3.211 \(\int (f+g x)^m \log (c (d+e x^n)^p) \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left ((f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ),x\right ) \]

[Out]

Unintegrable((g*x+f)^m*ln(c*(d+e*x^n)^p),x)

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Rubi [A]  time = 0.01, 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 = {} \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 0.56, size = 0, normalized size = 0.00 \[ \int (f+g x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x + f\right )}^{m} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [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((g*x+f)^m*log(c*(d+e*x^n)^p),x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to check s
ign: (4/(sign(t_nostep)-1)/2)>(-4/(sign(t_nostep)-1)/2)Unable to check sign: (4/(sign(x)-1)/2)>(-4/(sign(x)-1)
/2)Simplification assuming f near 0Unable to divide, perhaps due to rounding error%%%{-1,[0,0,6,3,6,0,2,2,0,1,
0]%%%}+%%%{1,[0,0,6,2,6,1,2,2,0,0,1]%%%}+%%%{1,[0,0,6,2,6,0,2,2,0,0,1]%%%}+%%%{-6,[0,0,5,3,5,0,2,2,1,1,0]%%%}+
%%%{5,[0,0,5,2,5,1,2,2,1,0,1]%%%}+%%%{5,[0,0,5,2,5,0,2,2,1,0,1]%%%}+%%%{-15,[0,0,4,3,4,0,2,2,2,1,0]%%%}+%%%{10
,[0,0,4,2,4,1,2,2,2,0,1]%%%}+%%%{10,[0,0,4,2,4,0,2,2,2,0,1]%%%}+%%%{-20,[0,0,3,3,3,0,2,2,3,1,0]%%%}+%%%{10,[0,
0,3,2,3,1,2,2,3,0,1]%%%}+%%%{10,[0,0,3,2,3,0,2,2,3,0,1]%%%}+%%%{-15,[0,0,2,3,2,0,2,2,4,1,0]%%%}+%%%{5,[0,0,2,2
,2,1,2,2,4,0,1]%%%}+%%%{5,[0,0,2,2,2,0,2,2,4,0,1]%%%}+%%%{-6,[0,0,1,3,1,0,2,2,5,1,0]%%%}+%%%{1,[0,0,1,2,1,1,2,
2,5,0,1]%%%}+%%%{1,[0,0,1,2,1,0,2,2,5,0,1]%%%}+%%%{-1,[0,0,0,3,0,0,2,2,6,1,0]%%%} / %%%{1,[0,0,6,2,5,0,1,2,0,0
,0]%%%}+%%%{5,[0,0,5,2,4,0,1,2,1,0,0]%%%}+%%%{10,[0,0,4,2,3,0,1,2,2,0,0]%%%}+%%%{10,[0,0,3,2,2,0,1,2,3,0,0]%%%
}+%%%{5,[0,0,2,2,1,0,1,2,4,0,0]%%%}+%%%{1,[0,0,1,2,0,0,1,2,5,0,0]%%%} Error: Bad Argument Value

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maple [A]  time = 1.77, size = 0, normalized size = 0.00 \[ \int \left (g x +f \right )^{m} \ln \left (c \left (e \,x^{n}+d \right )^{p}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (g x + f\right )} {\left (g x + f\right )}^{m} \log \left ({\left (e x^{n} + d\right )}^{p}\right )}{g {\left (m + 1\right )}} + \int \frac {{\left (d g {\left (m + 1\right )} x \log \relax (c) - {\left (e f n p + {\left (e g n p - e g {\left (m + 1\right )} \log \relax (c)\right )} x\right )} x^{n}\right )} {\left (g x + f\right )}^{m}}{e g {\left (m + 1\right )} x x^{n} + d g {\left (m + 1\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(g*x + f)*(g*x + f)^m*log((e*x^n + d)^p)/(g*(m + 1)) + integrate((d*g*(m + 1)*x*log(c) - (e*f*n*p + (e*g*n*p -
e*g*(m + 1)*log(c))*x)*x^n)*(g*x + f)^m/(e*g*(m + 1)*x*x^n + d*g*(m + 1)*x), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f+g\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(log(c*(d + e*x^n)^p)*(f + g*x)^m, 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((g*x+f)**m*ln(c*(d+e*x**n)**p),x)

[Out]

Timed out

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