Integrand size = 27, antiderivative size = 344 \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {b d e \sqrt {f} n \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{3/2} \left (e^2 f+d^2 g\right )}-\frac {b e^2 f n \log (d+e x)}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}+\frac {b e^2 f n \log \left (f+g x^2\right )}{4 g^2 \left (e^2 f+d^2 g\right )}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2} \]
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Time = 0.31 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {272, 45, 2463, 2460, 720, 31, 649, 211, 266, 2441, 2440, 2438} \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}-\frac {b d e \sqrt {f} n \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{3/2} \left (d^2 g+e^2 f\right )}+\frac {b e^2 f n \log \left (f+g x^2\right )}{4 g^2 \left (d^2 g+e^2 f\right )}-\frac {b e^2 f n \log (d+e x)}{2 g^2 \left (d^2 g+e^2 f\right )}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 g^2} \]
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Rule 31
Rule 45
Rule 211
Rule 266
Rule 272
Rule 649
Rule 720
Rule 2438
Rule 2440
Rule 2441
Rule 2460
Rule 2463
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {f x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \left (f+g x^2\right )^2}+\frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \left (f+g x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx}{g}-\frac {f \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx}{g} \\ & = \frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\int \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {a+b \log \left (c (d+e x)^n\right )}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{g}-\frac {(b e f n) \int \frac {1}{(d+e x) \left (f+g x^2\right )} \, dx}{2 g^2} \\ & = \frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}-\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 g^{3/2}}+\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 g^{3/2}}-\frac {(b e f n) \int \frac {d g-e g x}{f+g x^2} \, dx}{2 g^2 \left (e^2 f+d^2 g\right )}-\frac {\left (b e^3 f n\right ) \int \frac {1}{d+e x} \, dx}{2 g^2 \left (e^2 f+d^2 g\right )} \\ & = -\frac {b e^2 f n \log (d+e x)}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}-\frac {(b e n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 g^2}-\frac {(b e n) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 g^2}-\frac {(b d e f n) \int \frac {1}{f+g x^2} \, dx}{2 g \left (e^2 f+d^2 g\right )}+\frac {\left (b e^2 f n\right ) \int \frac {x}{f+g x^2} \, dx}{2 g \left (e^2 f+d^2 g\right )} \\ & = -\frac {b d e \sqrt {f} n \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{3/2} \left (e^2 f+d^2 g\right )}-\frac {b e^2 f n \log (d+e x)}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}+\frac {b e^2 f n \log \left (f+g x^2\right )}{4 g^2 \left (e^2 f+d^2 g\right )}-\frac {(b n) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 g^2}-\frac {(b n) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 g^2} \\ & = -\frac {b d e \sqrt {f} n \tan ^{-1}\left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{2 g^{3/2} \left (e^2 f+d^2 g\right )}-\frac {b e^2 f n \log (d+e x)}{2 g^2 \left (e^2 f+d^2 g\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2 \left (f+g x^2\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}+\frac {b e^2 f n \log \left (f+g x^2\right )}{4 g^2 \left (e^2 f+d^2 g\right )}+\frac {b n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^2}+\frac {b n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.32 \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\frac {2 f \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2}+2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \log \left (f+g x^2\right )+b n \left (\frac {\sqrt {f} \left (-i \sqrt {g} (d+e x) \log (d+e x)+e \left (\sqrt {f}+i \sqrt {g} x\right ) \log \left (i \sqrt {f}-\sqrt {g} x\right )\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}+\frac {\sqrt {f} \left (i \sqrt {g} (d+e x) \log (d+e x)+e \left (\sqrt {f}-i \sqrt {g} x\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+2 \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}+i \sqrt {g} x\right )}{e \sqrt {f}-i d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )\right )+2 \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{e \sqrt {f}+i d \sqrt {g}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )\right )\right )}{4 g^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.18 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.44
method | result | size |
risch | \(\frac {b \ln \left (\left (e x +d \right )^{n}\right ) f}{2 g^{2} \left (g \,x^{2}+f \right )}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g \,x^{2}+f \right )}{2 g^{2}}-\frac {b n \ln \left (e x +d \right ) \ln \left (g \,x^{2}+f \right )}{2 g^{2}}+\frac {b n \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 g^{2}}+\frac {b n \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 g^{2}}+\frac {b n \operatorname {dilog}\left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 g^{2}}+\frac {b n \operatorname {dilog}\left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 g^{2}}-\frac {b \,e^{2} f n \ln \left (e x +d \right )}{2 g^{2} \left (d^{2} g +f \,e^{2}\right )}+\frac {b \,e^{2} f n \ln \left (g \,x^{2}+f \right )}{4 g^{2} \left (d^{2} g +f \,e^{2}\right )}-\frac {b e n f d \arctan \left (\frac {g x}{\sqrt {f g}}\right )}{2 g \left (d^{2} g +f \,e^{2}\right ) \sqrt {f g}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{2}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{2}+b \ln \left (c \right )+a \right ) \left (\frac {f}{2 g^{2} \left (g \,x^{2}+f \right )}+\frac {\ln \left (g \,x^{2}+f \right )}{2 g^{2}}\right )\) | \(496\) |
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\[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{3}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{3}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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\[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{3}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]
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