Optimal. Leaf size=163 \[ -\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0432572, antiderivative size = 163, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 x^7 \left (a+b x^2\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )} \]
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^8} \, 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^8} \, 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^8}+\frac{3 a^2 b^4}{x^6}+\frac{3 a b^5}{x^4}+\frac{b^6}{x^2}\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}}{7 x^7 \left (a+b x^2\right )}-\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4}}{5 x^5 \left (a+b x^2\right )}-\frac{a b^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x^3 \left (a+b x^2\right )}-\frac{b^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{x \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0125719, size = 61, normalized size = 0.37 \[ -\frac{\sqrt{\left (a+b x^2\right )^2} \left (21 a^2 b x^2+5 a^3+35 a b^2 x^4+35 b^3 x^6\right )}{35 x^7 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.166, size = 58, normalized size = 0.4 \begin{align*} -{\frac{35\,{b}^{3}{x}^{6}+35\,a{x}^{4}{b}^{2}+21\,{a}^{2}b{x}^{2}+5\,{a}^{3}}{35\,{x}^{7} \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 = 1.20409, size = 50, normalized size = 0.31 \begin{align*} -\frac{35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45199, size = 84, normalized size = 0.52 \begin{align*} -\frac{35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \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^{8}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1118, size = 93, normalized size = 0.57 \begin{align*} -\frac{35 \, b^{3} x^{6} \mathrm{sgn}\left (b x^{2} + a\right ) + 35 \, a b^{2} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + 21 \, a^{2} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + 5 \, a^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{35 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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