3.4.82 \(\int \frac {(f+g x^n)^2 \log ^q(c (d+e x^n)^p)}{x} \, dx\) [382]

Optimal. Leaf size=307 \[ \frac {2^{-1-q} g^2 \left (d+e x^n\right )^2 \left (c \left (d+e x^n\right )^p\right )^{-2/p} \Gamma \left (1+q,-\frac {2 \log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^2 n}+\frac {2 f g \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \Gamma \left (1+q,-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e n}-\frac {d g^2 \left (d+e x^n\right ) \left (c \left (d+e x^n\right )^p\right )^{-1/p} \Gamma \left (1+q,-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right ) \log ^q\left (c \left (d+e x^n\right )^p\right ) \left (-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{p}\right )^{-q}}{e^2 n}+f^2 \text {Int}\left (\frac {\log ^q\left (c \left (d+e x^n\right )^p\right )}{x},x\right ) \]

[Out]

2^(-1-q)*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^2/n/((c*(d+e*x^n)^p)^(2/p))/(
(-ln(c*(d+e*x^n)^p)/p)^q)+2*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/n/((c*(d+e*x^n
)^p)^(1/p))/((-ln(c*(d+e*x^n)^p)/p)^q)-d*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^2
/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 = 0.05, 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 = {} \begin {gather*} \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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 = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (f+g x^n\right )^2 \log ^q\left (c \left (d+e x^n\right )^p\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.29, size = 0, normalized size = 0.00 \[\int \frac {\left (f +g \,x^{n}\right )^{2} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )^{q}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f+g*x^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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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