Optimal. Leaf size=164 \[ \frac{b^3 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
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Rubi [A] time = 0.0470999, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1112, 266, 43} \[ \frac{b^3 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 x^2 \left (a+b x^2\right )}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}{x^3} \, 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^3} \, dx}{b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^3}{x^2} \, dx,x,x^2\right )}{2 b^2 \left (a b+b^2 x^2\right )}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \operatorname{Subst}\left (\int \left (3 a b^5+\frac{a^3 b^3}{x^2}+\frac{3 a^2 b^4}{x}+b^6 x\right ) \, dx,x,x^2\right )}{2 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}}{2 x^2 \left (a+b x^2\right )}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}{2 \left (a+b x^2\right )}+\frac{b^3 x^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}{4 \left (a+b x^2\right )}+\frac{3 a^2 b \sqrt{a^2+2 a b x^2+b^2 x^4} \log (x)}{a+b x^2}\\ \end{align*}
Mathematica [A] time = 0.020755, size = 62, normalized size = 0.38 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (12 a^2 b x^2 \log (x)-2 a^3+6 a b^2 x^4+b^3 x^6\right )}{4 x^2 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.223, size = 59, normalized size = 0.4 \begin{align*}{\frac{{b}^{3}{x}^{6}+6\,a{x}^{4}{b}^{2}+12\,b{a}^{2}\ln \left ( x \right ){x}^{2}-2\,{a}^{3}}{4\, \left ( b{x}^{2}+a \right ) ^{3}{x}^{2}} \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 [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 = 1.4878, size = 85, normalized size = 0.52 \begin{align*} \frac{b^{3} x^{6} + 6 \, a b^{2} x^{4} + 12 \, a^{2} b x^{2} \log \left (x\right ) - 2 \, a^{3}}{4 \, x^{2}} \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^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10966, size = 117, normalized size = 0.71 \begin{align*} \frac{1}{4} \, b^{3} x^{4} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{3}{2} \, a b^{2} x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{3}{2} \, a^{2} b \log \left (x^{2}\right ) \mathrm{sgn}\left (b x^{2} + a\right ) - \frac{3 \, a^{2} b x^{2} \mathrm{sgn}\left (b x^{2} + a\right ) + a^{3} \mathrm{sgn}\left (b x^{2} + a\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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