Optimal. Leaf size=34 \[ \frac {a}{6 b^2 \left (a+b x^2\right )^3}-\frac {1}{4 b^2 \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 43} \begin {gather*} \frac {a}{6 b^2 \left (a+b x^2\right )^3}-\frac {1}{4 b^2 \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {x^3}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{2} b^4 \operatorname {Subst}\left (\int \frac {x}{\left (a b+b^2 x\right )^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^4 \operatorname {Subst}\left (\int \left (-\frac {a}{b^5 (a+b x)^4}+\frac {1}{b^5 (a+b x)^3}\right ) \, dx,x,x^2\right )\\ &=\frac {a}{6 b^2 \left (a+b x^2\right )^3}-\frac {1}{4 b^2 \left (a+b x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.71 \begin {gather*} -\frac {a+3 b x^2}{12 b^2 \left (a+b x^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.54, size = 47, normalized size = 1.38 \begin {gather*} -\frac {3 \, b x^{2} + a}{12 \, {\left (b^{5} x^{6} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 22, normalized size = 0.65 \begin {gather*} -\frac {3 \, b x^{2} + a}{12 \, {\left (b x^{2} + a\right )}^{3} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 31, normalized size = 0.91 \begin {gather*} \frac {a}{6 \left (b \,x^{2}+a \right )^{3} b^{2}}-\frac {1}{4 \left (b \,x^{2}+a \right )^{2} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 47, normalized size = 1.38 \begin {gather*} -\frac {3 \, b x^{2} + a}{12 \, {\left (b^{5} x^{6} + 3 \, a b^{4} x^{4} + 3 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.23, size = 48, normalized size = 1.41 \begin {gather*} -\frac {\frac {a}{12\,b^2}+\frac {x^2}{4\,b}}{a^3+3\,a^2\,b\,x^2+3\,a\,b^2\,x^4+b^3\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 48, normalized size = 1.41 \begin {gather*} \frac {- a - 3 b x^{2}}{12 a^{3} b^{2} + 36 a^{2} b^{3} x^{2} + 36 a b^{4} x^{4} + 12 b^{5} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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