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 = 0.04, antiderivative size = 102, normalized size of antiderivative = 1.00, 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 245
Rule 246
Rule 2314
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*}
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Mathematica [A] time = 0.59, size = 143, normalized size = 1.40 \[ \frac {x^{-\frac {1}{q+1}} \left (d+e x^{-\frac {1}{q+1}}\right )^q \left (\frac {d x^{\frac {1}{q+1}}}{e}+1\right )^{-q} \left (-b d n (q+1)^2 x^{\frac {q+2}{q+1}} \, _3F_2\left (1,1,-q;2,2;-\frac {d x^{\frac {1}{q+1}}}{e}\right )+\left (d x^{\frac {q+2}{q+1}}+e x\right ) \left (\frac {d x^{\frac {1}{q+1}}}{e}+1\right )^q \left (a+b \log \left (c x^n\right )\right )-b e n x \log (x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} \left (\frac {d x^{\left (\frac {1}{q + 1}\right )} + e}{x^{\left (\frac {1}{q + 1}\right )}}\right )^{q}, x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (d + \frac {e}{x^{\left (\frac {1}{q + 1}\right )}}\right )}^{q}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \left (e \,x^{-\frac {1}{q +1}}+d \right )^{q}\, dx \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (d + \frac {e}{x^{\left (\frac {1}{q + 1}\right )}}\right )}^{q}\,{d x} \]
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 \[ \int {\left (d+\frac {e}{x^{\frac {1}{q+1}}}\right )}^q\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]
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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