Integrand size = 25, antiderivative size = 139 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {b d e n \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g} \left (e^2 f+d^2 g\right )}+\frac {b e^2 n \log (d+e x)}{2 g \left (e^2 f+d^2 g\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g \left (f+g x^2\right )}-\frac {b e^2 n \log \left (f+g x^2\right )}{4 g \left (e^2 f+d^2 g\right )} \]
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Time = 0.06 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2460, 720, 31, 649, 211, 266} \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g \left (f+g x^2\right )}+\frac {b d e n \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g} \left (d^2 g+e^2 f\right )}-\frac {b e^2 n \log \left (f+g x^2\right )}{4 g \left (d^2 g+e^2 f\right )}+\frac {b e^2 n \log (d+e x)}{2 g \left (d^2 g+e^2 f\right )} \]
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Rule 31
Rule 211
Rule 266
Rule 649
Rule 720
Rule 2460
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \log \left (c (d+e x)^n\right )}{2 g \left (f+g x^2\right )}+\frac {(b e n) \int \frac {1}{(d+e x) \left (f+g x^2\right )} \, dx}{2 g} \\ & = -\frac {a+b \log \left (c (d+e x)^n\right )}{2 g \left (f+g x^2\right )}+\frac {(b e n) \int \frac {d g-e g x}{f+g x^2} \, dx}{2 g \left (e^2 f+d^2 g\right )}+\frac {\left (b e^3 n\right ) \int \frac {1}{d+e x} \, dx}{2 g \left (e^2 f+d^2 g\right )} \\ & = \frac {b e^2 n \log (d+e x)}{2 g \left (e^2 f+d^2 g\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g \left (f+g x^2\right )}+\frac {(b d e n) \int \frac {1}{f+g x^2} \, dx}{2 \left (e^2 f+d^2 g\right )}-\frac {\left (b e^2 n\right ) \int \frac {x}{f+g x^2} \, dx}{2 \left (e^2 f+d^2 g\right )} \\ & = \frac {b d e n \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 \sqrt {f} \sqrt {g} \left (e^2 f+d^2 g\right )}+\frac {b e^2 n \log (d+e x)}{2 g \left (e^2 f+d^2 g\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g \left (f+g x^2\right )}-\frac {b e^2 n \log \left (f+g x^2\right )}{4 g \left (e^2 f+d^2 g\right )} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.19 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {2 b d e \sqrt {g} n \left (f+g x^2\right ) \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )-\sqrt {f} \left (2 a e^2 f+2 a d^2 g-2 b e^2 n \left (f+g x^2\right ) \log (d+e x)+2 b \left (e^2 f+d^2 g\right ) \log \left (c (d+e x)^n\right )+b e^2 f n \log \left (f+g x^2\right )+b e^2 g n x^2 \log \left (f+g x^2\right )\right )}{4 \sqrt {f} g \left (e^2 f+d^2 g\right ) \left (f+g x^2\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.36 (sec) , antiderivative size = 969, normalized size of antiderivative = 6.97
method | result | size |
risch | \(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{2 g \left (g \,x^{2}+f \right )}-\frac {i \pi b \,e^{2} f^{2} \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,e^{2} f^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i f g \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i f g \pi b \,d^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-i f g \pi b \,d^{2} \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i f g \pi b \,d^{2} \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-i \pi b \,e^{2} f^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b \,e^{2} f^{2} \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-\ln \left (\left (-\sqrt {-f g}\, d^{2} g +3 \sqrt {-f g}\, e^{2} f +4 d e f g \right ) x +4 \sqrt {-f g}\, d e f +d^{2} f g -3 e^{2} f^{2}\right ) \sqrt {-f g}\, b d e g n \,x^{2}+\ln \left (\left (\sqrt {-f g}\, d^{2} g -3 \sqrt {-f g}\, e^{2} f +4 d e f g \right ) x -4 \sqrt {-f g}\, d e f +d^{2} f g -3 e^{2} f^{2}\right ) \sqrt {-f g}\, b d e g n \,x^{2}+\ln \left (\left (-\sqrt {-f g}\, d^{2} g +3 \sqrt {-f g}\, e^{2} f +4 d e f g \right ) x +4 \sqrt {-f g}\, d e f +d^{2} f g -3 e^{2} f^{2}\right ) b \,e^{2} f g n \,x^{2}+\ln \left (\left (\sqrt {-f g}\, d^{2} g -3 \sqrt {-f g}\, e^{2} f +4 d e f g \right ) x -4 \sqrt {-f g}\, d e f +d^{2} f g -3 e^{2} f^{2}\right ) b \,e^{2} f g n \,x^{2}-2 \ln \left (e x +d \right ) b \,e^{2} f g n \,x^{2}-\ln \left (\left (-\sqrt {-f g}\, d^{2} g +3 \sqrt {-f g}\, e^{2} f +4 d e f g \right ) x +4 \sqrt {-f g}\, d e f +d^{2} f g -3 e^{2} f^{2}\right ) \sqrt {-f g}\, b d e f n +\ln \left (\left (\sqrt {-f g}\, d^{2} g -3 \sqrt {-f g}\, e^{2} f +4 d e f g \right ) x -4 \sqrt {-f g}\, d e f +d^{2} f g -3 e^{2} f^{2}\right ) \sqrt {-f g}\, b d e f n +\ln \left (\left (-\sqrt {-f g}\, d^{2} g +3 \sqrt {-f g}\, e^{2} f +4 d e f g \right ) x +4 \sqrt {-f g}\, d e f +d^{2} f g -3 e^{2} f^{2}\right ) b \,e^{2} f^{2} n +\ln \left (\left (\sqrt {-f g}\, d^{2} g -3 \sqrt {-f g}\, e^{2} f +4 d e f g \right ) x -4 \sqrt {-f g}\, d e f +d^{2} f g -3 e^{2} f^{2}\right ) b \,e^{2} f^{2} n -2 b \,e^{2} f^{2} n \ln \left (e x +d \right )+2 \ln \left (c \right ) b \,d^{2} f g +2 \ln \left (c \right ) b \,e^{2} f^{2}+2 a \,d^{2} f g +2 a \,e^{2} f^{2}}{4 f \left (g \,x^{2}+f \right ) \left (d g -e \sqrt {-f g}\right ) \left (e \sqrt {-f g}+d g \right )}\) | \(969\) |
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Time = 0.36 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.68 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\left [-\frac {2 \, a e^{2} f^{2} + 2 \, a d^{2} f g + {\left (b d e g n x^{2} + b d e f n\right )} \sqrt {-f g} \log \left (\frac {g x^{2} - 2 \, \sqrt {-f g} x - f}{g x^{2} + f}\right ) + {\left (b e^{2} f g n x^{2} + b e^{2} f^{2} n\right )} \log \left (g x^{2} + f\right ) - 2 \, {\left (b e^{2} f g n x^{2} - b d^{2} f g n\right )} \log \left (e x + d\right ) + 2 \, {\left (b e^{2} f^{2} + b d^{2} f g\right )} \log \left (c\right )}{4 \, {\left (e^{2} f^{3} g + d^{2} f^{2} g^{2} + {\left (e^{2} f^{2} g^{2} + d^{2} f g^{3}\right )} x^{2}\right )}}, -\frac {2 \, a e^{2} f^{2} + 2 \, a d^{2} f g - 2 \, {\left (b d e g n x^{2} + b d e f n\right )} \sqrt {f g} \arctan \left (\frac {\sqrt {f g} x}{f}\right ) + {\left (b e^{2} f g n x^{2} + b e^{2} f^{2} n\right )} \log \left (g x^{2} + f\right ) - 2 \, {\left (b e^{2} f g n x^{2} - b d^{2} f g n\right )} \log \left (e x + d\right ) + 2 \, {\left (b e^{2} f^{2} + b d^{2} f g\right )} \log \left (c\right )}{4 \, {\left (e^{2} f^{3} g + d^{2} f^{2} g^{2} + {\left (e^{2} f^{2} g^{2} + d^{2} f g^{3}\right )} x^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {1}{4} \, b e n {\left (\frac {e \log \left (g x^{2} + f\right )}{e^{2} f g + d^{2} g^{2}} - \frac {2 \, e \log \left (e x + d\right )}{e^{2} f g + d^{2} g^{2}} - \frac {2 \, d \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{{\left (e^{2} f + d^{2} g\right )} \sqrt {f g}}\right )} - \frac {b \log \left ({\left (e x + d\right )}^{n} c\right )}{2 \, {\left (g^{2} x^{2} + f g\right )}} - \frac {a}{2 \, {\left (g^{2} x^{2} + f g\right )}} \]
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Time = 0.32 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.40 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {b e^{2} n \log \left (g x^{2} + f\right )}{4 \, {\left (e^{2} f g + d^{2} g^{2}\right )}} + \frac {b e^{2} n \log \left (e x + d\right )}{2 \, {\left (e^{2} f g + d^{2} g^{2}\right )}} + \frac {b d e n \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{2 \, {\left (e^{2} f + d^{2} g\right )} \sqrt {f g}} - \frac {b n \log \left (e x + d\right )}{2 \, {\left (g^{2} x^{2} + f g\right )}} - \frac {b \log \left (c\right ) + a}{g^{2} x^{2} + f g} - \frac {b e^{2} f \log \left (c\right ) + b d^{2} g \log \left (c\right ) + a e^{2} f + a d^{2} g}{2 \, {\left (e^{2} f + d^{2} g\right )} {\left (g x^{2} + f\right )} g} \]
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Time = 1.84 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.63 \[ \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {b\,e^2\,n\,\ln \left (d+e\,x\right )}{2\,d^2\,g^2+2\,f\,e^2\,g}-\frac {\ln \left (\frac {\left (b\,e^2\,f\,g\,n+b\,d\,e\,n\,\sqrt {-f\,g^3}\right )\,\left (x\,\left (2\,d^2\,e\,g^3-6\,e^3\,f\,g^2\right )-8\,d\,e^2\,f\,g^2\right )}{4\,\left (d^2\,f\,g^3+e^2\,f^2\,g^2\right )}+\frac {b\,d\,e^2\,g\,n}{2}+\frac {3\,b\,e^3\,g\,n\,x}{2}\right )\,\left (b\,e^2\,f\,g\,n+b\,d\,e\,n\,\sqrt {-f\,g^3}\right )}{4\,\left (d^2\,f\,g^3+e^2\,f^2\,g^2\right )}-\frac {\ln \left (\frac {\left (b\,e^2\,f\,g\,n-b\,d\,e\,n\,\sqrt {-f\,g^3}\right )\,\left (x\,\left (2\,d^2\,e\,g^3-6\,e^3\,f\,g^2\right )-8\,d\,e^2\,f\,g^2\right )}{4\,\left (d^2\,f\,g^3+e^2\,f^2\,g^2\right )}+\frac {b\,d\,e^2\,g\,n}{2}+\frac {3\,b\,e^3\,g\,n\,x}{2}\right )\,\left (b\,e^2\,f\,g\,n-b\,d\,e\,n\,\sqrt {-f\,g^3}\right )}{4\,\left (d^2\,f\,g^3+e^2\,f^2\,g^2\right )}-\frac {b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{2\,g\,\left (g\,x^2+f\right )}-\frac {a}{2\,g^2\,x^2+2\,f\,g} \]
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