Integrand size = 23, antiderivative size = 371 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {2 \sqrt {b} e p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d^3} \]
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Time = 0.27 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {2516, 2504, 2442, 36, 29, 31, 2505, 211, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=-\frac {2 \sqrt {b} e p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^3}+\frac {e^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}+\frac {b p \log (x)}{a d} \]
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Rule 29
Rule 31
Rule 36
Rule 211
Rule 266
Rule 2352
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rule 2504
Rule 2505
Rule 2512
Rule 2516
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x^3}-\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x^2}+\frac {e^2 \log \left (c \left (a+b x^2\right )^p\right )}{d^3 x}-\frac {e^3 \log \left (c \left (a+b x^2\right )^p\right )}{d^3 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x} \, dx}{d^3}-\frac {e^3 \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx}{d^3} \\ & = \frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {\text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^2} \, dx,x,x^2\right )}{2 d}+\frac {e^2 \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{2 d^3}-\frac {(2 b e p) \int \frac {1}{a+b x^2} \, dx}{d^2}+\frac {\left (2 b e^2 p\right ) \int \frac {x \log (d+e x)}{a+b x^2} \, dx}{d^3} \\ & = -\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {(b p) \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^2\right )}{2 d}-\frac {\left (b e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^2\right )}{2 d^3}+\frac {\left (2 b e^2 p\right ) \int \left (-\frac {\log (d+e x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x\right )}+\frac {\log (d+e x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x\right )}\right ) \, dx}{d^3} \\ & = -\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3}+\frac {(b p) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 a d}-\frac {\left (b^2 p\right ) \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^2\right )}{2 a d}-\frac {\left (\sqrt {b} e^2 p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{d^3}+\frac {\left (\sqrt {b} e^2 p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{d^3} \\ & = -\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right )}{d+e x} \, dx}{d^3}-\frac {\left (e^3 p\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right )}{d+e x} \, dx}{d^3} \\ & = -\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} d+\sqrt {-a} e}\right )}{x} \, dx,x,d+e x\right )}{d^3}-\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} d+\sqrt {-a} e}\right )}{x} \, dx,x,d+e x\right )}{d^3} \\ & = -\frac {2 \sqrt {b} e p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d^2}+\frac {b p \log (x)}{a d}+\frac {e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^3}+\frac {e^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^3}-\frac {b p \log \left (a+b x^2\right )}{2 a d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac {e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^3}+\frac {e^2 p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^3} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.94 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\frac {-\frac {4 \sqrt {b} d e p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {2 b d^2 p \log (x)}{a}+2 e^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+2 e^2 p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)-\frac {b d^2 p \log \left (a+b x^2\right )}{a}-\frac {d^2 \log \left (c \left (a+b x^2\right )^p\right )}{x^2}+\frac {2 d e \log \left (c \left (a+b x^2\right )^p\right )}{x}-2 e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+2 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+2 e^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+e^2 \left (\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )+p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )\right )}{2 d^3} \]
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Time = 1.40 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.11
method | result | size |
parts | \(-\frac {e^{2} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d^{3}}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {e \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d^{2} x}-p b \left (-\frac {\ln \left (x \right )}{d a}+\frac {\ln \left (b \,x^{2}+a \right )}{2 d a}+\frac {2 e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d^{2} \sqrt {a b}}+\frac {2 e^{2} \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 b}\right )}{d^{3}}-\frac {2 e^{2} \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{d^{3}}\right )\) | \(410\) |
risch | \(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{2 d \,x^{2}}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e}{d^{2} x}+\frac {b p \ln \left (x \right )}{a d}-\frac {b p \ln \left (b \,x^{2}+a \right )}{2 a d}-\frac {2 p b e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d^{2} \sqrt {a b}}-\frac {p \,e^{2} \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{3}}-\frac {p \,e^{2} \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{3}}-\frac {p \,e^{2} \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{3}}-\frac {p \,e^{2} \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{3}}+\frac {p \,e^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{3}}+\frac {p \,e^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{3}}+\frac {p \,e^{2} \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{3}}+\frac {p \,e^{2} \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{3}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}\right )\) | \(594\) |
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^3 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{x^3\,\left (d+e\,x\right )} \,d x \]
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