Optimal. Leaf size=114 \[ \frac {x \left (d x^q\right )^m e^{-\frac {a m q+a}{b n}} \left (c x^n\right )^{-\frac {m q+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m q+1} \]
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Rubi [A] time = 0.11, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {15, 2310, 2181} \[ \frac {x \left (d x^q\right )^m e^{-\frac {a m q+a}{b n}} \left (c x^n\right )^{-\frac {m q+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m q+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 2181
Rule 2310
Rubi steps
\begin {align*} \int \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \, dx &=\left (x^{-m q} \left (d x^q\right )^m\right ) \int x^{m q} \left (a+b \log \left (c x^n\right )\right )^p \, dx\\ &=\frac {\left (x \left (c x^n\right )^{-\frac {1+m q}{n}} \left (d x^q\right )^m\right ) \operatorname {Subst}\left (\int e^{\frac {(1+m q) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {e^{-\frac {a+a m q}{b n}} x \left (c x^n\right )^{-\frac {1+m q}{n}} \left (d x^q\right )^m \Gamma \left (1+p,-\frac {(1+m q) \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 {(1+m q) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m q}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 118, normalized size = 1.04 \[ \frac {x^{-m q} \left (d x^q\right )^m \left (a+b \log \left (c x^n\right )\right )^p \exp \left (-\frac {(m q+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \Gamma \left (p+1,-\frac {(m q+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m q+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d x^{q}\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}, x\right ) \]
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 \[ \int \left (d x^{q}\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.13, size = 0, normalized size = 0.00 \[ \int \left (d \,x^{q}\right )^{m} \left (b \ln \left (c \,x^{n}\right )+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x^{q}\right )^{m} {\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (d\,x^q\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \]
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 \[ \int \left (d x^{q}\right )^{m} \left (a + b \log {\left (c x^{n} \right )}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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