Optimal. Leaf size=59 \[ -\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]
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Rubi [A] time = 0.030736, 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} \[ -\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^{10}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x}} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt{a+b x}}-\frac{2 a \sqrt{a+b x}}{b^2}+\frac{(a+b x)^{3/2}}{b^2}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{3 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}\\ \end{align*}
Mathematica [A] time = 0.0173782, size = 42, normalized size = 0.71 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}} \left (8 a^2 x^6-4 a b x^3+3 b^2\right )}{45 b^3 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, 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+3\,{b}^{2} \right ) }{45\,{b}^{3}{x}^{9}}{\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 = 0.952835, size = 63, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a}{9 \, b^{3}} - \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a^{2}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46216, size = 96, normalized size = 1.63 \begin{align*} -\frac{2 \,{\left (8 \, a^{2} x^{6} - 4 \, a b x^{3} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{45 \, b^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.41594, size = 824, normalized size = 13.97 \begin{align*} \text{result too large to display} \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^{10}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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