Integrand size = 22, antiderivative size = 210 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {1}{6 a^2 c x^6}+\frac {2 b c+a d}{4 a^3 c^2 x^4}-\frac {3 b^2 c^2+2 a b c d+a^2 d^2}{2 a^4 c^3 x^2}-\frac {b^4}{2 a^4 (b c-a d) \left (a+b x^2\right )}-\frac {\left (4 b^3 c^3+3 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \log (x)}{a^5 c^4}+\frac {b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}+\frac {d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2} \]
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Time = 0.21 (sec) , antiderivative size = 210, 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} \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}-\frac {b^4}{2 a^4 \left (a+b x^2\right ) (b c-a d)}+\frac {a d+2 b c}{4 a^3 c^2 x^4}-\frac {1}{6 a^2 c x^6}-\frac {a^2 d^2+2 a b c d+3 b^2 c^2}{2 a^4 c^3 x^2}-\frac {\log (x) \left (a^3 d^3+2 a^2 b c d^2+3 a b^2 c^2 d+4 b^3 c^3\right )}{a^5 c^4}+\frac {d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2} \]
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Rule 90
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x^4 (a+b x)^2 (c+d x)} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 c x^4}+\frac {-2 b c-a d}{a^3 c^2 x^3}+\frac {3 b^2 c^2+2 a b c d+a^2 d^2}{a^4 c^3 x^2}+\frac {-4 b^3 c^3-3 a b^2 c^2 d-2 a^2 b c d^2-a^3 d^3}{a^5 c^4 x}-\frac {b^5}{a^4 (-b c+a d) (a+b x)^2}-\frac {b^5 (-4 b c+5 a d)}{a^5 (-b c+a d)^2 (a+b x)}+\frac {d^6}{c^4 (b c-a d)^2 (c+d x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {1}{6 a^2 c x^6}+\frac {2 b c+a d}{4 a^3 c^2 x^4}-\frac {3 b^2 c^2+2 a b c d+a^2 d^2}{2 a^4 c^3 x^2}-\frac {b^4}{2 a^4 (b c-a d) \left (a+b x^2\right )}-\frac {\left (4 b^3 c^3+3 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \log (x)}{a^5 c^4}+\frac {b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{2 a^5 (b c-a d)^2}+\frac {d^5 \log \left (c+d x^2\right )}{2 c^4 (b c-a d)^2} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {1}{12} \left (-\frac {2}{a^2 c x^6}+\frac {6 b c+3 a d}{a^3 c^2 x^4}-\frac {6 \left (3 b^2 c^2+2 a b c d+a^2 d^2\right )}{a^4 c^3 x^2}+\frac {6 b^4}{a^4 (-b c+a d) \left (a+b x^2\right )}-\frac {12 \left (4 b^3 c^3+3 a b^2 c^2 d+2 a^2 b c d^2+a^3 d^3\right ) \log (x)}{a^5 c^4}+\frac {6 b^4 (4 b c-5 a d) \log \left (a+b x^2\right )}{a^5 (b c-a d)^2}+\frac {6 d^5 \log \left (c+d x^2\right )}{c^4 (b c-a d)^2}\right ) \]
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Time = 2.67 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {1}{6 a^{2} c \,x^{6}}-\frac {-a d -2 b c}{4 x^{4} a^{3} c^{2}}-\frac {a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}}{2 a^{4} c^{3} x^{2}}+\frac {\left (-a^{3} d^{3}-2 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{5} c^{4}}-\frac {b^{5} \left (\frac {\left (5 a d -4 b c \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {\left (a d -b c \right ) a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{5} \left (a d -b c \right )^{2}}+\frac {d^{5} \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a d -b c \right )^{2}}\) | \(201\) |
norman | \(\frac {-\frac {1}{6 a c}+\frac {\left (3 a d +4 b c \right ) x^{2}}{12 a^{2} c^{2}}-\frac {\left (2 a^{2} d^{2}+3 a b c d +4 b^{2} c^{2}\right ) x^{4}}{4 c^{3} a^{3}}+\frac {\left (a^{3} b \,d^{3}+a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d -4 b^{4} c^{3}\right ) b \,x^{8}}{2 c^{3} \left (a d -b c \right ) a^{5}}}{x^{6} \left (b \,x^{2}+a \right )}+\frac {d^{5} \ln \left (d \,x^{2}+c \right )}{2 c^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) \ln \left (x \right )}{a^{5} c^{4}}-\frac {b^{4} \left (5 a d -4 b c \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(270\) |
risch | \(\frac {-\frac {b \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{6}}{2 a^{4} c^{3} \left (a d -b c \right )}-\frac {\left (2 a^{2} d^{2}+3 a b c d +4 b^{2} c^{2}\right ) x^{4}}{4 c^{3} a^{3}}+\frac {\left (3 a d +4 b c \right ) x^{2}}{12 a^{2} c^{2}}-\frac {1}{6 a c}}{x^{6} \left (b \,x^{2}+a \right )}-\frac {\ln \left (x \right ) d^{3}}{a^{2} c^{4}}-\frac {2 \ln \left (x \right ) b \,d^{2}}{a^{3} c^{3}}-\frac {3 \ln \left (x \right ) b^{2} d}{a^{4} c^{2}}-\frac {4 \ln \left (x \right ) b^{3}}{a^{5} c}+\frac {d^{5} \ln \left (-d \,x^{2}-c \right )}{2 c^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {5 b^{4} \ln \left (b \,x^{2}+a \right ) d}{2 a^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 b^{5} \ln \left (b \,x^{2}+a \right ) c}{a^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\) | \(310\) |
parallelrisch | \(-\frac {2 a^{6} b \,c^{3} d^{2}-4 a^{5} b^{2} c^{4} d +24 x^{6} a \,b^{6} c^{5}+12 x^{4} a^{2} b^{5} c^{5}-4 x^{2} a^{3} b^{4} c^{5}+48 \ln \left (x \right ) x^{8} b^{7} c^{5}-24 \ln \left (b \,x^{2}+a \right ) x^{8} b^{7} c^{5}-60 \ln \left (x \right ) x^{8} a \,b^{6} c^{4} d +30 \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{6} c^{4} d -60 \ln \left (x \right ) x^{6} a^{2} b^{5} c^{4} d +30 \ln \left (b \,x^{2}+a \right ) x^{6} a^{2} b^{5} c^{4} d +2 a^{4} b^{3} c^{5}+2 x^{2} a^{5} b^{2} c^{3} d^{2}+5 x^{2} a^{4} b^{3} c^{4} d +12 \ln \left (x \right ) x^{8} a^{5} b^{2} d^{5}-6 \ln \left (d \,x^{2}+c \right ) x^{8} a^{5} b^{2} d^{5}+12 \ln \left (x \right ) x^{6} a^{6} b \,d^{5}+48 \ln \left (x \right ) x^{6} a \,b^{6} c^{5}-24 \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{6} c^{5}-6 \ln \left (d \,x^{2}+c \right ) x^{6} a^{6} b \,d^{5}+6 x^{6} a^{5} b^{2} c \,d^{4}-30 x^{6} a^{2} b^{5} c^{4} d +6 x^{4} a^{6} b c \,d^{4}-3 x^{4} a^{5} b^{2} c^{2} d^{3}-15 x^{4} a^{3} b^{4} c^{4} d -3 x^{2} a^{6} b \,c^{2} d^{3}}{12 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b \,x^{2}+a \right ) x^{6} c^{4} b \,a^{5}}\) | \(459\) |
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Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (198) = 396\).
Time = 5.02 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.95 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=-\frac {2 \, a^{4} b^{2} c^{5} - 4 \, a^{5} b c^{4} d + 2 \, a^{6} c^{3} d^{2} + 6 \, {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d + a^{5} b c d^{4}\right )} x^{6} + 3 \, {\left (4 \, a^{2} b^{4} c^{5} - 5 \, a^{3} b^{3} c^{4} d - a^{5} b c^{2} d^{3} + 2 \, a^{6} c d^{4}\right )} x^{4} - {\left (4 \, a^{3} b^{3} c^{5} - 5 \, a^{4} b^{2} c^{4} d - 2 \, a^{5} b c^{3} d^{2} + 3 \, a^{6} c^{2} d^{3}\right )} x^{2} - 6 \, {\left ({\left (4 \, b^{6} c^{5} - 5 \, a b^{5} c^{4} d\right )} x^{8} + {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) - 6 \, {\left (a^{5} b d^{5} x^{8} + a^{6} d^{5} x^{6}\right )} \log \left (d x^{2} + c\right ) + 12 \, {\left ({\left (4 \, b^{6} c^{5} - 5 \, a b^{5} c^{4} d + a^{5} b d^{5}\right )} x^{8} + {\left (4 \, a b^{5} c^{5} - 5 \, a^{2} b^{4} c^{4} d + a^{6} d^{5}\right )} x^{6}\right )} \log \left (x\right )}{12 \, {\left ({\left (a^{5} b^{3} c^{6} - 2 \, a^{6} b^{2} c^{5} d + a^{7} b c^{4} d^{2}\right )} x^{8} + {\left (a^{6} b^{2} c^{6} - 2 \, a^{7} b c^{5} d + a^{8} c^{4} d^{2}\right )} x^{6}\right )}} \]
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Timed out. \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.61 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {d^{5} \log \left (d x^{2} + c\right )}{2 \, {\left (b^{2} c^{6} - 2 \, a b c^{5} d + a^{2} c^{4} d^{2}\right )}} + \frac {{\left (4 \, b^{5} c - 5 \, a b^{4} d\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2}\right )}} - \frac {2 \, a^{3} b c^{3} - 2 \, a^{4} c^{2} d + 6 \, {\left (4 \, b^{4} c^{3} - a b^{3} c^{2} d - a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{6} + 3 \, {\left (4 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} - 2 \, a^{4} d^{3}\right )} x^{4} - {\left (4 \, a^{2} b^{2} c^{3} - a^{3} b c^{2} d - 3 \, a^{4} c d^{2}\right )} x^{2}}{12 \, {\left ({\left (a^{4} b^{2} c^{4} - a^{5} b c^{3} d\right )} x^{8} + {\left (a^{5} b c^{4} - a^{6} c^{3} d\right )} x^{6}\right )}} - \frac {{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5} c^{4}} \]
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Time = 0.27 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.69 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {d^{6} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b^{2} c^{6} d - 2 \, a b c^{5} d^{2} + a^{2} c^{4} d^{3}\right )}} + \frac {{\left (4 \, b^{6} c - 5 \, a b^{5} d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a^{5} b^{3} c^{2} - 2 \, a^{6} b^{2} c d + a^{7} b d^{2}\right )}} - \frac {4 \, b^{6} c x^{2} - 5 \, a b^{5} d x^{2} + 5 \, a b^{5} c - 6 \, a^{2} b^{4} d}{2 \, {\left (a^{5} b^{2} c^{2} - 2 \, a^{6} b c d + a^{7} d^{2}\right )} {\left (b x^{2} + a\right )}} - \frac {{\left (4 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5} c^{4}} + \frac {44 \, b^{3} c^{3} x^{6} + 33 \, a b^{2} c^{2} d x^{6} + 22 \, a^{2} b c d^{2} x^{6} + 11 \, a^{3} d^{3} x^{6} - 18 \, a b^{2} c^{3} x^{4} - 12 \, a^{2} b c^{2} d x^{4} - 6 \, a^{3} c d^{2} x^{4} + 6 \, a^{2} b c^{3} x^{2} + 3 \, a^{3} c^{2} d x^{2} - 2 \, a^{3} c^{3}}{12 \, a^{5} c^{4} x^{6}} \]
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Time = 6.02 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.32 \[ \int \frac {1}{x^7 \left (a+b x^2\right )^2 \left (c+d x^2\right )} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (4\,b^5\,c-5\,a\,b^4\,d\right )}{2\,a^7\,d^2-4\,a^6\,b\,c\,d+2\,a^5\,b^2\,c^2}-\frac {\frac {1}{6\,a\,c}-\frac {x^2\,\left (3\,a\,d+4\,b\,c\right )}{12\,a^2\,c^2}+\frac {x^4\,\left (2\,a^2\,d^2+3\,a\,b\,c\,d+4\,b^2\,c^2\right )}{4\,a^3\,c^3}+\frac {x^6\,\left (a^3\,b\,d^3+a^2\,b^2\,c\,d^2+a\,b^3\,c^2\,d-4\,b^4\,c^3\right )}{2\,a^4\,c^3\,\left (a\,d-b\,c\right )}}{b\,x^8+a\,x^6}+\frac {d^5\,\ln \left (d\,x^2+c\right )}{2\,\left (a^2\,c^4\,d^2-2\,a\,b\,c^5\,d+b^2\,c^6\right )}-\frac {\ln \left (x\right )\,\left (a^3\,d^3+2\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+4\,b^3\,c^3\right )}{a^5\,c^4} \]
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