Optimal. Leaf size=87 \[ \frac {3 a^2 x^2}{b^5}-\frac {3 a x^4}{4 b^4}+\frac {x^6}{6 b^3}+\frac {a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac {5 a^4}{2 b^6 \left (a+b x^2\right )}-\frac {5 a^3 \log \left (a+b x^2\right )}{b^6} \]
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Rubi [A]
time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} \frac {a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac {5 a^4}{2 b^6 \left (a+b x^2\right )}-\frac {5 a^3 \log \left (a+b x^2\right )}{b^6}+\frac {3 a^2 x^2}{b^5}-\frac {3 a x^4}{4 b^4}+\frac {x^6}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^5}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {6 a^2}{b^5}-\frac {3 a x}{b^4}+\frac {x^2}{b^3}-\frac {a^5}{b^5 (a+b x)^3}+\frac {5 a^4}{b^5 (a+b x)^2}-\frac {10 a^3}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {3 a^2 x^2}{b^5}-\frac {3 a x^4}{4 b^4}+\frac {x^6}{6 b^3}+\frac {a^5}{4 b^6 \left (a+b x^2\right )^2}-\frac {5 a^4}{2 b^6 \left (a+b x^2\right )}-\frac {5 a^3 \log \left (a+b x^2\right )}{b^6}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 75, normalized size = 0.86 \begin {gather*} \frac {36 a^2 b x^2-9 a b^2 x^4+2 b^3 x^6+\frac {3 a^5}{\left (a+b x^2\right )^2}-\frac {30 a^4}{a+b x^2}-60 a^3 \log \left (a+b x^2\right )}{12 b^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 84, normalized size = 0.97
method | result | size |
norman | \(\frac {\frac {x^{10}}{6 b}-\frac {5 a \,x^{8}}{12 b^{2}}+\frac {5 a^{2} x^{6}}{3 b^{3}}-\frac {15 a^{5}}{2 b^{6}}-\frac {10 a^{4} x^{2}}{b^{5}}}{\left (b \,x^{2}+a \right )^{2}}-\frac {5 a^{3} \ln \left (b \,x^{2}+a \right )}{b^{6}}\) | \(76\) |
risch | \(\frac {x^{6}}{6 b^{3}}-\frac {3 a \,x^{4}}{4 b^{4}}+\frac {3 a^{2} x^{2}}{b^{5}}+\frac {-\frac {5 a^{4} x^{2}}{2}-\frac {9 a^{5}}{4 b}}{b^{5} \left (b \,x^{2}+a \right )^{2}}-\frac {5 a^{3} \ln \left (b \,x^{2}+a \right )}{b^{6}}\) | \(76\) |
default | \(\frac {\frac {1}{6} b^{2} x^{6}-\frac {3}{4} a b \,x^{4}+3 a^{2} x^{2}}{b^{5}}-\frac {a^{3} \left (\frac {5 a}{b \left (b \,x^{2}+a \right )}+\frac {10 \ln \left (b \,x^{2}+a \right )}{b}-\frac {a^{2}}{2 b \left (b \,x^{2}+a \right )^{2}}\right )}{2 b^{5}}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 89, normalized size = 1.02 \begin {gather*} -\frac {10 \, a^{4} b x^{2} + 9 \, a^{5}}{4 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} - \frac {5 \, a^{3} \log \left (b x^{2} + a\right )}{b^{6}} + \frac {2 \, b^{2} x^{6} - 9 \, a b x^{4} + 36 \, a^{2} x^{2}}{12 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.57, size = 115, normalized size = 1.32 \begin {gather*} \frac {2 \, b^{5} x^{10} - 5 \, a b^{4} x^{8} + 20 \, a^{2} b^{3} x^{6} + 63 \, a^{3} b^{2} x^{4} + 6 \, a^{4} b x^{2} - 27 \, a^{5} - 60 \, {\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{12 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 92, normalized size = 1.06 \begin {gather*} - \frac {5 a^{3} \log {\left (a + b x^{2} \right )}}{b^{6}} + \frac {3 a^{2} x^{2}}{b^{5}} - \frac {3 a x^{4}}{4 b^{4}} + \frac {- 9 a^{5} - 10 a^{4} b x^{2}}{4 a^{2} b^{6} + 8 a b^{7} x^{2} + 4 b^{8} x^{4}} + \frac {x^{6}}{6 b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.71, size = 92, normalized size = 1.06 \begin {gather*} -\frac {5 \, a^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac {30 \, a^{3} b^{2} x^{4} + 50 \, a^{4} b x^{2} + 21 \, a^{5}}{4 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {2 \, b^{6} x^{6} - 9 \, a b^{5} x^{4} + 36 \, a^{2} b^{4} x^{2}}{12 \, b^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.49, size = 90, normalized size = 1.03 \begin {gather*} \frac {x^6}{6\,b^3}-\frac {\frac {9\,a^5}{4\,b}+\frac {5\,a^4\,x^2}{2}}{a^2\,b^5+2\,a\,b^6\,x^2+b^7\,x^4}-\frac {3\,a\,x^4}{4\,b^4}-\frac {5\,a^3\,\ln \left (b\,x^2+a\right )}{b^6}+\frac {3\,a^2\,x^2}{b^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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