Optimal. Leaf size=196 \[ -\frac{a^4}{8 b^5 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a^3}{3 b^5 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a^2}{2 b^5 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a}{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.162137, antiderivative size = 196, 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 = {1111, 646, 43} \[ -\frac{a^4}{8 b^5 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a^3}{3 b^5 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a^2}{2 b^5 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a}{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,x^2\right )\\ &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (a b+b^2 x\right )^5} \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \operatorname{Subst}\left (\int \left (\frac{a^4}{b^9 (a+b x)^5}-\frac{4 a^3}{b^9 (a+b x)^4}+\frac{6 a^2}{b^9 (a+b x)^3}-\frac{4 a}{b^9 (a+b x)^2}+\frac{1}{b^9 (a+b x)}\right ) \, dx,x,x^2\right )}{2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 a}{b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{a^4}{8 b^5 \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{2 a^3}{3 b^5 \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{3 a^2}{2 b^5 \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b^5 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0292026, size = 83, normalized size = 0.42 \[ \frac{a \left (88 a^2 b x^2+25 a^3+108 a b^2 x^4+48 b^3 x^6\right )+12 \left (a+b x^2\right )^4 \log \left (a+b x^2\right )}{24 b^5 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.23, size = 141, normalized size = 0.7 \begin{align*}{\frac{ \left ( 12\,\ln \left ( b{x}^{2}+a \right ){x}^{8}{b}^{4}+48\,\ln \left ( b{x}^{2}+a \right ){x}^{6}a{b}^{3}+48\,a{b}^{3}{x}^{6}+72\,\ln \left ( b{x}^{2}+a \right ){x}^{4}{a}^{2}{b}^{2}+108\,{a}^{2}{b}^{2}{x}^{4}+48\,\ln \left ( b{x}^{2}+a \right ){x}^{2}{a}^{3}b+88\,{a}^{3}b{x}^{2}+12\,\ln \left ( b{x}^{2}+a \right ){a}^{4}+25\,{a}^{4} \right ) \left ( b{x}^{2}+a \right ) }{24\,{b}^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02962, size = 134, normalized size = 0.68 \begin{align*} \frac{48 \, a b^{3} x^{6} + 108 \, a^{2} b^{2} x^{4} + 88 \, a^{3} b x^{2} + 25 \, a^{4}}{24 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26654, size = 282, normalized size = 1.44 \begin{align*} \frac{48 \, a b^{3} x^{6} + 108 \, a^{2} b^{2} x^{4} + 88 \, a^{3} b x^{2} + 25 \, a^{4} + 12 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{24 \,{\left (b^{9} x^{8} + 4 \, a b^{8} x^{6} + 6 \, a^{2} b^{7} x^{4} + 4 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\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{x^{9}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{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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