Optimal. Leaf size=159 \[ -\frac {a^2 p \log (a+b x)}{2 b^2 e}+\frac {d^2 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^3}-\frac {d (a+b x) \log \left (c (a+b x)^p\right )}{b e^2}+\frac {x^2 \log \left (c (a+b x)^p\right )}{2 e}+\frac {d^2 p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e^3}+\frac {a p x}{2 b e}+\frac {d p x}{e^2}-\frac {p x^2}{4 e} \]
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Rubi [A] time = 0.17, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {43, 2416, 2389, 2295, 2395, 2394, 2393, 2391} \[ \frac {d^2 p \text {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^3}-\frac {a^2 p \log (a+b x)}{2 b^2 e}+\frac {d^2 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^3}-\frac {d (a+b x) \log \left (c (a+b x)^p\right )}{b e^2}+\frac {x^2 \log \left (c (a+b x)^p\right )}{2 e}+\frac {a p x}{2 b e}+\frac {d p x}{e^2}-\frac {p x^2}{4 e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rubi steps
\begin {align*} \int \frac {x^2 \log \left (c (a+b x)^p\right )}{d+e x} \, dx &=\int \left (-\frac {d \log \left (c (a+b x)^p\right )}{e^2}+\frac {x \log \left (c (a+b x)^p\right )}{e}+\frac {d^2 \log \left (c (a+b x)^p\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac {d \int \log \left (c (a+b x)^p\right ) \, dx}{e^2}+\frac {d^2 \int \frac {\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{e^2}+\frac {\int x \log \left (c (a+b x)^p\right ) \, dx}{e}\\ &=\frac {x^2 \log \left (c (a+b x)^p\right )}{2 e}+\frac {d^2 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^3}-\frac {d \operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x\right )}{b e^2}-\frac {\left (b d^2 p\right ) \int \frac {\log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{e^3}-\frac {(b p) \int \frac {x^2}{a+b x} \, dx}{2 e}\\ &=\frac {d p x}{e^2}+\frac {x^2 \log \left (c (a+b x)^p\right )}{2 e}-\frac {d (a+b x) \log \left (c (a+b x)^p\right )}{b e^2}+\frac {d^2 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^3}-\frac {\left (d^2 p\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{e^3}-\frac {(b p) \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{2 e}\\ &=\frac {d p x}{e^2}+\frac {a p x}{2 b e}-\frac {p x^2}{4 e}-\frac {a^2 p \log (a+b x)}{2 b^2 e}+\frac {x^2 \log \left (c (a+b x)^p\right )}{2 e}-\frac {d (a+b x) \log \left (c (a+b x)^p\right )}{b e^2}+\frac {d^2 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^3}+\frac {d^2 p \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 127, normalized size = 0.80 \[ \frac {-2 a^2 e^2 p \log (a+b x)+4 b^2 d^2 p \text {Li}_2\left (\frac {e (a+b x)}{a e-b d}\right )+b \log \left (c (a+b x)^p\right ) \left (4 b d^2 \log \left (\frac {b (d+e x)}{b d-a e}\right )-4 a d e+2 b e x (e x-2 d)\right )+b e p x (2 a e+4 b d-b e x)}{4 b^2 e^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}, x\right ) \]
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 \[ \int \frac {x^{2} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 666, normalized size = 4.19 \[ -\frac {p \,x^{2}}{4 e}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{4 e}+\frac {x^{2} \ln \relax (c )}{2 e}-\frac {a d p \ln \left (a e -b d +\left (e x +d \right ) b \right )}{b \,e^{2}}-\frac {a^{2} p \ln \left (a e -b d +\left (e x +d \right ) b \right )}{2 b^{2} e}+\frac {a d p}{2 b \,e^{2}}-\frac {d^{2} p \ln \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right ) \ln \left (e x +d \right )}{e^{3}}+\frac {5 d^{2} p}{4 e^{3}}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{4 e}+\frac {i \pi \,x^{2} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{4 e}+\frac {d p x}{e^{2}}-\frac {d x \ln \relax (c )}{e^{2}}+\frac {d^{2} \ln \relax (c ) \ln \left (e x +d \right )}{e^{3}}+\frac {x^{2} \ln \left (\left (b x +a \right )^{p}\right )}{2 e}-\frac {d^{2} p \dilog \left (\frac {a e -b d +\left (e x +d \right ) b}{a e -b d}\right )}{e^{3}}+\frac {d^{2} \ln \left (\left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{e^{3}}-\frac {d x \ln \left (\left (b x +a \right )^{p}\right )}{e^{2}}-\frac {i \pi \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right ) \ln \left (e x +d \right )}{2 e^{3}}+\frac {i \pi d x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )}{2 e^{2}}+\frac {i \pi d x \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2 e^{2}}-\frac {i \pi \,d^{2} \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3} \ln \left (e x +d \right )}{2 e^{3}}+\frac {a p x}{2 b e}+\frac {i \pi \,d^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 e^{3}}+\frac {i \pi \,d^{2} \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \ln \left (e x +d \right )}{2 e^{3}}-\frac {i \pi d x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2 e^{2}}-\frac {i \pi d x \,\mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2 e^{2}}-\frac {i \pi \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (b x +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b x +a \right )^{p}\right )}{4 e} \]
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 \[ \int \frac {x^{2} \log \left ({\left (b x + a\right )}^{p} c\right )}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{d+e\,x} \,d x \]
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 \[ \int \frac {x^{2} \log {\left (c \left (a + b x\right )^{p} \right )}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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