Optimal. Leaf size=68 \[ \frac{2 x}{3 a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{x \left (a+b x^2\right )}{3 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \]
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Rubi [A] time = 0.0161971, antiderivative size = 70, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1089, 192, 191} \[ \frac{x}{3 a \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{2 x}{3 a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1089
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \, dx &=\frac{\sqrt{1+\frac{b x^2}{a}} \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/2}} \, dx}{a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{x}{3 a \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (2 \sqrt{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/2}} \, dx}{3 a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{2 x}{3 a^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{3 a \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.012402, size = 40, normalized size = 0.59 \[ \frac{x \left (3 a+2 b x^2\right )}{3 a^2 \left (a+b x^2\right ) \sqrt [4]{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 44, normalized size = 0.7 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) x \left ( 2\,b{x}^{2}+3\,a \right ) }{3\,{a}^{2}} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{-{\frac{5}{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{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28432, size = 123, normalized size = 1.81 \begin{align*} \frac{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}{\left (2 \, b x^{3} + 3 \, a x\right )}}{3 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\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}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{4}}}\, 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*} \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{5}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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