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

Optimal. Leaf size=607 \[ \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} \text{Gamma}\left (q+1,-\frac{2 \log \left (c \left (d+e x^n\right )^p\right )}{p}\right )}{e^4 n}-\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} \text{Gamma}\left (q+1,-\frac{\log \left (c \left (d+e x^n\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} \text{Gamma}\left (q+1,-\frac{2 \log \left (c \left (d+e x^n\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} \text{Gamma}\left (q+1,-\frac{\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )}{e^2 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} \text{Gamma}\left (q+1,-\frac{4 \log \left (c \left (d+e x^n\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} \text{Gamma}\left (q+1,-\frac{3 \log \left (c \left (d+e x^n\right )^p\right )}{p}\right )}{e^4 n}+f^2 \text{Unintegrable}\left (\frac{\log ^q\left (c \left (d+e x^n\right )^p\right )}{x},x\right ) \]

[Out]

(4^(-1 - q)*g^2*(d + e*x^n)^4*Gamma[1 + q, (-4*Log[c*(d + e*x^n)^p])/p]*Log[c*(d + e*x^n)^p]^q)/(e^4*n*(c*(d +
e*x^n)^p)^(4/p)*(-(Log[c*(d + e*x^n)^p]/p))^q) - (d*g^2*(d + e*x^n)^3*Gamma[1 + q, (-3*Log[c*(d + e*x^n)^p])/
p]*Log[c*(d + e*x^n)^p]^q)/(3^q*e^4*n*(c*(d + e*x^n)^p)^(3/p)*(-(Log[c*(d + e*x^n)^p]/p))^q) + (f*g*(d + e*x^n
)^2*Gamma[1 + q, (-2*Log[c*(d + e*x^n)^p])/p]*Log[c*(d + e*x^n)^p]^q)/(2^q*e^2*n*(c*(d + e*x^n)^p)^(2/p)*(-(Lo
g[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*Log[c*(d + e*x^n)^p])/p]*Log[
c*(d + e*x^n)^p]^q)/(e^4*n*(c*(d + e*x^n)^p)^(2/p)*(-(Log[c*(d + e*x^n)^p]/p))^q) - (2*d*f*g*(d + e*x^n)*Gamma
[1 + q, -(Log[c*(d + e*x^n)^p]/p)]*Log[c*(d + e*x^n)^p]^q)/(e^2*n*(c*(d + e*x^n)^p)^p^(-1)*(-(Log[c*(d + e*x^n
)^p]/p))^q) - (d^3*g^2*(d + e*x^n)*Gamma[1 + q, -(Log[c*(d + e*x^n)^p]/p)]*Log[c*(d + e*x^n)^p]^q)/(e^4*n*(c*(
d + e*x^n)^p)^p^(-1)*(-(Log[c*(d + e*x^n)^p]/p))^q) + f^2*Unintegrable[Log[c*(d + e*x^n)^p]^q/x, x]

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Rubi [A]  time = 0.0845421, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.416753, size = 0, normalized size = 0. \[ \int \frac{\left (f+g x^{2 n}\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \]

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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Maple [A]  time = 29.864, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( f+g{x}^{2\,n} \right ) ^{2} \left ( \ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{q}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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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.

[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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)