3.71 \(\int \frac{\log (h (f+g x)^m)}{(a+b x) (c+d x) \log ^2(e (\frac{a+b x}{c+d x})^n)} \, dx\)

Optimal. Leaf size=92 \[ \frac{g m \text{Unintegrable}\left (\frac{1}{(f+g x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )},x\right )}{n (b c-a d)}-\frac{\log \left (h (f+g x)^m\right )}{n (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \]

[Out]

-(Log[h*(f + g*x)^m]/((b*c - a*d)*n*Log[e*((a + b*x)/(c + d*x))^n])) + (g*m*Unintegrable[1/((f + g*x)*Log[e*((
a + b*x)/(c + d*x))^n]), x])/((b*c - a*d)*n)

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Rubi [A]  time = 0.128186, 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 = {} \[ \int \frac{\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]^2),x]

[Out]

-(Log[h*(f + g*x)^m]/((b*c - a*d)*n*Log[e*((a + b*x)/(c + d*x))^n])) + (g*m*Defer[Int][1/((f + g*x)*Log[e*((a
+ b*x)/(c + d*x))^n]), x])/((b*c - a*d)*n)

Rubi steps

\begin{align*} \int \frac{\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=-\frac{\log \left (h (f+g x)^m\right )}{(b c-a d) n \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}+\frac{(g m) \int \frac{1}{(f+g x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx}{(b c-a d) n}\\ \end{align*}

Mathematica [A]  time = 1.28668, size = 0, normalized size = 0. \[ \int \frac{\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]^2),x]

[Out]

Integrate[Log[h*(f + g*x)^m]/((a + b*x)*(c + d*x)*Log[e*((a + b*x)/(c + d*x))^n]^2), x]

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Maple [A]  time = 48.165, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( h \left ( gx+f \right ) ^{m} \right ) }{ \left ( bx+a \right ) \left ( dx+c \right ) } \left ( \ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/ln(e*((b*x+a)/(d*x+c))^n)^2,x)

[Out]

int(ln(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/ln(e*((b*x+a)/(d*x+c))^n)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} g m \int \frac{1}{b c f n \log \left (e\right ) - a d f n \log \left (e\right ) +{\left (b c g n \log \left (e\right ) - a d g n \log \left (e\right )\right )} x +{\left (b c f n - a d f n +{\left (b c g n - a d g n\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left (b c f n - a d f n +{\left (b c g n - a d g n\right )} x\right )} \log \left ({\left (d x + c\right )}^{n}\right )}\,{d x} - \frac{\log \left ({\left (g x + f\right )}^{m}\right ) + \log \left (h\right )}{b c n \log \left (e\right ) - a d n \log \left (e\right ) +{\left (b c n - a d n\right )} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left (b c n - a d n\right )} \log \left ({\left (d x + c\right )}^{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/log(e*((b*x+a)/(d*x+c))^n)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/log(e*((b*x+a)/(d*x+c))^n)^2,x, algorithm="fricas")

[Out]

integral(log((g*x + f)^m*h)/((b*d*x^2 + a*c + (b*c + a*d)*x)*log(e*((b*x + a)/(d*x + c))^n)^2), 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(ln(h*(g*x+f)**m)/(b*x+a)/(d*x+c)/ln(e*((b*x+a)/(d*x+c))**n)**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )}{\left (d x + c\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(h*(g*x+f)^m)/(b*x+a)/(d*x+c)/log(e*((b*x+a)/(d*x+c))^n)^2,x, algorithm="giac")

[Out]

integrate(log((g*x + f)^m*h)/((b*x + a)*(d*x + c)*log(e*((b*x + a)/(d*x + c))^n)^2), x)