Optimal. Leaf size=34 \[ \frac{x \left (a+b x^2\right )}{a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}} \]
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Rubi [A] time = 0.0084038, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1089, 191} \[ \frac{x \left (a+b x^2\right )}{a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 1089
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}} \, dx &=\frac{\left (1+\frac{b x^2}{a}\right )^{3/2} \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/2}} \, dx}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}}\\ &=\frac{x \left (a+b x^2\right )}{a \left (a^2+2 a b x^2+b^2 x^4\right )^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0110457, size = 25, normalized size = 0.74 \[ \frac{x \left (a+b x^2\right )}{a \left (\left (a+b x^2\right )^2\right )^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 33, normalized size = 1. \begin{align*}{\frac{x \left ( b{x}^{2}+a \right ) }{a} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{-{\frac{3}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28183, size = 72, normalized size = 2.12 \begin{align*} \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}} x}{a b x^{2} + a^{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{1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{4}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16254, size = 32, normalized size = 0.94 \begin{align*} -\frac{x^{2}}{\sqrt{-b x^{2} - a} a{\left | x \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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