Optimal. Leaf size=295 \[ -\frac {2^{-2-p} e e^{\frac {2 a}{b n}} r \left (c x^n\right )^{2/n} \Gamma \left (2+p,\frac {2 a}{b n}+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{x^2}+\frac {2^{-1-p} e e^{\frac {2 a}{b n}} r \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac {2 a}{b n}+\frac {2 \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^{1+p} \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p}}{b n x^2}-\frac {2^{-1-p} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{x^2} \]
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Rubi [A]
time = 0.16, antiderivative size = 295, 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 = {2347, 2212,
2413, 12, 15, 19, 6692} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 15
Rule 19
Rule 2212
Rule 2347
Rule 2413
Rule 6692
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 ) \text {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 ) \text {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*}
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Mathematica [A]
time = 0.26, size = 154, normalized size = 0.52 \begin {gather*} -\frac {2^{-2-p} e^{\frac {2 a}{b n}} \left (c x^n\right )^{2/n} \left (a+b \log \left (c x^n\right )\right )^{-1+p} \left (\frac {a+b \log \left (c x^n\right )}{b n}\right )^{1-p} \left (b e n r \Gamma \left (2+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+2 \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (b d n-a e r-b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right )}{x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{p} \left (d +e \ln \left (f \,x^{r}\right )\right )}{x^{3}}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (d+e\,\ln \left (f\,x^r\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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