Optimal. Leaf size=167 \[ -\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0394533, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1112, 270} \[ -\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 270
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^{10}} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \frac{\left (a b+b^2 x^2\right )^3}{x^{10}} \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (\frac{a^3 b^3}{x^{10}}+\frac{3 a^2 b^4}{x^8}+\frac{3 a b^5}{x^6}+\frac{b^6}{x^4}\right ) \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{9 x^9 \left (a+b x^2\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{3 a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 x^3 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0127908, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (135 a^2 b x^2+35 a^3+189 a b^2 x^4+105 b^3 x^6\right )}{315 x^9 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.162, size = 58, normalized size = 0.4 \begin{align*} -{\frac{105\,{b}^{3}{x}^{6}+189\,a{x}^{4}{b}^{2}+135\,{a}^{2}b{x}^{2}+35\,{a}^{3}}{315\,{x}^{9} \left ( b{x}^{2}+a \right ) ^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01958, size = 50, normalized size = 0.3 \begin{align*} -\frac{105 \, b^{3} x^{6} + 189 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} + 35 \, a^{3}}{315 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48877, size = 90, normalized size = 0.54 \begin{align*} -\frac{105 \, b^{3} x^{6} + 189 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} + 35 \, a^{3}}{315 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}{x^{10}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11884, size = 93, normalized size = 0.56 \begin{align*} -\frac{105 \, b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 189 \, a b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 135 \, a^{2} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 35 \, a^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{315 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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