Integrand size = 22, antiderivative size = 93 \[ \int \frac {x^5}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {a^2}{2 b^2 (b c-a d) \left (a+b x^2\right )}-\frac {a (2 b c-a d) \log \left (a+b x^2\right )}{2 b^2 (b c-a d)^2}+\frac {c^2 \log \left (c+d x^2\right )}{2 d (b c-a d)^2} \]
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Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {457, 90} \[ \int \frac {x^5}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {a^2}{2 b^2 \left (a+b x^2\right ) (b c-a d)}-\frac {a (2 b c-a d) \log \left (a+b x^2\right )}{2 b^2 (b c-a d)^2}+\frac {c^2 \log \left (c+d x^2\right )}{2 d (b c-a d)^2} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^2 (c+d x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {a^2}{b (b c-a d) (a+b x)^2}+\frac {a (-2 b c+a d)}{b (b c-a d)^2 (a+b x)}+\frac {c^2}{(b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {a^2}{2 b^2 (b c-a d) \left (a+b x^2\right )}-\frac {a (2 b c-a d) \log \left (a+b x^2\right )}{2 b^2 (b c-a d)^2}+\frac {c^2 \log \left (c+d x^2\right )}{2 d (b c-a d)^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \frac {x^5}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {a^2 d (-b c+a d)+a d (-2 b c+a d) \left (a+b x^2\right ) \log \left (a+b x^2\right )+b^2 c^2 \left (a+b x^2\right ) \log \left (c+d x^2\right )}{2 b^2 d (b c-a d)^2 \left (a+b x^2\right )} \]
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Time = 2.72 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {a \left (\frac {\left (-a d +2 b c \right ) \ln \left (b \,x^{2}+a \right )}{b^{2}}-\frac {\left (a d -b c \right ) a}{b^{2} \left (b \,x^{2}+a \right )}\right )}{2 \left (a d -b c \right )^{2}}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 \left (a d -b c \right )^{2} d}\) | \(86\) |
norman | \(\frac {a^{2}}{2 b^{2} \left (a d -b c \right ) \left (b \,x^{2}+a \right )}+\frac {c^{2} \ln \left (d \,x^{2}+c \right )}{2 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {a \left (a d -2 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2}}\) | \(113\) |
risch | \(\frac {a^{2}}{2 b^{2} \left (a d -b c \right ) \left (b \,x^{2}+a \right )}+\frac {a^{2} \ln \left (b \,x^{2}+a \right ) d}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2}}-\frac {a \ln \left (b \,x^{2}+a \right ) c}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b}+\frac {c^{2} \ln \left (-d \,x^{2}-c \right )}{2 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(149\) |
parallelrisch | \(\frac {\ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b \,d^{2}-2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{2} c d +\ln \left (d \,x^{2}+c \right ) x^{2} b^{3} c^{2}+\ln \left (b \,x^{2}+a \right ) a^{3} d^{2}-2 \ln \left (b \,x^{2}+a \right ) a^{2} b c d +\ln \left (d \,x^{2}+c \right ) a \,b^{2} c^{2}+a^{3} d^{2}-a^{2} b c d}{2 d \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right ) b^{2}}\) | \(160\) |
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Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.74 \[ \int \frac {x^5}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {a^{2} b c d - a^{3} d^{2} + {\left (2 \, a^{2} b c d - a^{3} d^{2} + {\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - {\left (b^{3} c^{2} x^{2} + a b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (a b^{4} c^{2} d - 2 \, a^{2} b^{3} c d^{2} + a^{3} b^{2} d^{3} + {\left (b^{5} c^{2} d - 2 \, a b^{4} c d^{2} + a^{2} b^{3} d^{3}\right )} x^{2}\right )}} \]
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Timed out. \[ \int \frac {x^5}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Timed out} \]
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Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.40 \[ \int \frac {x^5}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {c^{2} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} - \frac {a^{2}}{2 \, {\left (a b^{3} c - a^{2} b^{2} d + {\left (b^{4} c - a b^{3} d\right )} x^{2}\right )}} - \frac {{\left (2 \, a b c - a^{2} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}} \]
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Time = 0.31 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.63 \[ \int \frac {x^5}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {c^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} - \frac {{\left (2 \, a b c - a^{2} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )}} + \frac {2 \, a b c x^{2} - a^{2} d x^{2} + a^{2} c}{2 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (b x^{2} + a\right )}} \]
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Time = 5.50 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.82 \[ \int \frac {x^5}{\left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {a^2}{2\,\left (d\,a^2\,b^2+d\,a\,b^3\,x^2-c\,a\,b^3-c\,b^4\,x^2\right )}+\frac {c^2\,\ln \left (d\,x^2+c\right )}{2\,a^2\,d^3-4\,a\,b\,c\,d^2+2\,b^2\,c^2\,d}+\frac {a^2\,d\,\ln \left (b\,x^2+a\right )}{2\,a^2\,b^2\,d^2-4\,a\,b^3\,c\,d+2\,b^4\,c^2}-\frac {2\,a\,b\,c\,\ln \left (b\,x^2+a\right )}{2\,a^2\,b^2\,d^2-4\,a\,b^3\,c\,d+2\,b^4\,c^2} \]
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