Optimal. Leaf size=271 \[ x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {a+b \log \left (c x^n\right )}{b n}\right )-e r x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+2,-\frac {a}{b n}-\frac {\log \left (c x^n\right )}{n}\right )-\frac {e r x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {a}{b n}-\frac {\log \left (c x^n\right )}{n}\right )}{b n} \]
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Rubi [A] time = 0.17, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2300, 2181, 2361, 12, 15, 19, 6557} \[ x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a+b \log \left (c x^n\right )}{b n}\right )-e r x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text {Gamma}\left (p+2,-\frac {a}{b n}-\frac {\log \left (c x^n\right )}{n}\right )-\frac {e r x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {a}{b n}-\frac {\log \left (c x^n\right )}{n}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 15
Rule 19
Rule 2181
Rule 2300
Rule 2361
Rule 6557
Rubi steps
\begin {align*} \int \left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right ) \, dx &=e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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 )-(e r) \int e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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} \, dx\\ &=e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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 )-\left (e e^{-\frac {a}{b n}} r\right ) \int \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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} \, dx\\ &=e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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 )-\left (e e^{-\frac {a}{b n}} r x \left (c x^n\right )^{-1/n}\right ) \int \frac {\Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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\\ &=e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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 )-\left (e e^{-\frac {a}{b n}} r x \left (c x^n\right )^{-1/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 {a+b \log \left (c x^n\right )}{b n}\right )}{x} \, dx\\ &=e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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 )-\frac {\left (e e^{-\frac {a}{b n}} r x \left (c x^n\right )^{-1/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 {a+b x}{b n}\right ) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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 )+\left (e e^{-\frac {a}{b n}} r x \left (c x^n\right )^{-1/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 {a}{b n}-\frac {\log \left (c x^n\right )}{n}\right )\\ &=-e e^{-\frac {a}{b n}} r x \left (c x^n\right )^{-1/n} \Gamma \left (2+p,-\frac {a}{b n}-\frac {\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}-e e^{-\frac {a}{b n}} r x \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a}{b n}-\frac {\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 )+e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \Gamma \left (1+p,-\frac {a+b \log \left (c x^n\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 )\\ \end {align*}
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Mathematica [A] time = 0.31, size = 146, normalized size = 0.54 \[ x \left (-e^{-\frac {a}{b n}}\right ) \left (c x^n\right )^{-1/n} \left (a+b \log \left (c x^n\right )\right )^{p-1} \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (\Gamma \left (p+1,-\frac {a+b \log \left (c x^n\right )}{b n}\right ) \left (-a e r-b e r \log \left (c x^n\right )+b d n+b e n \log \left (f x^r\right )\right )-b e n r \Gamma \left (p+2,-\frac {a+b \log \left (c x^n\right )}{b n}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 131, normalized size = 0.48 \[ -\frac {{\left (b e r \log \relax (c) - b e n \log \relax (f) - b d n + {\left (b e n p + b e n + a e\right )} r\right )} e^{\left (-\frac {b n p \log \left (-\frac {1}{b n}\right ) + b \log \relax (c) + a}{b n}\right )} \Gamma \left (p + 1, -\frac {b n \log \relax (x) + b \log \relax (c) + a}{b n}\right ) - {\left (b e n r x \log \relax (x) + b e r x \log \relax (c) + a e r x\right )} {\left (b n \log \relax (x) + b \log \relax (c) + a\right )}^{p}}{b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e \log \left (f x^{r}\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.42, size = 0, normalized size = 0.00 \[ \int \left (e \ln \left (f \,x^{r}\right )+d \right ) \left (b \ln \left (c \,x^{n}\right )+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (d+e\,\ln \left (f\,x^r\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \log {\left (c x^{n} \right )}\right )^{p} \left (d + e \log {\left (f x^{r} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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