Optimal. Leaf size=71 \[ \frac {a^3}{6 b^4 \left (a+b x^2\right )^3}-\frac {3 a^2}{4 b^4 \left (a+b x^2\right )^2}+\frac {3 a}{2 b^4 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^4} \]
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Rubi [A]
time = 0.04, antiderivative size = 71, 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, 272, 45}
\begin {gather*} \frac {a^3}{6 b^4 \left (a+b x^2\right )^3}-\frac {3 a^2}{4 b^4 \left (a+b x^2\right )^2}+\frac {3 a}{2 b^4 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {x^7}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{2} b^4 \text {Subst}\left (\int \frac {x^3}{\left (a b+b^2 x\right )^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^4 \text {Subst}\left (\int \left (-\frac {a^3}{b^7 (a+b x)^4}+\frac {3 a^2}{b^7 (a+b x)^3}-\frac {3 a}{b^7 (a+b x)^2}+\frac {1}{b^7 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a^3}{6 b^4 \left (a+b x^2\right )^3}-\frac {3 a^2}{4 b^4 \left (a+b x^2\right )^2}+\frac {3 a}{2 b^4 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 0.70 \begin {gather*} \frac {\frac {a \left (11 a^2+27 a b x^2+18 b^2 x^4\right )}{\left (a+b x^2\right )^3}+6 \log \left (a+b x^2\right )}{12 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 64, normalized size = 0.90
method | result | size |
norman | \(\frac {\frac {11 a^{3}}{12 b^{4}}+\frac {3 a \,x^{4}}{2 b^{2}}+\frac {9 a^{2} x^{2}}{4 b^{3}}}{\left (b \,x^{2}+a \right )^{3}}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(54\) |
default | \(\frac {a^{3}}{6 b^{4} \left (b \,x^{2}+a \right )^{3}}-\frac {3 a^{2}}{4 b^{4} \left (b \,x^{2}+a \right )^{2}}+\frac {3 a}{2 b^{4} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(64\) |
risch | \(\frac {\frac {11 a^{3}}{12 b^{4}}+\frac {3 a \,x^{4}}{2 b^{2}}+\frac {9 a^{2} x^{2}}{4 b^{3}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{4}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 77, normalized size = 1.08 \begin {gather*} \frac {18 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 11 \, a^{3}}{12 \, {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 102, normalized size = 1.44 \begin {gather*} \frac {18 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 11 \, a^{3} + 6 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right )}{12 \, {\left (b^{7} x^{6} + 3 \, a b^{6} x^{4} + 3 \, a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 76, normalized size = 1.07 \begin {gather*} \frac {11 a^{3} + 27 a^{2} b x^{2} + 18 a b^{2} x^{4}}{12 a^{3} b^{4} + 36 a^{2} b^{5} x^{2} + 36 a b^{6} x^{4} + 12 b^{7} x^{6}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.94, size = 53, normalized size = 0.75 \begin {gather*} \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} - \frac {11 \, b^{2} x^{6} + 15 \, a b x^{4} + 6 \, a^{2} x^{2}}{12 \, {\left (b x^{2} + a\right )}^{3} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.33, size = 75, normalized size = 1.06 \begin {gather*} \frac {\frac {11\,a^3}{12\,b^4}+\frac {3\,a\,x^4}{2\,b^2}+\frac {9\,a^2\,x^2}{4\,b^3}}{a^3+3\,a^2\,b\,x^2+3\,a\,b^2\,x^4+b^3\,x^6}+\frac {\ln \left (b\,x^2+a\right )}{2\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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