Optimal. Leaf size=69 \[ \frac{a}{8 b^2 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}-\frac{1}{6 b^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]
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Rubi [A] time = 0.0537152, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1111, 640, 607} \[ \frac{a}{8 b^2 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}-\frac{1}{6 b^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1111
Rule 640
Rule 607
Rubi steps
\begin{align*} \int \frac{x^3}{\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}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,x^2\right )\\ &=-\frac{1}{6 b^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx,x,x^2\right )}{2 b}\\ &=-\frac{1}{6 b^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{a}{8 b^2 \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0136127, size = 39, normalized size = 0.57 \[ \frac{-a-4 b x^2}{24 b^2 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.167, size = 32, normalized size = 0.5 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 4\,b{x}^{2}+a \right ) }{24\,{b}^{2}} \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 = 0.963772, size = 65, normalized size = 0.94 \begin{align*} -\frac{1}{6 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{3}{2}} b^{2}} + \frac{a}{8 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{4} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22864, size = 117, normalized size = 1.7 \begin{align*} -\frac{4 \, b x^{2} + a}{24 \,{\left (b^{6} x^{8} + 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{2} + a^{4} b^{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{x^{3}}{\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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