Optimal. Leaf size=102 \[ \frac{x \left (d+e x^{-\frac{1}{q+1}}\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d}-b n x \left (d+e x^{-\frac{1}{q+1}}\right )^q \left (\frac{e x^{-\frac{1}{q+1}}}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac{e x^{-\frac{1}{q+1}}}{d}\right ) \]
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Rubi [A] time = 0.0440655, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {2314, 246, 245} \[ \frac{x \left (d+e x^{-\frac{1}{q+1}}\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d}-b n x \left (d+e x^{-\frac{1}{q+1}}\right )^q \left (\frac{e x^{-\frac{1}{q+1}}}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac{e x^{-\frac{1}{q+1}}}{d}\right ) \]
Antiderivative was successfully verified.
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Rule 2314
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \left (d+e x^{-\frac{1}{1+q}}\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{x \left (d+e x^{-\frac{1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d}-\frac{(b n) \int \left (d+e x^{-\frac{1}{1+q}}\right )^{1+q} \, dx}{d}\\ &=\frac{x \left (d+e x^{-\frac{1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d}-\left (b n \left (d+e x^{-\frac{1}{1+q}}\right )^q \left (1+\frac{e x^{-\frac{1}{1+q}}}{d}\right )^{-q}\right ) \int \left (1+\frac{e x^{-\frac{1}{1+q}}}{d}\right )^{1+q} \, dx\\ &=-b n x \left (d+e x^{-\frac{1}{1+q}}\right )^q \left (1+\frac{e x^{-\frac{1}{1+q}}}{d}\right )^{-q} \, _2F_1\left (-1-q,-1-q;-q;-\frac{e x^{-\frac{1}{1+q}}}{d}\right )+\frac{x \left (d+e x^{-\frac{1}{1+q}}\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.571386, size = 143, normalized size = 1.4 \[ \frac{x^{-\frac{1}{q+1}} \left (d+e x^{-\frac{1}{q+1}}\right )^q \left (\frac{d x^{\frac{1}{q+1}}}{e}+1\right )^{-q} \left (-b d n (q+1)^2 x^{\frac{q+2}{q+1}} \, _3F_2\left (1,1,-q;2,2;-\frac{d x^{\frac{1}{q+1}}}{e}\right )+\left (d x^{\frac{q+2}{q+1}}+e x\right ) \left (\frac{d x^{\frac{1}{q+1}}}{e}+1\right )^q \left (a+b \log \left (c x^n\right )\right )-b e n x \log (x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.829, size = 0, normalized size = 0. \begin{align*} \int \left ( d+{\frac{e}{{x}^{ \left ( 1+q \right ) ^{-1}}}} \right ) ^{q} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \, 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 (b \log \left (c x^{n}\right ) + a\right )}{\left (d + \frac{e}{x^{\left (\frac{1}{q + 1}\right )}}\right )}^{q}\,{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 (b \log \left (c x^{n}\right ) + a\right )} \left (\frac{d x^{\left (\frac{1}{q + 1}\right )} + e}{x^{\left (\frac{1}{q + 1}\right )}}\right )^{q}, 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 (b \log \left (c x^{n}\right ) + a\right )}{\left (d + \frac{e}{x^{\left (\frac{1}{q + 1}\right )}}\right )}^{q}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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