Optimal. Leaf size=59 \[ \frac{2 a^2}{3 b^3 \sqrt{a+\frac{b}{x^3}}}+\frac{4 a \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]
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Rubi [A] time = 0.0309743, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2 a^2}{3 b^3 \sqrt{a+\frac{b}{x^3}}}+\frac{4 a \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x^{10}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=\frac{2 a^2}{3 b^3 \sqrt{a+\frac{b}{x^3}}}+\frac{4 a \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.0145023, size = 42, normalized size = 0.71 \[ \frac{2 \left (8 a^2 x^6+4 a b x^3-b^2\right )}{9 b^3 x^6 \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 50, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}+4\,{x}^{3}ab-{b}^{2} \right ) }{9\,{b}^{3}{x}^{9}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.986495, size = 63, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b^{3}} + \frac{4 \, \sqrt{a + \frac{b}{x^{3}}} a}{3 \, b^{3}} + \frac{2 \, a^{2}}{3 \, \sqrt{a + \frac{b}{x^{3}}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52856, size = 107, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (8 \, a^{2} x^{6} + 4 \, a b x^{3} - b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{9 \,{\left (a b^{3} x^{6} + b^{4} x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.41568, size = 466, normalized size = 7.9 \begin{align*} \frac{16 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{9} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} + \frac{24 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} + \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{16 a^{5} b^{3} x^{\frac{21}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{32 a^{4} b^{4} x^{\frac{15}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{16 a^{3} b^{5} x^{\frac{9}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} \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{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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