Integrand size = 22, antiderivative size = 107 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {a}{2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {c}{2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {(b c+a d) \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {(b c+a d) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
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Time = 0.08 (sec) , antiderivative size = 107, 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, 78} \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {a}{2 \left (a+b x^2\right ) (b c-a d)^2}+\frac {c}{2 \left (c+d x^2\right ) (b c-a d)^2}+\frac {(a d+b c) \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {(a d+b c) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
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Rule 78
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^2 (c+d x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {a b}{(b c-a d)^2 (a+b x)^2}+\frac {b (b c+a d)}{(b c-a d)^3 (a+b x)}-\frac {c d}{(b c-a d)^2 (c+d x)^2}-\frac {d (b c+a d)}{(b c-a d)^3 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {a}{2 (b c-a d)^2 \left (a+b x^2\right )}+\frac {c}{2 (b c-a d)^2 \left (c+d x^2\right )}+\frac {(b c+a d) \log \left (a+b x^2\right )}{2 (b c-a d)^3}-\frac {(b c+a d) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.80 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {\frac {a (b c-a d)}{a+b x^2}+\frac {c (b c-a d)}{c+d x^2}+(b c+a d) \log \left (a+b x^2\right )-(b c+a d) \log \left (c+d x^2\right )}{2 (b c-a d)^3} \]
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Time = 2.70 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {b \left (\frac {\left (a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 \left (a d -b c \right )^{3}}+\frac {d \left (\frac {\left (a d +b c \right ) \ln \left (d \,x^{2}+c \right )}{d}+\frac {\left (a d -b c \right ) c}{d \left (d \,x^{2}+c \right )}\right )}{2 \left (a d -b c \right )^{3}}\) | \(113\) |
norman | \(\frac {\frac {a c}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {\left (a d +b c \right ) x^{2}}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}-\frac {\left (a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (a d +b c \right ) \ln \left (d \,x^{2}+c \right )}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}\) | \(193\) |
risch | \(\frac {\frac {a c}{a^{2} d^{2}-2 a b c d +b^{2} c^{2}}+\frac {\left (a d +b c \right ) x^{2}}{2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}}}{\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}+\frac {d \ln \left (d \,x^{2}+c \right ) a}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}+\frac {c \ln \left (d \,x^{2}+c \right ) b}{2 a^{3} d^{3}-6 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -2 b^{3} c^{3}}-\frac {\ln \left (-b \,x^{2}-a \right ) a d}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {\ln \left (-b \,x^{2}-a \right ) b c}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(289\) |
parallelrisch | \(-\frac {\ln \left (b \,x^{2}+a \right ) x^{4} a \,b^{2} d^{3}+\ln \left (b \,x^{2}+a \right ) x^{4} b^{3} c \,d^{2}-\ln \left (d \,x^{2}+c \right ) x^{4} a \,b^{2} d^{3}-\ln \left (d \,x^{2}+c \right ) x^{4} b^{3} c \,d^{2}+\ln \left (b \,x^{2}+a \right ) x^{2} a^{2} b \,d^{3}+\ln \left (b \,x^{2}+a \right ) x^{2} b^{3} c^{2} d -2 a^{2} b c \,d^{2}+2 a \,b^{2} c^{2} d +2 \ln \left (b \,x^{2}+a \right ) x^{2} a \,b^{2} c \,d^{2}-\ln \left (d \,x^{2}+c \right ) x^{2} a^{2} b \,d^{3}-\ln \left (d \,x^{2}+c \right ) x^{2} b^{3} c^{2} d -x^{2} a^{2} b \,d^{3}+x^{2} b^{3} c^{2} d +\ln \left (b \,x^{2}+a \right ) a^{2} b c \,d^{2}+\ln \left (b \,x^{2}+a \right ) a \,b^{2} c^{2} d -2 \ln \left (d \,x^{2}+c \right ) x^{2} a \,b^{2} c \,d^{2}-\ln \left (d \,x^{2}+c \right ) a^{2} b c \,d^{2}-\ln \left (d \,x^{2}+c \right ) a \,b^{2} c^{2} d}{2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) b d}\) | \(377\) |
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (99) = 198\).
Time = 0.25 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.77 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {2 \, a b c^{2} - 2 \, a^{2} c d + {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x^{2} + {\left ({\left (b^{2} c d + a b d^{2}\right )} x^{4} + a b c^{2} + a^{2} c d + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - {\left ({\left (b^{2} c d + a b d^{2}\right )} x^{4} + a b c^{2} + a^{2} c d + {\left (b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} + {\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{4} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x^{2}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 507 vs. \(2 (90) = 180\).
Time = 2.58 (sec) , antiderivative size = 507, normalized size of antiderivative = 4.74 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {2 a c + x^{2} \left (a d + b c\right )}{2 a^{3} c d^{2} - 4 a^{2} b c^{2} d + 2 a b^{2} c^{3} + x^{4} \cdot \left (2 a^{2} b d^{3} - 4 a b^{2} c d^{2} + 2 b^{3} c^{2} d\right ) + x^{2} \cdot \left (2 a^{3} d^{3} - 2 a^{2} b c d^{2} - 2 a b^{2} c^{2} d + 2 b^{3} c^{3}\right )} + \frac {\left (a d + b c\right ) \log {\left (x^{2} + \frac {- \frac {a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} + \frac {4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d - \frac {b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{2 \left (a d - b c\right )^{3}} - \frac {\left (a d + b c\right ) \log {\left (x^{2} + \frac {\frac {a^{4} d^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c d^{3} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{2} d^{2} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + a^{2} d^{2} - \frac {4 a b^{3} c^{3} d \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + 2 a b c d + \frac {b^{4} c^{4} \left (a d + b c\right )}{\left (a d - b c\right )^{3}} + b^{2} c^{2}}{2 a b d^{2} + 2 b^{2} c d} \right )}}{2 \left (a d - b c\right )^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (99) = 198\).
Time = 0.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.13 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {{\left (b c + a d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} - \frac {{\left (b c + a d\right )} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}} + \frac {{\left (b c + a d\right )} x^{2} + 2 \, a c}{2 \, {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{4} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2}\right )}} \]
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Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.66 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {\frac {a b^{3}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (b x^{2} + a\right )}} - \frac {{\left (b^{3} c + a b^{2} d\right )} \log \left ({\left | \frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d \right |}\right )}{b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}} - \frac {b^{2} c d}{{\left (b c - a d\right )}^{3} {\left (\frac {b c}{b x^{2} + a} - \frac {a d}{b x^{2} + a} + d\right )}}}{2 \, b} \]
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Time = 5.10 (sec) , antiderivative size = 522, normalized size of antiderivative = 4.88 \[ \int \frac {x^3}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {b^2\,c^2\,x^2-a^2\,d^2\,x^2+2\,a\,b\,c^2-2\,a^2\,c\,d+a^2\,d^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+b^2\,c^2\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+a\,b\,c^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+a^2\,c\,d\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+a\,b\,d^2\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+b^2\,c\,d\,x^4\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,2{}\mathrm {i}+a\,b\,c\,d\,x^2\,\mathrm {atan}\left (\frac {a\,d\,x^2\,1{}\mathrm {i}-b\,c\,x^2\,1{}\mathrm {i}}{2\,a\,c+a\,d\,x^2+b\,c\,x^2}\right )\,4{}\mathrm {i}}{-2\,a^4\,c\,d^3-2\,a^4\,d^4\,x^2+6\,a^3\,b\,c^2\,d^2+4\,a^3\,b\,c\,d^3\,x^2-2\,a^3\,b\,d^4\,x^4-6\,a^2\,b^2\,c^3\,d+6\,a^2\,b^2\,c\,d^3\,x^4+2\,a\,b^3\,c^4-4\,a\,b^3\,c^3\,d\,x^2-6\,a\,b^3\,c^2\,d^2\,x^4+2\,b^4\,c^4\,x^2+2\,b^4\,c^3\,d\,x^4} \]
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