Integrand size = 20, antiderivative size = 201 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=-\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e} \]
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Time = 0.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2512, 266, 2463, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e}-\frac {p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{e}-\frac {p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{e} \]
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Rule 266
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rule 2512
Rubi steps \begin{align*} \text {integral}& = \frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {(2 b p) \int \frac {x \log (d+e x)}{a+b x^2} \, dx}{e} \\ & = \frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {(2 b 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}{e} \\ & = \frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}+\frac {\left (\sqrt {b} p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{e}-\frac {\left (\sqrt {b} p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{e} \\ & = -\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}+p \int \frac {\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right )}{d+e x} \, dx+p \int \frac {\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right )}{d+e x} \, dx \\ & = -\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}+\frac {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 )}{e}+\frac {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 )}{e} \\ & = -\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e}-\frac {p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=-\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e} \]
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Time = 1.19 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.98
method | result | size |
parts | \(\frac {\ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e}-\frac {2 p b \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{e}\) | \(196\) |
risch | \(\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (e x +d \right )}{e}-\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e}-\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (e x +d \right )}{e}\) | \(331\) |
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{d + e x}\, dx \]
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \]
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\[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int \frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{d+e\,x} \,d x \]
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