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 = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 191
Rule 1089
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.86, size = 34, normalized size = 1.00 \[ \frac {{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}} x}{a b x^{2} + a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 19, normalized size = 0.56 \[ -\frac {1}{a \sqrt {-\frac {b x^{2} + a}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 33, normalized size = 0.97 \[ \frac {\left (b \,x^{2}+a \right ) x}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {3}{4}} a} \]
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 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.14, size = 34, normalized size = 1.00 \[ \frac {x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{1/4}}{a\,\left (b\,x^2+a\right )} \]
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 \[ \int \frac {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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