Integrand size = 17, antiderivative size = 49 \[ \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac {b}{c^3 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^3} \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 45} \[ \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac {b}{c^3 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^3} \]
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Rule 45
Rule 272
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5}{\left (b+c x^2\right )^3} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(b+c x)^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {b^2}{c^2 (b+c x)^3}-\frac {2 b}{c^2 (b+c x)^2}+\frac {1}{c^2 (b+c x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b^2}{4 c^3 \left (b+c x^2\right )^2}+\frac {b}{c^3 \left (b+c x^2\right )}+\frac {\log \left (b+c x^2\right )}{2 c^3} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {b \left (3 b+4 c x^2\right )}{\left (b+c x^2\right )^2}+2 \log \left (b+c x^2\right )}{4 c^3} \]
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Time = 0.06 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86
method | result | size |
risch | \(\frac {\frac {b \,x^{2}}{c^{2}}+\frac {3 b^{2}}{4 c^{3}}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\ln \left (c \,x^{2}+b \right )}{2 c^{3}}\) | \(42\) |
default | \(-\frac {b^{2}}{4 c^{3} \left (c \,x^{2}+b \right )^{2}}+\frac {b}{c^{3} \left (c \,x^{2}+b \right )}+\frac {\ln \left (c \,x^{2}+b \right )}{2 c^{3}}\) | \(46\) |
norman | \(\frac {\frac {b \,x^{7}}{c^{2}}+\frac {3 b^{2} x^{5}}{4 c^{3}}}{x^{5} \left (c \,x^{2}+b \right )^{2}}+\frac {\ln \left (c \,x^{2}+b \right )}{2 c^{3}}\) | \(48\) |
parallelrisch | \(\frac {2 c^{2} \ln \left (c \,x^{2}+b \right ) x^{4}+4 \ln \left (c \,x^{2}+b \right ) x^{2} b c +4 b c \,x^{2}+2 b^{2} \ln \left (c \,x^{2}+b \right )+3 b^{2}}{4 c^{3} \left (c \,x^{2}+b \right )^{2}}\) | \(72\) |
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Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.41 \[ \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {4 \, b c x^{2} + 3 \, b^{2} + 2 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \log \left (c x^{2} + b\right )}{4 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.08 \[ \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {3 b^{2} + 4 b c x^{2}}{4 b^{2} c^{3} + 8 b c^{4} x^{2} + 4 c^{5} x^{4}} + \frac {\log {\left (b + c x^{2} \right )}}{2 c^{3}} \]
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Time = 0.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12 \[ \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {4 \, b c x^{2} + 3 \, b^{2}}{4 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} + \frac {\log \left (c x^{2} + b\right )}{2 \, c^{3}} \]
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} - \frac {3 \, c x^{4} + 2 \, b x^{2}}{4 \, {\left (c x^{2} + b\right )}^{2} c^{2}} \]
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Time = 12.88 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06 \[ \int \frac {x^{11}}{\left (b x^2+c x^4\right )^3} \, dx=\frac {\frac {3\,b^2}{4\,c^3}+\frac {b\,x^2}{c^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}+\frac {\ln \left (c\,x^2+b\right )}{2\,c^3} \]
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