Integrand size = 17, antiderivative size = 86 \[ \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac {3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}-\frac {6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4}+\frac {\left (a+b x^{3/2}\right )^{10/3}}{5 b^4} \]
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Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45} \[ \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac {3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}+\frac {\left (a+b x^{3/2}\right )^{10/3}}{5 b^4}-\frac {6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \text {Subst}\left (\int \frac {x^3}{(a+b x)^{2/3}} \, dx,x,x^{3/2}\right ) \\ & = \frac {2}{3} \text {Subst}\left (\int \left (-\frac {a^3}{b^3 (a+b x)^{2/3}}+\frac {3 a^2 \sqrt [3]{a+b x}}{b^3}-\frac {3 a (a+b x)^{4/3}}{b^3}+\frac {(a+b x)^{7/3}}{b^3}\right ) \, dx,x,x^{3/2}\right ) \\ & = -\frac {2 a^3 \sqrt [3]{a+b x^{3/2}}}{b^4}+\frac {3 a^2 \left (a+b x^{3/2}\right )^{4/3}}{2 b^4}-\frac {6 a \left (a+b x^{3/2}\right )^{7/3}}{7 b^4}+\frac {\left (a+b x^{3/2}\right )^{10/3}}{5 b^4} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.65 \[ \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {\sqrt [3]{a+b x^{3/2}} \left (-81 a^3+27 a^2 b x^{3/2}-18 a b^2 x^3+14 b^3 x^{9/2}\right )}{70 b^4} \]
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\[\int \frac {x^{5}}{\left (a +b \,x^{\frac {3}{2}}\right )^{\frac {2}{3}}}d x\]
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none
Time = 0.48 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.58 \[ \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {{\left (18 \, a b^{2} x^{3} + 81 \, a^{3} - {\left (14 \, b^{3} x^{4} + 27 \, a^{2} b x\right )} \sqrt {x}\right )} {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{70 \, b^{4}} \]
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Time = 2.39 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.19 \[ \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\begin {cases} - \frac {81 a^{3} \sqrt [3]{a + b x^{\frac {3}{2}}}}{70 b^{4}} + \frac {27 a^{2} x^{\frac {3}{2}} \sqrt [3]{a + b x^{\frac {3}{2}}}}{70 b^{3}} - \frac {9 a x^{3} \sqrt [3]{a + b x^{\frac {3}{2}}}}{35 b^{2}} + \frac {x^{\frac {9}{2}} \sqrt [3]{a + b x^{\frac {3}{2}}}}{5 b} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 a^{\frac {2}{3}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {10}{3}}}{5 \, b^{4}} - \frac {6 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {7}{3}} a}{7 \, b^{4}} + \frac {3 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} a^{2}}{2 \, b^{4}} - \frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{3}}{b^{4}} \]
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Time = 0.32 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.71 \[ \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=-\frac {2 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} a^{3}}{b^{4}} + \frac {14 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {10}{3}} - 60 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {7}{3}} a + 105 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}} a^{2}}{70 \, b^{4}} \]
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Time = 5.85 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.74 \[ \int \frac {x^5}{\left (a+b x^{3/2}\right )^{2/3}} \, dx=\frac {{\left (a+b\,x^{3/2}\right )}^{10/3}}{5\,b^4}-\frac {6\,a\,{\left (a+b\,x^{3/2}\right )}^{7/3}}{7\,b^4}-\frac {2\,a^3\,{\left (a+b\,x^{3/2}\right )}^{1/3}}{b^4}+\frac {3\,a^2\,{\left (a+b\,x^{3/2}\right )}^{4/3}}{2\,b^4} \]
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