Optimal. Leaf size=189 \[ -\frac{b}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 b \log (x) \left (a+b x^2\right )}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.095797, antiderivative size = 189, 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, 44} \[ -\frac{b}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 b \log (x) \left (a+b x^2\right )}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{x^3 \left (a b+b^2 x^2\right )^3} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a b+b^2 x\right )^3} \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^3 b^3 x^2}-\frac{3}{a^4 b^2 x}+\frac{1}{a^2 b (a+b x)^3}+\frac{2}{a^3 b (a+b x)^2}+\frac{3}{a^4 b (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{b}{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b}{4 a^2 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a+b x^2}{2 a^3 x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 b \left (a+b x^2\right ) \log (x)}{a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{3 b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^4 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0386016, size = 97, normalized size = 0.51 \[ \frac{-a \left (2 a^2+9 a b x^2+6 b^2 x^4\right )-12 b x^2 \log (x) \left (a+b x^2\right )^2+6 b x^2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )}{4 a^4 x^2 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.223, size = 133, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 12\,{b}^{3}\ln \left ( x \right ){x}^{6}-6\,\ln \left ( b{x}^{2}+a \right ){x}^{6}{b}^{3}+24\,{b}^{2}a\ln \left ( x \right ){x}^{4}-12\,\ln \left ( b{x}^{2}+a \right ){x}^{4}a{b}^{2}+6\,a{x}^{4}{b}^{2}+12\,b{a}^{2}\ln \left ( x \right ){x}^{2}-6\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{2}b+9\,{a}^{2}b{x}^{2}+2\,{a}^{3} \right ) \left ( b{x}^{2}+a \right ) }{4\,{x}^{2}{a}^{4}} \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 = 1.29989, size = 247, normalized size = 1.31 \begin{align*} -\frac{6 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3} - 6 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 12 \,{\left (b^{3} x^{6} + 2 \, a b^{2} x^{4} + a^{2} b x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \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{1}{x^{3} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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