Optimal. Leaf size=61 \[ -\frac{b^2 p x}{3 a^2}+\frac{b^3 p \log (a x+b)}{3 a^3}+\frac{1}{3} x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^2}{6 a} \]
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Rubi [A] time = 0.0316943, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2455, 263, 43} \[ -\frac{b^2 p x}{3 a^2}+\frac{b^3 p \log (a x+b)}{3 a^3}+\frac{1}{3} x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^2}{6 a} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 263
Rule 43
Rubi steps
\begin{align*} \int x^2 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=\frac{1}{3} x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{3} (b p) \int \frac{x}{a+\frac{b}{x}} \, dx\\ &=\frac{1}{3} x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{3} (b p) \int \frac{x^2}{b+a x} \, dx\\ &=\frac{1}{3} x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{3} (b p) \int \left (-\frac{b}{a^2}+\frac{x}{a}+\frac{b^2}{a^2 (b+a x)}\right ) \, dx\\ &=-\frac{b^2 p x}{3 a^2}+\frac{b p x^2}{6 a}+\frac{1}{3} x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b^3 p \log (b+a x)}{3 a^3}\\ \end{align*}
Mathematica [A] time = 0.0240148, size = 62, normalized size = 1.02 \[ \frac{2 a^3 x^3 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+2 b^3 p \log \left (a+\frac{b}{x}\right )+a b p x (a x-2 b)+2 b^3 p \log (x)}{6 a^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\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 [A] time = 1.20513, size = 69, normalized size = 1.13 \begin{align*} \frac{1}{3} \, x^{3} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) + \frac{1}{6} \, b p{\left (\frac{2 \, b^{2} \log \left (a x + b\right )}{a^{3}} + \frac{a x^{2} - 2 \, b x}{a^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14907, size = 149, normalized size = 2.44 \begin{align*} \frac{2 \, a^{3} p x^{3} \log \left (\frac{a x + b}{x}\right ) + 2 \, a^{3} x^{3} \log \left (c\right ) + a^{2} b p x^{2} - 2 \, a b^{2} p x + 2 \, b^{3} p \log \left (a x + b\right )}{6 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.51316, size = 95, normalized size = 1.56 \begin{align*} \begin{cases} \frac{p x^{3} \log{\left (a + \frac{b}{x} \right )}}{3} + \frac{x^{3} \log{\left (c \right )}}{3} + \frac{b p x^{2}}{6 a} - \frac{b^{2} p x}{3 a^{2}} + \frac{b^{3} p \log{\left (a x + b \right )}}{3 a^{3}} & \text{for}\: a \neq 0 \\\frac{p x^{3} \log{\left (b \right )}}{3} - \frac{p x^{3} \log{\left (x \right )}}{3} + \frac{p x^{3}}{9} + \frac{x^{3} \log{\left (c \right )}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28979, size = 85, normalized size = 1.39 \begin{align*} \frac{1}{3} \, p x^{3} \log \left (a x + b\right ) - \frac{1}{3} \, p x^{3} \log \left (x\right ) + \frac{1}{3} \, x^{3} \log \left (c\right ) + \frac{b p x^{2}}{6 \, a} - \frac{b^{2} p x}{3 \, a^{2}} + \frac{b^{3} p \log \left (a x + b\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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