Optimal. Leaf size=38 \[ \frac{2 a \sqrt{a+\frac{b}{x^3}}}{3 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2} \]
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Rubi [A] time = 0.0204016, 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} \[ \frac{2 a \sqrt{a+\frac{b}{x^3}}}{3 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^7} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b x}} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=\frac{2 a \sqrt{a+\frac{b}{x^3}}}{3 b^2}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.0152488, size = 31, normalized size = 0.82 \[ \frac{2 \sqrt{a+\frac{b}{x^3}} \left (2 a x^3-b\right )}{9 b^2 x^3} \]
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}-b \right ) }{9\,{b}^{2}{x}^{6}}{\frac{1}{\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 = 0.952738, size = 41, normalized size = 1.08 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b^{2}} + \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50814, size = 69, normalized size = 1.82 \begin{align*} \frac{2 \,{\left (2 \, a x^{3} - b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{9 \, b^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.05548, size = 255, normalized size = 6.71 \begin{align*} \frac{4 a^{\frac{7}{2}} b^{\frac{3}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} + \frac{2 a^{\frac{5}{2}} b^{\frac{5}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{7}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} - \frac{4 a^{4} b x^{\frac{15}{2}}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} x^{\frac{9}{2}}} - \frac{4 a^{3} b^{2} x^{\frac{9}{2}}}{9 a^{\frac{5}{2}} b^{3} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{4} 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}{\sqrt{a + \frac{b}{x^{3}}} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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