\(\int \frac {a g+b g x}{A+B \log (e (\frac {a+b x}{c+d x})^n)} \, dx\) [20]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {Int}\left (\frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right ) \]

[Out]

Unintegrable((b*g*x+a*g)/(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

[In]

Int[(a*g + b*g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

Defer[Int][(a*g + b*g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

[In]

Integrate[(a*g + b*g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]

[Out]

Integrate[(a*g + b*g*x)/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]

Maple [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00

\[\int \frac {b g x +a g}{A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}d x\]

[In]

int((b*g*x+a*g)/(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

int((b*g*x+a*g)/(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {b g x + a g}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

integrate((b*g*x+a*g)/(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

integral((b*g*x + a*g)/(B*log(e*((b*x + a)/(d*x + c))^n) + A), x)

Sympy [N/A]

Not integrable

Time = 16.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=g \left (\int \frac {a}{A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}\, dx + \int \frac {b x}{A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}\, dx\right ) \]

[In]

integrate((b*g*x+a*g)/(A+B*ln(e*((b*x+a)/(d*x+c))**n)),x)

[Out]

g*(Integral(a/(A + B*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)), x) + Integral(b*x/(A + B*log(e*(a/(c + d*x) + b
*x/(c + d*x))**n)), x))

Maxima [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {b g x + a g}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

integrate((b*g*x+a*g)/(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")

[Out]

integrate((b*g*x + a*g)/(B*log(e*((b*x + a)/(d*x + c))^n) + A), x)

Giac [N/A]

Not integrable

Time = 16.77 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {b g x + a g}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

integrate((b*g*x+a*g)/(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

integrate((b*g*x + a*g)/(B*log(e*((b*x + a)/(d*x + c))^n) + A), x)

Mupad [N/A]

Not integrable

Time = 0.67 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {a g+b g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {a\,g+b\,g\,x}{A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )} \,d x \]

[In]

int((a*g + b*g*x)/(A + B*log(e*((a + b*x)/(c + d*x))^n)),x)

[Out]

int((a*g + b*g*x)/(A + B*log(e*((a + b*x)/(c + d*x))^n)), x)