Integrand size = 23, antiderivative size = 306 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d^2} \]
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Time = 0.23 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2516, 2505, 211, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e p \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2} \]
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Rule 211
Rule 266
Rule 2352
Rule 2438
Rule 2440
Rule 2441
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^2}-\frac {e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}+\frac {e^2 \log \left (c \left (a+b x^2\right )^p\right )}{d^2 (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx}{d}-\frac {e \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx}{d^2} \\ & = -\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x} \, dx,x,x^2\right )}{2 d^2}+\frac {(2 b p) \int \frac {1}{a+b x^2} \, dx}{d}-\frac {(2 b e p) \int \frac {x \log (d+e x)}{a+b x^2} \, dx}{d^2} \\ & = \frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}+\frac {(b e p) \text {Subst}\left (\int \frac {\log \left (-\frac {b x}{a}\right )}{a+b x} \, dx,x,x^2\right )}{2 d^2}-\frac {(2 b e p) \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^2} \\ & = \frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^2}+\frac {\left (\sqrt {b} e p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{d^2}-\frac {\left (\sqrt {b} e p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{d^2} \\ & = \frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^2}+\frac {\left (e^2 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^2}+\frac {\left (e^2 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^2} \\ & = \frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^2}+\frac {(e p) \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^2}+\frac {(e p) \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^2} \\ & = \frac {2 \sqrt {b} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^2}-\frac {e p \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 d^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=-\frac {-\frac {4 \sqrt {b} d p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+2 e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+2 e p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+\frac {2 d \log \left (c \left (a+b x^2\right )^p\right )}{x}+e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )-2 e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+e p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d^2} \]
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Time = 0.94 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.16
method | result | size |
parts | \(\frac {e \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d^{2}}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d x}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-2 p b \left (\frac {e \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{d^{2}}-\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d \sqrt {a b}}-\frac {e \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 b}\right )}{d^{2}}\right )\) | \(354\) |
risch | \(\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{d x}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-\frac {p e \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{2}}-\frac {p e \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{2}}-\frac {p e \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{2}}-\frac {p e \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{2}}+\frac {2 p b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d \sqrt {a b}}+\frac {p e \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\) | \(514\) |
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{x^2\,\left (d+e\,x\right )} \,d x \]
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