Integrand size = 20, antiderivative size = 70 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{2 (b c-a d) \left (c+d x^2\right )}+\frac {b \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac {b \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
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Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {455, 46} \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {1}{2 \left (c+d x^2\right ) (b c-a d)}+\frac {b \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac {b \log \left (c+d x^2\right )}{2 (b c-a d)^2} \]
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Rule 46
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b x) (c+d x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{2 (b c-a d) \left (c+d x^2\right )}+\frac {b \log \left (a+b x^2\right )}{2 (b c-a d)^2}-\frac {b \log \left (c+d x^2\right )}{2 (b c-a d)^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b c-a d+b \left (c+d x^2\right ) \log \left (a+b x^2\right )-b \left (c+d x^2\right ) \log \left (c+d x^2\right )}{2 (b c-a d)^2 \left (c+d x^2\right )} \]
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Time = 2.71 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04
method | result | size |
default | \(\frac {b \ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )^{2}}+\frac {d \left (-\frac {b \ln \left (d \,x^{2}+c \right )}{d}-\frac {a d -b c}{d \left (d \,x^{2}+c \right )}\right )}{2 \left (a d -b c \right )^{2}}\) | \(73\) |
risch | \(-\frac {1}{2 \left (a d -b c \right ) \left (d \,x^{2}+c \right )}-\frac {b \ln \left (-d \,x^{2}-c \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}\) | \(94\) |
norman | \(\frac {d \,x^{2}}{2 c \left (a d -b c \right ) \left (d \,x^{2}+c \right )}+\frac {b \ln \left (b \,x^{2}+a \right )}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}-\frac {b \ln \left (d \,x^{2}+c \right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(98\) |
parallelrisch | \(\frac {\ln \left (b \,x^{2}+a \right ) x^{2} b \,d^{2}-\ln \left (d \,x^{2}+c \right ) x^{2} b \,d^{2}+\ln \left (b \,x^{2}+a \right ) b c d -\ln \left (d \,x^{2}+c \right ) b c d -a \,d^{2}+b c d}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d \,x^{2}+c \right ) d}\) | \(107\) |
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Time = 0.26 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.47 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b c - a d + {\left (b d x^{2} + b c\right )} \log \left (b x^{2} + a\right ) - {\left (b d x^{2} + b c\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (56) = 112\).
Time = 0.98 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.54 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=- \frac {b \log {\left (x^{2} + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{2 \left (a d - b c\right )^{2}} + \frac {b \log {\left (x^{2} + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{2 \left (a d - b c\right )^{2}} - \frac {1}{2 a c d - 2 b c^{2} + x^{2} \cdot \left (2 a d^{2} - 2 b c d\right )} \]
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Time = 0.21 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.41 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b \log \left (b x^{2} + a\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} - \frac {b \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}} + \frac {1}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.21 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {b d \log \left ({\left | b - \frac {b c}{d x^{2} + c} + \frac {a d}{d x^{2} + c} \right |}\right )}{2 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )}} + \frac {d}{2 \, {\left (b c d - a d^{2}\right )} {\left (d x^{2} + c\right )}} \]
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Time = 0.20 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.29 \[ \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx=\frac {-a\,d+c\,\left (b+b\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}\right )+b\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}}{2\,a^2\,c\,d^2+2\,a^2\,d^3\,x^2-4\,a\,b\,c^2\,d-4\,a\,b\,c\,d^2\,x^2+2\,b^2\,c^3+2\,b^2\,c^2\,d\,x^2} \]
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