Optimal. Leaf size=298 \[ 3^{-p-1} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+e \left (-3^{-p-2}\right ) r x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+2,-\frac {3 a}{b n}-\frac {3 \log \left (c x^n\right )}{n}\right )-\frac {e 3^{-p-1} r x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {3 a}{b n}-\frac {3 \log \left (c x^n\right )}{n}\right )}{b n} \]
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Rubi [A] time = 0.25, antiderivative size = 298, normalized size of antiderivative = 1.00, 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} \[ 3^{-p-1} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+e \left (-3^{-p-2}\right ) r x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text {Gamma}\left (p+2,-\frac {3 a}{b n}-\frac {3 \log \left (c x^n\right )}{n}\right )-\frac {e 3^{-p-1} r x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {3 a}{b n}-\frac {3 \log \left (c x^n\right )}{n}\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 12
Rule 15
Rule 19
Rule 2181
Rule 2310
Rule 2366
Rule 6557
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right ) \, dx &=3^{-1-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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 )-(e r) \int 3^{-1-p} e^{-\frac {3 a}{b n}} x^2 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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} \, dx\\ &=3^{-1-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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 )-\left (3^{-1-p} e e^{-\frac {3 a}{b n}} r\right ) \int x^2 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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} \, dx\\ &=3^{-1-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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 )-\left (3^{-1-p} e e^{-\frac {3 a}{b n}} r x^3 \left (c x^n\right )^{-3/n}\right ) \int \frac {\Gamma \left (1+p,-\frac {3 \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\\ &=3^{-1-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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 )-\left (3^{-1-p} e e^{-\frac {3 a}{b n}} r x^3 \left (c x^n\right )^{-3/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 {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x} \, dx\\ &=3^{-1-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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 )-\frac {\left (3^{-1-p} e e^{-\frac {3 a}{b n}} r x^3 \left (c x^n\right )^{-3/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 {3 (a+b x)}{b n}\right ) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=3^{-1-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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 )+\left (3^{-2-p} e e^{-\frac {3 a}{b n}} r x^3 \left (c x^n\right )^{-3/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 {3 a}{b n}-\frac {3 \log \left (c x^n\right )}{n}\right )\\ &=-3^{-2-p} e e^{-\frac {3 a}{b n}} r x^3 \left (c x^n\right )^{-3/n} \Gamma \left (2+p,-\frac {3 a}{b n}-\frac {3 \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}-3^{-1-p} e e^{-\frac {3 a}{b n}} r x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 a}{b n}-\frac {3 \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 )+3^{-1-p} e^{-\frac {3 a}{b n}} x^3 \left (c x^n\right )^{-3/n} \Gamma \left (1+p,-\frac {3 \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 )\\ \end {align*}
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Mathematica [A] time = 0.40, size = 156, normalized size = 0.52 \[ -3^{-p-2} x^3 e^{-\frac {3 a}{b n}} \left (c x^n\right )^{-3/n} \left (a+b \log \left (c x^n\right )\right )^{p-1} \left (-\frac {a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (3 \Gamma \left (p+1,-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-a e r-b e r \log \left (c x^n\right )+b d n+b e n \log \left (f x^r\right )\right )-b e n r \Gamma \left (p+2,-\frac {3 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x^{2} \log \left (f x^{r}\right ) + d x^{2}\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}, x\right ) \]
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 \[ \int {\left (e \log \left (f x^{r}\right ) + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.44, size = 0, normalized size = 0.00 \[ \int \left (e \ln \left (f \,x^{r}\right )+d \right ) x^{2} \left (b \ln \left (c \,x^{n}\right )+a \right )^{p}\, dx \]
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 \[ \int x^2\,\left (d+e\,\ln \left (f\,x^r\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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