Optimal. Leaf size=89 \[ \frac{b^3 p x^2}{10 a^3}-\frac{b^2 p x^3}{15 a^2}-\frac{b^4 p x}{5 a^4}+\frac{b^5 p \log (a x+b)}{5 a^5}+\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^4}{20 a} \]
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Rubi [A] time = 0.0549788, antiderivative size = 89, 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} \[ \frac{b^3 p x^2}{10 a^3}-\frac{b^2 p x^3}{15 a^2}-\frac{b^4 p x}{5 a^4}+\frac{b^5 p \log (a x+b)}{5 a^5}+\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b p x^4}{20 a} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 263
Rule 43
Rubi steps
\begin{align*} \int x^4 \log \left (c \left (a+\frac{b}{x}\right )^p\right ) \, dx &=\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{5} (b p) \int \frac{x^3}{a+\frac{b}{x}} \, dx\\ &=\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{5} (b p) \int \frac{x^4}{b+a x} \, dx\\ &=\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{1}{5} (b p) \int \left (-\frac{b^3}{a^4}+\frac{b^2 x}{a^3}-\frac{b x^2}{a^2}+\frac{x^3}{a}+\frac{b^4}{a^4 (b+a x)}\right ) \, dx\\ &=-\frac{b^4 p x}{5 a^4}+\frac{b^3 p x^2}{10 a^3}-\frac{b^2 p x^3}{15 a^2}+\frac{b p x^4}{20 a}+\frac{1}{5} x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+\frac{b^5 p \log (b+a x)}{5 a^5}\\ \end{align*}
Mathematica [A] time = 0.0457851, size = 85, normalized size = 0.96 \[ \frac{a b p x \left (-4 a^2 b x^2+3 a^3 x^3+6 a b^2 x-12 b^3\right )+12 a^5 x^5 \log \left (c \left (a+\frac{b}{x}\right )^p\right )+12 b^5 p \log \left (a+\frac{b}{x}\right )+12 b^5 p \log (x)}{60 a^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.358, size = 0, normalized size = 0. \begin{align*} \int{x}^{4}\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 = 1.06121, size = 100, normalized size = 1.12 \begin{align*} \frac{1}{5} \, x^{5} \log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right ) + \frac{1}{60} \, b p{\left (\frac{12 \, b^{4} \log \left (a x + b\right )}{a^{5}} + \frac{3 \, a^{3} x^{4} - 4 \, a^{2} b x^{3} + 6 \, a b^{2} x^{2} - 12 \, b^{3} x}{a^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33376, size = 207, normalized size = 2.33 \begin{align*} \frac{12 \, a^{5} p x^{5} \log \left (\frac{a x + b}{x}\right ) + 12 \, a^{5} x^{5} \log \left (c\right ) + 3 \, a^{4} b p x^{4} - 4 \, a^{3} b^{2} p x^{3} + 6 \, a^{2} b^{3} p x^{2} - 12 \, a b^{4} p x + 12 \, b^{5} p \log \left (a x + b\right )}{60 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 34.2231, size = 122, normalized size = 1.37 \begin{align*} \begin{cases} \frac{p x^{5} \log{\left (a + \frac{b}{x} \right )}}{5} + \frac{x^{5} \log{\left (c \right )}}{5} + \frac{b p x^{4}}{20 a} - \frac{b^{2} p x^{3}}{15 a^{2}} + \frac{b^{3} p x^{2}}{10 a^{3}} - \frac{b^{4} p x}{5 a^{4}} + \frac{b^{5} p \log{\left (a x + b \right )}}{5 a^{5}} & \text{for}\: a \neq 0 \\\frac{p x^{5} \log{\left (b \right )}}{5} - \frac{p x^{5} \log{\left (x \right )}}{5} + \frac{p x^{5}}{25} + \frac{x^{5} \log{\left (c \right )}}{5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33013, size = 117, normalized size = 1.31 \begin{align*} \frac{1}{5} \, p x^{5} \log \left (a x + b\right ) - \frac{1}{5} \, p x^{5} \log \left (x\right ) + \frac{1}{5} \, x^{5} \log \left (c\right ) + \frac{b p x^{4}}{20 \, a} - \frac{b^{2} p x^{3}}{15 \, a^{2}} + \frac{b^{3} p x^{2}}{10 \, a^{3}} - \frac{b^{4} p x}{5 \, a^{4}} + \frac{b^{5} p \log \left (a x + b\right )}{5 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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