3.138 \(\int (g x)^q (a+b \log (c x^n)) \log (d (e+f x^m)^k) \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left ((g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ),x\right ) \]

[Out]

Unintegrable[(g*x)^q*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k], x]

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Rubi [A]  time = 0.0204296, 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 = {} \[ \int (g x)^q \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^k\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(g*x)^q*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

Defer[Int][(g*x)^q*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k], x]

Rubi steps

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

Mathematica [A]  time = 0.329968, size = 304, normalized size = 10.13 \[ \frac{x (g x)^q \left (-b k m n \, _3F_2\left (1,\frac{q}{m}+\frac{1}{m},\frac{q}{m}+\frac{1}{m};\frac{q}{m}+\frac{1}{m}+1,\frac{q}{m}+\frac{1}{m}+1;-\frac{f x^m}{e}\right )+k m \, _2F_1\left (1,\frac{q+1}{m};\frac{m+q+1}{m};-\frac{f x^m}{e}\right ) \left (a q+a+b (q+1) \log \left (c x^n\right )-b n\right )+a q^2 \log \left (d \left (e+f x^m\right )^k\right )+2 a q \log \left (d \left (e+f x^m\right )^k\right )+a \log \left (d \left (e+f x^m\right )^k\right )-a k m q-a k m+b q^2 \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+2 b q \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )+b \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^k\right )-b k m q \log \left (c x^n\right )-b k m \log \left (c x^n\right )-b n q \log \left (d \left (e+f x^m\right )^k\right )-b n \log \left (d \left (e+f x^m\right )^k\right )+2 b k m n\right )}{(q+1)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(g*x)^q*(a + b*Log[c*x^n])*Log[d*(e + f*x^m)^k],x]

[Out]

(x*(g*x)^q*(-(a*k*m) + 2*b*k*m*n - a*k*m*q - b*k*m*n*HypergeometricPFQ[{1, m^(-1) + q/m, m^(-1) + q/m}, {1 + m
^(-1) + q/m, 1 + m^(-1) + q/m}, -((f*x^m)/e)] - b*k*m*Log[c*x^n] - b*k*m*q*Log[c*x^n] + k*m*Hypergeometric2F1[
1, (1 + q)/m, (1 + m + q)/m, -((f*x^m)/e)]*(a - b*n + a*q + b*(1 + q)*Log[c*x^n]) + a*Log[d*(e + f*x^m)^k] - b
*n*Log[d*(e + f*x^m)^k] + 2*a*q*Log[d*(e + f*x^m)^k] - b*n*q*Log[d*(e + f*x^m)^k] + a*q^2*Log[d*(e + f*x^m)^k]
+ b*Log[c*x^n]*Log[d*(e + f*x^m)^k] + 2*b*q*Log[c*x^n]*Log[d*(e + f*x^m)^k] + b*q^2*Log[c*x^n]*Log[d*(e + f*x
^m)^k]))/(1 + q)^3

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Maple [A]  time = 0.25, size = 0, normalized size = 0. \begin{align*} \int \left ( gx \right ) ^{q} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f{x}^{m} \right ) ^{k} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x)^q*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)

[Out]

int((g*x)^q*(a+b*ln(c*x^n))*ln(d*(e+f*x^m)^k),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\left (g x\right )^{q} b \log \left (c x^{n}\right ) + \left (g x\right )^{q} a\right )} \log \left ({\left (f x^{m} + e\right )}^{k} d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((g*x)^q*b*log(c*x^n) + (g*x)^q*a)*log((f*x^m + e)^k*d), 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((g*x)**q*(a+b*ln(c*x**n))*ln(d*(e+f*x**m)**k),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} \left (g x\right )^{q} \log \left ({\left (f x^{m} + e\right )}^{k} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^q*(a+b*log(c*x^n))*log(d*(e+f*x^m)^k),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*(g*x)^q*log((f*x^m + e)^k*d), x)