3.381 \(\int \frac {(f+g x^{2 n})^2 \log ^q(c (d+e x^n)^p)}{x} \, dx\)

Optimal. Leaf size=608 \[ f^2 \text {Int}\left (\frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x},x\right )-\frac {d^3 g^2 \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {\log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e^4 n}+\frac {3 d^2 g^2 2^{-q-1} \left (d+e x^n\right )^2 \left (c \left (d+e x^n\right )^p\right )^{-2/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {2 \log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e^4 n}+\frac {g^2 4^{-q-1} \left (d+e x^n\right )^4 \left (c \left (d+e x^n\right )^p\right )^{-4/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {4 \log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e^4 n}-\frac {d g^2 3^{-q} \left (d+e x^n\right )^3 \left (c \left (d+e x^n\right )^p\right )^{-3/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {3 \log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e^4 n}+\frac {f g 2^{-q} \left (d+e x^n\right )^2 \left (c \left (d+e x^n\right )^p\right )^{-2/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {2 \log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e^2 n}-\frac {2 d f g \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q} \Gamma \left (q+1,-\frac {\log \left (c \left (e x^n+d\right )^p\right )}{p}\right )}{e^2 n} \]

[Out]

4^(-1-q)*g^2*(d+e*x^n)^4*GAMMA(1+q,-4*ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e*x^n)^p)^q/e^4/n/((c*(d+e*x^n)^p)^(4/p))/(
(-ln(c*(d+e*x^n)^p)/p)^q)-d*g^2*(d+e*x^n)^3*GAMMA(1+q,-3*ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e*x^n)^p)^q/(3^q)/e^4/n/
((c*(d+e*x^n)^p)^(3/p))/((-ln(c*(d+e*x^n)^p)/p)^q)+f*g*(d+e*x^n)^2*GAMMA(1+q,-2*ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e
*x^n)^p)^q/(2^q)/e^2/n/((c*(d+e*x^n)^p)^(2/p))/((-ln(c*(d+e*x^n)^p)/p)^q)+3*2^(-1-q)*d^2*g^2*(d+e*x^n)^2*GAMMA
(1+q,-2*ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e*x^n)^p)^q/e^4/n/((c*(d+e*x^n)^p)^(2/p))/((-ln(c*(d+e*x^n)^p)/p)^q)-2*d*
f*g*(d+e*x^n)*GAMMA(1+q,-ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e*x^n)^p)^q/e^2/n/((c*(d+e*x^n)^p)^(1/p))/((-ln(c*(d+e*x
^n)^p)/p)^q)-d^3*g^2*(d+e*x^n)*GAMMA(1+q,-ln(c*(d+e*x^n)^p)/p)*ln(c*(d+e*x^n)^p)^q/e^4/n/((c*(d+e*x^n)^p)^(1/p
))/((-ln(c*(d+e*x^n)^p)/p)^q)+f^2*Unintegrable(ln(c*(d+e*x^n)^p)^q/x,x)

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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 = {} \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((f + g*x^(2*n))^2*Log[c*(d + e*x^n)^p]^q)/x,x]

[Out]

Defer[Int][((f + g*x^(2*n))^2*Log[c*(d + e*x^n)^p]^q)/x, x]

Rubi steps

\begin {align*} \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\int \frac {\left (f+g x^{2 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 = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((f + g*x^(2*n))^2*Log[c*(d + e*x^n)^p]^q)/x,x]

[Out]

Integrate[((f + g*x^(2*n))^2*Log[c*(d + e*x^n)^p]^q)/x, x]

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fricas [A]  time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g^{2} x^{4 \, n} + 2 \, f g x^{2 \, n} + f^{2}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f+g*x^(2*n))^2*log(c*(d+e*x^n)^p)^q/x,x, algorithm="fricas")

[Out]

integral((g^2*x^(4*n) + 2*f*g*x^(2*n) + f^2)*log((e*x^n + d)^p*c)^q/x, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x^{2 \, n} + f\right )}^{2} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{q}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f+g*x^(2*n))^2*log(c*(d+e*x^n)^p)^q/x,x, algorithm="giac")

[Out]

integrate((g*x^(2*n) + f)^2*log((e*x^n + d)^p*c)^q/x, x)

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maple [A]  time = 50.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \,x^{2 n}+f \right )^{2} \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )^{q}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x^(2*n)+f)^2*ln(c*(e*x^n+d)^p)^q/x,x)

[Out]

int((g*x^(2*n)+f)^2*ln(c*(e*x^n+d)^p)^q/x,x)

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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.

[In]

integrate((f+g*x^(2*n))^2*log(c*(d+e*x^n)^p)^q/x,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}^q\,{\left (f+g\,x^{2\,n}\right )}^2}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(c*(d + e*x^n)^p)^q*(f + g*x^(2*n))^2)/x,x)

[Out]

int((log(c*(d + e*x^n)^p)^q*(f + g*x^(2*n))^2)/x, x)

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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.

[In]

integrate((f+g*x**(2*n))**2*ln(c*(d+e*x**n)**p)**q/x,x)

[Out]

Timed out

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