Optimal. Leaf size=136 \[ \frac{x \left (\text{d1} x^{\text{q1}}\right )^{\text{m1}} \left (\text{d2} x^{\text{q2}}\right )^{\text{m2}} \left (a+b \log \left (c x^n\right )\right )^p e^{-\frac{a (\text{m1} \text{q1}+\text{m2} \text{q2}+1)}{b n}} \left (c x^n\right )^{-\frac{\text{m1} \text{q1}+\text{m2} \text{q2}+1}{n}} \left (-\frac{(\text{m1} \text{q1}+\text{m2} \text{q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(\text{m1} \text{q1}+\text{m2} \text{q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{\text{m1} \text{q1}+\text{m2} \text{q2}+1} \]
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Rubi [A] time = 0.16834, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 2310, 2181} \[ \frac{x \left (\text{d1} x^{\text{q1}}\right )^{\text{m1}} \left (\text{d2} x^{\text{q2}}\right )^{\text{m2}} \left (a+b \log \left (c x^n\right )\right )^p e^{-\frac{a (\text{m1} \text{q1}+\text{m2} \text{q2}+1)}{b n}} \left (c x^n\right )^{-\frac{\text{m1} \text{q1}+\text{m2} \text{q2}+1}{n}} \left (-\frac{(\text{m1} \text{q1}+\text{m2} \text{q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(\text{m1} \text{q1}+\text{m2} \text{q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{\text{m1} \text{q1}+\text{m2} \text{q2}+1} \]
Antiderivative was successfully verified.
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Rule 15
Rule 2310
Rule 2181
Rubi steps
\begin{align*} \int \left (\text{d1} x^{\text{q1}}\right )^{\text{m1}} \left (\text{d2} x^{\text{q2}}\right )^{\text{m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx &=\left (x^{-\text{m1} \text{q1}} \left (\text{d1} x^{\text{q1}}\right )^{\text{m1}}\right ) \int x^{\text{m1} \text{q1}} \left (\text{d2} x^{\text{q2}}\right )^{\text{m2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx\\ &=\left (x^{-\text{m1} \text{q1}-\text{m2} \text{q2}} \left (\text{d1} x^{\text{q1}}\right )^{\text{m1}} \left (\text{d2} x^{\text{q2}}\right )^{\text{m2}}\right ) \int x^{\text{m1} \text{q1}+\text{m2} \text{q2}} \left (a+b \log \left (c x^n\right )\right )^p \, dx\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{1+\text{m1} \text{q1}+\text{m2} \text{q2}}{n}} \left (\text{d1} x^{\text{q1}}\right )^{\text{m1}} \left (\text{d2} x^{\text{q2}}\right )^{\text{m2}}\right ) \operatorname{Subst}\left (\int e^{\frac{(1+\text{m1} \text{q1}+\text{m2} \text{q2}) x}{n}} (a+b x)^p \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{e^{-\frac{a (1+\text{m1} \text{q1}+\text{m2} \text{q2})}{b n}} x \left (c x^n\right )^{-\frac{1+\text{m1} \text{q1}+\text{m2} \text{q2}}{n}} \left (\text{d1} x^{\text{q1}}\right )^{\text{m1}} \left (\text{d2} x^{\text{q2}}\right )^{\text{m2}} \Gamma \left (1+p,-\frac{(1+\text{m1} \text{q1}+\text{m2} \text{q2}) \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+\text{m1} \text{q1}+\text{m2} \text{q2}) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+\text{m1} \text{q1}+\text{m2} \text{q2}}\\ \end{align*}
Mathematica [A] time = 0.226893, size = 142, normalized size = 1.04 \[ \frac{\left (\text{d1} x^{\text{q1}}\right )^{\text{m1}} \left (\text{d2} x^{\text{q2}}\right )^{\text{m2}} x^{-\text{m1} \text{q1}-\text{m2} \text{q2}} \left (a+b \log \left (c x^n\right )\right )^p \exp \left (-\frac{(\text{m1} \text{q1}+\text{m2} \text{q2}+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (-\frac{(\text{m1} \text{q1}+\text{m2} \text{q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(\text{m1} \text{q1}+\text{m2} \text{q2}+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{\text{m1} \text{q1}+\text{m2} \text{q2}+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 19.344, size = 0, normalized size = 0. \begin{align*} \int \left ({\it d1}\,{x}^{{\it q1}} \right ) ^{{\it m1}} \left ({\it d2}\,{x}^{{\it q2}} \right ) ^{{\it m2}} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \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 (\left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}, 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 \left (d_{1} x^{q_{1}}\right )^{m_{1}} \left (d_{2} x^{q_{2}}\right )^{m_{2}}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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