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 = 0.02, 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 191
Rule 192
Rule 1089
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 1.14, size = 58, normalized size = 0.85 \[ \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}} {\left (2 \, b x^{3} + 3 \, a x\right )}}{3 \, {\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 44, normalized size = 0.65 \[ \frac {\left (b \,x^{2}+a \right ) \left (2 b \,x^{2}+3 a \right ) x}{3 \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {5}{4}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 45, normalized size = 0.66 \[ \frac {x\,\left (2\,b\,x^2+3\,a\right )\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/4}}{3\,a^2\,{\left (b\,x^2+a\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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