Optimal. Leaf size=19 \[ \frac {x^4}{4 b \left (a x^2+b\right )^2} \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1593, 264} \[ \frac {x^4}{4 b \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 264
Rule 1593
Rubi steps
\begin {align*} \int \frac {1}{\left (\frac {b}{x}+a x\right )^3} \, dx &=\int \frac {x^3}{\left (b+a x^2\right )^3} \, dx\\ &=\frac {x^4}{4 b \left (b+a x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 1.26 \[ -\frac {2 a x^2+b}{4 a^2 \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.37, size = 36, normalized size = 1.89 \[ -\frac {2 \, a x^{2} + b}{4 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 22, normalized size = 1.16 \[ -\frac {2 \, a x^{2} + b}{4 \, {\left (a x^{2} + b\right )}^{2} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 31, normalized size = 1.63 \[ \frac {b}{4 \left (a \,x^{2}+b \right )^{2} a^{2}}-\frac {1}{2 \left (a \,x^{2}+b \right ) a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, size = 36, normalized size = 1.89 \[ -\frac {2 \, a x^{2} + b}{4 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 37, normalized size = 1.95 \[ -\frac {\frac {b}{4\,a^2}+\frac {x^2}{2\,a}}{a^2\,x^4+2\,a\,b\,x^2+b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.28, size = 36, normalized size = 1.89 \[ \frac {- 2 a x^{2} - b}{4 a^{4} x^{4} + 8 a^{3} b x^{2} + 4 a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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