Optimal. Leaf size=295 \[ -\frac{2^{-p-1} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2}-\frac{e 2^{-p-2} r e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+2,\frac{2 a}{b n}+\frac{2 \log \left (c x^n\right )}{n}\right )}{x^2}+\frac{e 2^{-p-1} r e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 a}{b n}+\frac{2 \log \left (c x^n\right )}{n}\right )}{b n x^2} \]
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Rubi [A] time = 0.234354, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2310, 2181, 2366, 12, 15, 19, 6557} \[ -\frac{2^{-p-1} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x^2}-\frac{e 2^{-p-2} r e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+2,\frac{2 a}{b n}+\frac{2 \log \left (c x^n\right )}{n}\right )}{x^2}+\frac{e 2^{-p-1} r e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,\frac{2 a}{b n}+\frac{2 \log \left (c x^n\right )}{n}\right )}{b n x^2} \]
Antiderivative was successfully verified.
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Rule 2310
Rule 2181
Rule 2366
Rule 12
Rule 15
Rule 19
Rule 6557
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx &=-\frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2}+(e r) \int \frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^3} \, dx\\ &=-\frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2}+\left (2^{-1-p} e e^{\frac{2 a}{b n}} r\right ) \int \frac{\left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^3} \, dx\\ &=-\frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2}+\frac{\left (2^{-1-p} e e^{\frac{2 a}{b n}} r \left (c x^n\right )^{2/n}\right ) \int \frac{\Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x} \, dx}{x^2}\\ &=-\frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2}+\frac{\left (2^{-1-p} e e^{\frac{2 a}{b n}} r \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}\right ) \int \frac{\Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x} \, dx}{x^2}\\ &=-\frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2}+\frac{\left (2^{-1-p} e e^{\frac{2 a}{b n}} r \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}\right ) \operatorname{Subst}\left (\int \Gamma \left (1+p,\frac{2 (a+b x)}{b n}\right ) \, dx,x,\log \left (c x^n\right )\right )}{n x^2}\\ &=-\frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2}+\frac{\left (2^{-2-p} e e^{\frac{2 a}{b n}} r \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}\right ) \operatorname{Subst}\left (\int \Gamma (1+p,x) \, dx,x,\frac{2 a}{b n}+\frac{2 \log \left (c x^n\right )}{n}\right )}{x^2}\\ &=-\frac{2^{-2-p} e e^{\frac{2 a}{b n}} r \left (c x^n\right )^{2/n} \Gamma \left (2+p,\frac{2 a}{b n}+\frac{2 \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^2}+\frac{2^{-1-p} e e^{\frac{2 a}{b n}} r \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 a}{b n}+\frac{2 \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (\frac{a}{b n}+\frac{\log \left (c x^n\right )}{n}\right )}{x^2}-\frac{2^{-1-p} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2}\\ \end{align*}
Mathematica [A] time = 0.375355, size = 154, normalized size = 0.52 \[ -\frac{2^{-p-2} e^{\frac{2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^{p-1} \left (\frac{a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (2 \text{Gamma}\left (p+1,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-a e r-b e r \log \left (c x^n\right )+b d n+b e n \log \left (f x^r\right )\right )+b e n r \text{Gamma}\left (p+2,\frac{2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )}{x^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.215, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p} \left ( d+e\ln \left ( f{x}^{r} \right ) \right ) }{{x}^{3}}}\, 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 (e \log \left (f x^{r}\right ) + d\right )}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{3}}, 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{{\left (e \log \left (f x^{r}\right ) + d\right )}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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