Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^2} \]
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Rubi [A] time = 0.0226198, antiderivative size = 38, 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} \[ \frac{2 a \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^3}\right )^{3/2}}{x^7} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^{3/2} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^{3/2}}{b}+\frac{(a+b x)^{5/2}}{b}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=\frac{2 a \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{7/2}}{21 b^2}\\ \end{align*}
Mathematica [A] time = 0.011973, size = 40, normalized size = 1.05 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right )^2 \left (2 a x^3-5 b\right )}{105 b^2 x^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 39, normalized size = 1. \begin{align*}{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 2\,a{x}^{3}-5\,b \right ) }{105\,{b}^{2}{x}^{6}} \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 = 0.955871, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}}}{21 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a}{15 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48881, size = 115, normalized size = 3.03 \begin{align*} \frac{2 \,{\left (2 \, a^{3} x^{9} - a^{2} b x^{6} - 8 \, a b^{2} x^{3} - 5 \, b^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{105 \, b^{2} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.33394, size = 371, normalized size = 9.76 \begin{align*} \frac{4 a^{\frac{15}{2}} b^{\frac{3}{2}} x^{12} \sqrt{\frac{a x^{3}}{b} + 1}}{105 a^{\frac{9}{2}} b^{3} x^{\frac{27}{2}} + 105 a^{\frac{7}{2}} b^{4} x^{\frac{21}{2}}} + \frac{2 a^{\frac{13}{2}} b^{\frac{5}{2}} x^{9} \sqrt{\frac{a x^{3}}{b} + 1}}{105 a^{\frac{9}{2}} b^{3} x^{\frac{27}{2}} + 105 a^{\frac{7}{2}} b^{4} x^{\frac{21}{2}}} - \frac{18 a^{\frac{11}{2}} b^{\frac{7}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{105 a^{\frac{9}{2}} b^{3} x^{\frac{27}{2}} + 105 a^{\frac{7}{2}} b^{4} x^{\frac{21}{2}}} - \frac{26 a^{\frac{9}{2}} b^{\frac{9}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{105 a^{\frac{9}{2}} b^{3} x^{\frac{27}{2}} + 105 a^{\frac{7}{2}} b^{4} x^{\frac{21}{2}}} - \frac{10 a^{\frac{7}{2}} b^{\frac{11}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{105 a^{\frac{9}{2}} b^{3} x^{\frac{27}{2}} + 105 a^{\frac{7}{2}} b^{4} x^{\frac{21}{2}}} - \frac{4 a^{8} b x^{\frac{27}{2}}}{105 a^{\frac{9}{2}} b^{3} x^{\frac{27}{2}} + 105 a^{\frac{7}{2}} b^{4} x^{\frac{21}{2}}} - \frac{4 a^{7} b^{2} x^{\frac{21}{2}}}{105 a^{\frac{9}{2}} b^{3} x^{\frac{27}{2}} + 105 a^{\frac{7}{2}} b^{4} x^{\frac{21}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21181, size = 105, normalized size = 2.76 \begin{align*} -\frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} - 5 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a\right )} a}{b} + \frac{15 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{7}{2}} - 42 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}} a + 35 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a^{2}}{b}\right )}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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