Optimal. Leaf size=38 \[ -\frac{2 a}{3 b^2 \sqrt{a+\frac{b}{x^3}}}-\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b^2} \]
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Rubi [A] time = 0.0209723, 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} \[ -\frac{2 a}{3 b^2 \sqrt{a+\frac{b}{x^3}}}-\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b^2} \]
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^7} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{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}{b (a+b x)^{3/2}}+\frac{1}{b \sqrt{a+b x}}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\frac{2 a}{3 b^2 \sqrt{a+\frac{b}{x^3}}}-\frac{2 \sqrt{a+\frac{b}{x^3}}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.0101295, size = 29, normalized size = 0.76 \[ -\frac{2 \left (2 a x^3+b\right )}{3 b^2 x^3 \sqrt{a+\frac{b}{x^3}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 1. \begin{align*} -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 2\,a{x}^{3}+b \right ) }{3\,{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 = 0.973699, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \, \sqrt{a + \frac{b}{x^{3}}}}{3 \, b^{2}} - \frac{2 \, a}{3 \, \sqrt{a + \frac{b}{x^{3}}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51538, size = 81, normalized size = 2.13 \begin{align*} -\frac{2 \,{\left (2 \, a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{3 \,{\left (a b^{2} x^{3} + b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.06826, size = 51, normalized size = 1.34 \begin{align*} \begin{cases} - \frac{4 a}{3 b^{2} \sqrt{a + \frac{b}{x^{3}}}} - \frac{2}{3 b x^{3} \sqrt{a + \frac{b}{x^{3}}}} & \text{for}\: b \neq 0 \\- \frac{1}{6 a^{\frac{3}{2}} x^{6}} & \text{otherwise} \end{cases} \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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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