Integrand size = 17, antiderivative size = 139 \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac {x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b}-\frac {10 a^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a+b x^{3/2}}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a^2 \log \left (\sqrt [3]{b} \sqrt {x}-\sqrt [3]{a+b x^{3/2}}\right )}{9 b^{8/3}} \]
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Time = 0.05 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {348, 327, 337} \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {10 a^2 \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {5 a^2 \log \left (\sqrt [3]{b} \sqrt {x}-\sqrt [3]{a+b x^{3/2}}\right )}{9 b^{8/3}}-\frac {5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac {x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b} \]
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Rule 327
Rule 337
Rule 348
Rubi steps \begin{align*} \text {integral}& = 2 \text {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) \text {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 ) \text {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 {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 {5 a^2 \log \left (\sqrt [3]{b} \sqrt {x}-\sqrt [3]{a+b x^{3/2}}\right )}{9 b^{8/3}} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.40 \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\sqrt [3]{a+b x^{3/2}} \left (-5 a x+3 b x^{5/2}\right )}{9 b^2}-\frac {10 a^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt {x}}{\sqrt [3]{b} \sqrt {x}+2 \sqrt [3]{a+b x^{3/2}}}\right )}{9 \sqrt {3} b^{8/3}}-\frac {10 a^2 \log \left (-\sqrt [3]{b} \sqrt {x}+\sqrt [3]{a+b x^{3/2}}\right )}{27 b^{8/3}}+\frac {5 a^2 \log \left (b^{2/3} x+\sqrt [3]{b} \sqrt {x} \sqrt [3]{a+b x^{3/2}}+\left (a+b x^{3/2}\right )^{2/3}\right )}{27 b^{8/3}} \]
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\[\int \frac {x^{3}}{\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}}d x\]
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Timed out. \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 1.84 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.29 \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {2 x^{4} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {b x^{\frac {3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {11}{3}\right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.32 \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {10 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{\sqrt {x}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{27 \, b^{\frac {8}{3}}} + \frac {5 \, a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{\sqrt {x}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}}}{x}\right )}{27 \, b^{\frac {8}{3}}} - \frac {10 \, a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{\sqrt {x}}\right )}{27 \, b^{\frac {8}{3}}} + \frac {\frac {8 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{2} b}{\sqrt {x}} - \frac {5 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} a^{2}}{x^{2}}}{9 \, {\left (b^{4} - \frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )} b^{3}}{x^{\frac {3}{2}}} + \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{2} b^{2}}{x^{3}}\right )}} \]
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\[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\int { \frac {x^{3}}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\int \frac {x^3}{{\left (a+b\,x^{3/2}\right )}^{2/3}} \,d x \]
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