Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2} \]
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Rubi [A] time = 0.0210704, 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} \[ \frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{x^3}}}{x^7} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int x \sqrt{a+b x} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a \sqrt{a+b x}}{b}+\frac{(a+b x)^{3/2}}{b}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=\frac{2 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^2}\\ \end{align*}
Mathematica [A] time = 0.0104663, size = 38, normalized size = 1. \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (a x^3+b\right ) \left (2 a x^3-3 b\right )}{45 b^2 x^6} \]
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}-3\,b \right ) }{45\,{b}^{2}{x}^{6}}\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956515, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a}{9 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46503, size = 92, normalized size = 2.42 \begin{align*} \frac{2 \,{\left (2 \, a^{2} x^{6} - a b x^{3} - 3 \, b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{45 \, b^{2} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.87429, size = 313, normalized size = 8.24 \begin{align*} \frac{4 a^{\frac{11}{2}} b^{\frac{3}{2}} x^{9} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{5}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{7}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{9}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{4 a^{6} b x^{\frac{21}{2}}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} - \frac{4 a^{5} b^{2} x^{\frac{15}{2}}}{45 a^{\frac{7}{2}} b^{3} x^{\frac{21}{2}} + 45 a^{\frac{5}{2}} b^{4} x^{\frac{15}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19324, size = 39, normalized size = 1.03 \begin{align*} -\frac{2 \,{\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 )}}{45 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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