Optimal. Leaf size=44 \[ \frac{2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9} \]
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Rubi [A] time = 0.0105772, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {271, 264} \[ \frac{2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{5/2}}{x^{10}} \, dx &=-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9}-\frac{(2 b) \int \frac{\left (a+b x^2\right )^{5/2}}{x^8} \, dx}{9 a}\\ &=-\frac{\left (a+b x^2\right )^{7/2}}{9 a x^9}+\frac{2 b \left (a+b x^2\right )^{7/2}}{63 a^2 x^7}\\ \end{align*}
Mathematica [A] time = 0.0106238, size = 31, normalized size = 0.7 \[ \frac{\left (a+b x^2\right )^{7/2} \left (2 b x^2-7 a\right )}{63 a^2 x^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 28, normalized size = 0.6 \begin{align*} -{\frac{-2\,b{x}^{2}+7\,a}{63\,{x}^{9}{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60237, size = 130, normalized size = 2.95 \begin{align*} \frac{{\left (2 \, b^{4} x^{8} - a b^{3} x^{6} - 15 \, a^{2} b^{2} x^{4} - 19 \, a^{3} b x^{2} - 7 \, a^{4}\right )} \sqrt{b x^{2} + a}}{63 \, a^{2} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.16759, size = 121, normalized size = 2.75 \begin{align*} - \frac{a^{2} \sqrt{b} \sqrt{\frac{a}{b x^{2}} + 1}}{9 x^{8}} - \frac{19 a b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 x^{6}} - \frac{5 b^{\frac{5}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{21 x^{4}} - \frac{b^{\frac{7}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 a x^{2}} + \frac{2 b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{2}} + 1}}{63 a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.69998, size = 297, normalized size = 6.75 \begin{align*} \frac{4 \,{\left (63 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{14} b^{\frac{9}{2}} + 105 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{12} a b^{\frac{9}{2}} + 315 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{10} a^{2} b^{\frac{9}{2}} + 189 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{8} a^{3} b^{\frac{9}{2}} + 189 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{6} a^{4} b^{\frac{9}{2}} + 27 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{4} a^{5} b^{\frac{9}{2}} + 9 \,{\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} a^{6} b^{\frac{9}{2}} - a^{7} b^{\frac{9}{2}}\right )}}{63 \,{\left ({\left (\sqrt{b} x - \sqrt{b x^{2} + a}\right )}^{2} - a\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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