Optimal. Leaf size=307 \[ f^2 \text {Int}\left (\frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x},x\right )+\frac {g^2 2^{-q-1} \left (d+e x^n\right )^2 \left (c \left (d+e x^n\right )^p\right )^{-2/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {2 \log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e^2 n}-\frac {d g^2 \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {\log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e^2 n}+\frac {2 f g \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {\log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e n} \]
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Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx\\ \end {align*}
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Mathematica [A] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g^{2} x^{2 \, n} + 2 \, f g x^{n} + f^{2}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 47.95, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \,x^{n}+f \right )^{2} \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )^{q}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^q\,{\left (f+g\,x^n\right )}^2}{x} \,d x \]
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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