Optimal. Leaf size=256 \[ -\frac {d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}+\frac {d p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^2}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e}-\frac {2 p x}{e} \]
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Rubi [A] time = 0.27, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2466, 2448, 321, 205, 2462, 260, 2416, 2394, 2393, 2391} \[ \frac {d p \text {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^2}+\frac {d p \text {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {-a} e+\sqrt {b} d}\right )}{e^2}-\frac {d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}+\frac {d p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^2}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e}-\frac {2 p x}{e} \]
Antiderivative was successfully verified.
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Rule 205
Rule 260
Rule 321
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rule 2448
Rule 2462
Rule 2466
Rubi steps
\begin {align*} \int \frac {x \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx &=\int \left (\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {d \log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}\right ) \, dx\\ &=\frac {\int \log \left (c \left (a+b x^2\right )^p\right ) \, dx}{e}-\frac {d \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx}{e}\\ &=\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {(2 b d p) \int \frac {x \log (d+e x)}{a+b x^2} \, dx}{e^2}-\frac {(2 b p) \int \frac {x^2}{a+b x^2} \, dx}{e}\\ &=-\frac {2 p x}{e}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {(2 b d 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^2}+\frac {(2 a p) \int \frac {1}{a+b x^2} \, dx}{e}\\ &=-\frac {2 p x}{e}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^2}-\frac {\left (\sqrt {b} d p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}-\sqrt {b} x} \, dx}{e^2}+\frac {\left (\sqrt {b} d p\right ) \int \frac {\log (d+e x)}{\sqrt {-a}+\sqrt {b} x} \, dx}{e^2}\\ &=-\frac {2 p x}{e}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e}+\frac {d p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^2}+\frac {d p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^2}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^2}-\frac {(d 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}{e}-\frac {(d 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}{e}\\ &=-\frac {2 p x}{e}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e}+\frac {d p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^2}+\frac {d p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^2}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^2}-\frac {(d p) \operatorname {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^2}-\frac {(d p) \operatorname {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^2}\\ &=-\frac {2 p x}{e}+\frac {2 \sqrt {a} p \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e}+\frac {d p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^2}+\frac {d p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^2}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^2}+\frac {d p \text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 225, normalized size = 0.88 \[ \frac {-d \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+e x \log \left (c \left (a+b x^2\right )^p\right )+d p \left (\text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+\text {Li}_2\left (\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+\log (d+e x) \left (\log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )+\log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {-a} e-\sqrt {b} d}\right )\right )\right )-2 e p \left (x-\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}\right )}{e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.31, size = 576, normalized size = 2.25 \[ \frac {i \pi d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi d \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 e^{2}}+\frac {i \pi d \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3} \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}{2 e}+\frac {i \pi x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{2 e}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}}{2 e}-\frac {i \pi x \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}}{2 e}+\frac {2 a p \arctan \left (\frac {-2 b d +2 \left (e x +d \right ) b}{2 \sqrt {a b}\, e}\right )}{\sqrt {a b}\, e}+\frac {d p \ln \left (\frac {b d -\left (e x +d \right ) b +\sqrt {-a b}\, e}{b d +\sqrt {-a b}\, e}\right ) \ln \left (e x +d \right )}{e^{2}}+\frac {d p \ln \left (\frac {-b d +\left (e x +d \right ) b +\sqrt {-a b}\, e}{-b d +\sqrt {-a b}\, e}\right ) \ln \left (e x +d \right )}{e^{2}}+\frac {d p \dilog \left (\frac {b d -\left (e x +d \right ) b +\sqrt {-a b}\, e}{b d +\sqrt {-a b}\, e}\right )}{e^{2}}+\frac {d p \dilog \left (\frac {-b d +\left (e x +d \right ) b +\sqrt {-a b}\, e}{-b d +\sqrt {-a b}\, e}\right )}{e^{2}}-\frac {d \ln \relax (c ) \ln \left (e x +d \right )}{e^{2}}-\frac {d \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (e x +d \right )}{e^{2}}-\frac {2 p x}{e}+\frac {x \ln \relax (c )}{e}+\frac {x \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{e}-\frac {2 d p}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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