Optimal. Leaf size=91 \[ -\frac {2 a x^2}{b^5}+\frac {x^4}{4 b^4}+\frac {a^5}{6 b^6 \left (a+b x^2\right )^3}-\frac {5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac {5 a^3}{b^6 \left (a+b x^2\right )}+\frac {5 a^2 \log \left (a+b x^2\right )}{b^6} \]
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Rubi [A]
time = 0.06, antiderivative size = 91, 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^5}{6 b^6 \left (a+b x^2\right )^3}-\frac {5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac {5 a^3}{b^6 \left (a+b x^2\right )}+\frac {5 a^2 \log \left (a+b x^2\right )}{b^6}-\frac {2 a x^2}{b^5}+\frac {x^4}{4 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^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac {x^{11}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac {1}{2} b^4 \text {Subst}\left (\int \frac {x^5}{\left (a b+b^2 x\right )^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^4 \text {Subst}\left (\int \left (-\frac {4 a}{b^9}+\frac {x}{b^8}-\frac {a^5}{b^9 (a+b x)^4}+\frac {5 a^4}{b^9 (a+b x)^3}-\frac {10 a^3}{b^9 (a+b x)^2}+\frac {10 a^2}{b^9 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {2 a x^2}{b^5}+\frac {x^4}{4 b^4}+\frac {a^5}{6 b^6 \left (a+b x^2\right )^3}-\frac {5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac {5 a^3}{b^6 \left (a+b x^2\right )}+\frac {5 a^2 \log \left (a+b x^2\right )}{b^6}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 78, normalized size = 0.86 \begin {gather*} \frac {-24 a b x^2+3 b^2 x^4+\frac {2 a^5}{\left (a+b x^2\right )^3}-\frac {15 a^4}{\left (a+b x^2\right )^2}+\frac {60 a^3}{a+b x^2}+60 a^2 \log \left (a+b x^2\right )}{12 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 90, normalized size = 0.99
method | result | size |
norman | \(\frac {\frac {x^{10}}{4 b}-\frac {5 a \,x^{8}}{4 b^{2}}+\frac {55 a^{5}}{6 b^{6}}+\frac {15 a^{3} x^{4}}{b^{4}}+\frac {45 a^{4} x^{2}}{2 b^{5}}}{\left (b \,x^{2}+a \right )^{3}}+\frac {5 a^{2} \ln \left (b \,x^{2}+a \right )}{b^{6}}\) | \(76\) |
default | \(\frac {\left (-b \,x^{2}+4 a \right )^{2}}{4 b^{6}}+\frac {a^{2} \left (\frac {a^{3}}{3 b \left (b \,x^{2}+a \right )^{3}}+\frac {10 a}{b \left (b \,x^{2}+a \right )}-\frac {5 a^{2}}{2 b \left (b \,x^{2}+a \right )^{2}}+\frac {10 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 b^{5}}\) | \(90\) |
risch | \(\frac {x^{4}}{4 b^{4}}-\frac {2 a \,x^{2}}{b^{5}}+\frac {4 a^{2}}{b^{6}}+\frac {5 a^{3} b \,x^{4}+\frac {35 a^{4} x^{2}}{4}+\frac {47 a^{5}}{12 b}}{b^{5} \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}+\frac {5 a^{2} \ln \left (b \,x^{2}+a \right )}{b^{6}}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 99, normalized size = 1.09 \begin {gather*} \frac {60 \, a^{3} b^{2} x^{4} + 105 \, a^{4} b x^{2} + 47 \, a^{5}}{12 \, {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}} + \frac {5 \, a^{2} \log \left (b x^{2} + a\right )}{b^{6}} + \frac {b x^{4} - 8 \, a x^{2}}{4 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 137, normalized size = 1.51 \begin {gather*} \frac {3 \, b^{5} x^{10} - 15 \, a b^{4} x^{8} - 63 \, a^{2} b^{3} x^{6} - 9 \, a^{3} b^{2} x^{4} + 81 \, a^{4} b x^{2} + 47 \, a^{5} + 60 \, {\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{12 \, {\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 100, normalized size = 1.10 \begin {gather*} \frac {5 a^{2} \log {\left (a + b x^{2} \right )}}{b^{6}} - \frac {2 a x^{2}}{b^{5}} + \frac {47 a^{5} + 105 a^{4} b x^{2} + 60 a^{3} b^{2} x^{4}}{12 a^{3} b^{6} + 36 a^{2} b^{7} x^{2} + 36 a b^{8} x^{4} + 12 b^{9} x^{6}} + \frac {x^{4}}{4 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.54, size = 91, normalized size = 1.00 \begin {gather*} \frac {5 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac {b^{4} x^{4} - 8 \, a b^{3} x^{2}}{4 \, b^{8}} - \frac {110 \, a^{2} b^{3} x^{6} + 270 \, a^{3} b^{2} x^{4} + 225 \, a^{4} b x^{2} + 63 \, a^{5}}{12 \, {\left (b x^{2} + a\right )}^{3} b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.48, size = 98, normalized size = 1.08 \begin {gather*} \frac {\frac {47\,a^5}{12\,b}+\frac {35\,a^4\,x^2}{4}+5\,a^3\,b\,x^4}{a^3\,b^5+3\,a^2\,b^6\,x^2+3\,a\,b^7\,x^4+b^8\,x^6}+\frac {x^4}{4\,b^4}-\frac {2\,a\,x^2}{b^5}+\frac {5\,a^2\,\ln \left (b\,x^2+a\right )}{b^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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