Optimal. Leaf size=219 \[ -\frac{d^2 p \text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )}{e^3}+\frac{d^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e^3}-\frac{b^2 p \log (a x+b)}{2 a^2 e}+\frac{d^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^3}-\frac{d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}-\frac{d^2 p \log (d+e x) \log \left (-\frac{e (a x+b)}{a d-b e}\right )}{e^3}-\frac{b d p \log (a x+b)}{a e^2}+\frac{b p x}{2 a e}+\frac{d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3} \]
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Rubi [A] time = 0.265552, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.609, Rules used = {2466, 2448, 263, 31, 2455, 193, 43, 2462, 260, 2416, 2394, 2315, 2393, 2391} \[ -\frac{d^2 p \text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )}{e^3}+\frac{d^2 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e^3}-\frac{b^2 p \log (a x+b)}{2 a^2 e}+\frac{d^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^3}-\frac{d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}-\frac{d^2 p \log (d+e x) \log \left (-\frac{e (a x+b)}{a d-b e}\right )}{e^3}-\frac{b d p \log (a x+b)}{a e^2}+\frac{b p x}{2 a e}+\frac{d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3} \]
Antiderivative was successfully verified.
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Rule 2466
Rule 2448
Rule 263
Rule 31
Rule 2455
Rule 193
Rule 43
Rule 2462
Rule 260
Rule 2416
Rule 2394
Rule 2315
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d+e x} \, dx &=\int \left (-\frac{d \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2 (d+e x)}\right ) \, dx\\ &=-\frac{d \int \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx}{e^2}+\frac{d^2 \int \frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{d+e x} \, dx}{e^2}+\frac{\int x \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx}{e}\\ &=-\frac{d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac{\left (b d^2 p\right ) \int \frac{\log (d+e x)}{\left (a+\frac{b}{x}\right ) x^2} \, dx}{e^3}-\frac{(b d p) \int \frac{1}{\left (a+\frac{b}{x}\right ) x} \, dx}{e^2}+\frac{(b p) \int \frac{1}{a+\frac{b}{x}} \, dx}{2 e}\\ &=-\frac{d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac{\left (b d^2 p\right ) \int \left (\frac{\log (d+e x)}{b x}-\frac{a \log (d+e x)}{b (b+a x)}\right ) \, dx}{e^3}-\frac{(b d p) \int \frac{1}{b+a x} \, dx}{e^2}+\frac{(b p) \int \frac{x}{b+a x} \, dx}{2 e}\\ &=-\frac{d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}-\frac{b d p \log (b+a x)}{a e^2}+\frac{d^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac{\left (d^2 p\right ) \int \frac{\log (d+e x)}{x} \, dx}{e^3}-\frac{\left (a d^2 p\right ) \int \frac{\log (d+e x)}{b+a x} \, dx}{e^3}+\frac{(b p) \int \left (\frac{1}{a}-\frac{b}{a (b+a x)}\right ) \, dx}{2 e}\\ &=\frac{b p x}{2 a e}-\frac{d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}-\frac{b d p \log (b+a x)}{a e^2}-\frac{b^2 p \log (b+a x)}{2 a^2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac{d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^3}-\frac{\left (d^2 p\right ) \int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx}{e^2}+\frac{\left (d^2 p\right ) \int \frac{\log \left (\frac{e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{e^2}\\ &=\frac{b p x}{2 a e}-\frac{d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}-\frac{b d p \log (b+a x)}{a e^2}-\frac{b^2 p \log (b+a x)}{2 a^2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac{d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^3}+\frac{d^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{e^3}+\frac{\left (d^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{e^3}\\ &=\frac{b p x}{2 a e}-\frac{d x \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )}{2 e}-\frac{b d p \log (b+a x)}{a e^2}-\frac{b^2 p \log (b+a x)}{2 a^2 e}+\frac{d^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \log (d+e x)}{e^3}+\frac{d^2 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \log \left (-\frac{e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^3}-\frac{d^2 p \text{Li}_2\left (\frac{a (d+e x)}{a d-b e}\right )}{e^3}+\frac{d^2 p \text{Li}_2\left (1+\frac{e x}{d}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.124978, size = 183, normalized size = 0.84 \[ \frac{2 d^2 p \left (-\text{PolyLog}\left (2,\frac{a (d+e x)}{a d-b e}\right )+\text{PolyLog}\left (2,\frac{e x}{d}+1\right )+\log (d+e x) \left (\log \left (-\frac{e x}{d}\right )-\log \left (\frac{e (a x+b)}{b e-a d}\right )\right )\right )+\frac{b e^2 p (a x-b \log (a x+b))}{a^2}+2 d^2 \log (d+e x) \log \left (c \left (a+\frac{b}{x}\right )^p\right )-2 d e x \log \left (c \left (a+\frac{b}{x}\right )^p\right )+e^2 x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right )-\frac{2 b d e p \left (\log \left (a+\frac{b}{x}\right )+\log (x)\right )}{a}}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.723, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{ex+d}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \log \left (c \left (\frac{a x + b}{x}\right )^{p}\right )}{e x + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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