac{5 a^2 \log \left (\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1\right )}{27 b^{8/3}}-\frac{10 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3}}-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b} \]
Antiderivative was successfully verified.
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Rule 341
Rule 321
Rule 331
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{\left (a+b x^3\right )^{2/3}} \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}-\frac{(5 a) \operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^3\right )^{2/3}} \, dx,x,\sqrt{x}\right )}{3 b}\\ &=-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}+\frac{\left (10 a^2\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a+b x^3\right )^{2/3}} \, dx,x,\sqrt{x}\right )}{9 b^2}\\ &=-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}+\frac{\left (10 a^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-b x^3} \, dx,x,\frac{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{9 b^2}\\ &=-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}+\frac{\left (10 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{b} x} \, dx,x,\frac{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{7/3}}-\frac{\left (10 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{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{7/3}}\\ &=-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}-\frac{10 a^2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\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{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 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{\sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{9 b^{7/3}}\\ &=-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}-\frac{10 a^2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{8/3}}+\frac{5 a^2 \log \left (1+\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{8/3}}+\frac{\left (10 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{9 b^{8/3}}\\ &=-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}-\frac{10 a^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3}}-\frac{10 a^2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{8/3}}+\frac{5 a^2 \log \left (1+\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{8/3}}\\ \end{align*}
Mathematica [C] time = 0.0241633, size = 75, normalized size = 0.54 \[ \frac{x \left (5 a^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};\frac{b x^{3/2}}{b x^{3/2}+a}\right )-5 a^2-2 a b x^{3/2}+3 b^2 x^3\right )}{9 b^2 \left (a+b x^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( a+b{x}^{{\frac{3}{2}}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.90247, size = 41, normalized size = 0.29 \begin{align*} \frac{2 x^{4} \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^{\frac{3}{2}} 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^{3}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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