Optimal. Leaf size=119 \[ -\frac {(f x)^{-((q+1) r)} \left (d+e x^r\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d f (q+1) r}-\frac {b n (f x)^{-((q+1) r)} \left (d+e x^r\right )^q \left (\frac {e x^r}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac {e x^r}{d}\right )}{f (q+1)^2 r^2} \]
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Rubi [A] time = 0.13, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2335, 365, 364} \[ -\frac {(f x)^{-(q+1) r} \left (d+e x^r\right )^{q+1} \left (a+b \log \left (c x^n\right )\right )}{d f (q+1) r}-\frac {b n (f x)^{-(q+1) r} \left (d+e x^r\right )^q \left (\frac {e x^r}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac {e x^r}{d}\right )}{f (q+1)^2 r^2} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2335
Rubi steps
\begin {align*} \int (f x)^{-1-(1+q) r} \left (d+e x^r\right )^q \left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {(f x)^{-(1+q) r} \left (d+e x^r\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d f (1+q) r}+\frac {(b n) \int (f x)^{-1-(1+q) r} \left (d+e x^r\right )^{1+q} \, dx}{d (1+q) r}\\ &=-\frac {(f x)^{-(1+q) r} \left (d+e x^r\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d f (1+q) r}+\frac {\left (b n \left (d+e x^r\right )^q \left (1+\frac {e x^r}{d}\right )^{-q}\right ) \int (f x)^{-1-(1+q) r} \left (1+\frac {e x^r}{d}\right )^{1+q} \, dx}{(1+q) r}\\ &=-\frac {b n (f x)^{-(1+q) r} \left (d+e x^r\right )^q \left (1+\frac {e x^r}{d}\right )^{-q} \, _2F_1\left (-1-q,-1-q;-q;-\frac {e x^r}{d}\right )}{f (1+q)^2 r^2}-\frac {(f x)^{-(1+q) r} \left (d+e x^r\right )^{1+q} \left (a+b \log \left (c x^n\right )\right )}{d f (1+q) r}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 98, normalized size = 0.82 \[ -\frac {(f x)^{-((q+1) r)} \left (d+e x^r\right )^q \left (\frac {(q+1) r \left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )}{d}+b n \left (\frac {e x^r}{d}+1\right )^{-q} \, _2F_1\left (-q-1,-q-1;-q;-\frac {e x^r}{d}\right )\right )}{f (q+1)^2 r^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (f x\right )^{-{\left (q + 1\right )} r - 1} b \log \left (c x^{n}\right ) + \left (f x\right )^{-{\left (q + 1\right )} r - 1} a\right )} {\left (e x^{r} + d\right )}^{q}, x\right ) \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (e x^{r} + d\right )}^{q} \left (f x\right )^{-{\left (q + 1\right )} r - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.74, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \left (f x \right )^{-\left (q +1\right ) r -1} \left (e \,x^{r}+d \right )^{q}\, dx \]
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 \[ \int {\left (b \log \left (c x^{n}\right ) + a\right )} {\left (e x^{r} + d\right )}^{q} \left (f x\right )^{-{\left (q + 1\right )} r - 1}\,{d x} \]
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 \[ \int \frac {{\left (d+e\,x^r\right )}^q\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (f\,x\right )}^{r\,\left (q+1\right )+1}} \,d x \]
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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