Optimal. Leaf size=107 \[ -\frac{e^{\frac{a}{b m n}} \left (c \left (d x^m\right )^n\right )^{\frac{1}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (\frac{a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )}{x} \]
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Rubi [A] time = 0.159549, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2310, 2181, 2445} \[ -\frac{e^{\frac{a}{b m n}} \left (c \left (d x^m\right )^n\right )^{\frac{1}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (\frac{a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2181
Rule 2445
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^2} \, dx &=\operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c d^n x^{m n}\right )\right )^p}{x^2} \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\left (c d^n x^{m n}\right )^{\frac{1}{m n}} \operatorname{Subst}\left (\int e^{-\frac{x}{m n}} (a+b x)^p \, dx,x,\log \left (c d^n x^{m n}\right )\right )}{m n x},c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=-\frac{e^{\frac{a}{b m n}} \left (c \left (d x^m\right )^n\right )^{\frac{1}{m n}} \Gamma \left (1+p,\frac{a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (\frac{a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p}}{x}\\ \end{align*}
Mathematica [A] time = 0.128186, size = 107, normalized size = 1. \[ -\frac{e^{\frac{a}{b m n}} \left (c \left (d x^m\right )^n\right )^{\frac{1}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (\frac{a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p} \text{Gamma}\left (p+1,\frac{a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.048, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( d{x}^{m} \right ) ^{n} \right ) \right ) ^{p}}{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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{{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d x^{m}\right )^{n} \right )}\right )^{p}}{x^{2}}\, dx \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{{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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