\(\int \frac {(c g+d g x)^2}{A+B \log (e (\frac {a+b x}{c+d x})^n)} \, dx\) [47]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\text {Int}\left (\frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right ) \]

[Out]

Unintegrable((d*g*x+c*g)^2/(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 35, 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 = {} \[ \int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

[In]

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

[Out]

Defer[Int][(c*g + d*g*x)^2/(A + B*Log[e*((a + b*x)/(c + d*x))^n]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(c g+d g x)^2}{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 = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00

\[\int \frac {\left (d g x +c g \right )^{2}}{A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}d x\]

[In]

int((d*g*x+c*g)^2/(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

int((d*g*x+c*g)^2/(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {{\left (d g x + c g\right )}^{2}}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

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

[Out]

integral((d^2*g^2*x^2 + 2*c*d*g^2*x + c^2*g^2)/(B*log(e*((b*x + a)/(d*x + c))^n) + A), x)

Sympy [N/A]

Not integrable

Time = 24.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.71 \[ \int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=g^{2} \left (\int \frac {c^{2}}{A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}\, dx + \int \frac {d^{2} x^{2}}{A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}\, dx + \int \frac {2 c d x}{A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}\, dx\right ) \]

[In]

integrate((d*g*x+c*g)**2/(A+B*ln(e*((b*x+a)/(d*x+c))**n)),x)

[Out]

g**2*(Integral(c**2/(A + B*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)), x) + Integral(d**2*x**2/(A + B*log(e*(a/(
c + d*x) + b*x/(c + d*x))**n)), x) + Integral(2*c*d*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 = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {{\left (d g x + c g\right )}^{2}}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

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

[Out]

integrate((d*g*x + c*g)^2/(B*log(e*((b*x + a)/(d*x + c))^n) + A), x)

Giac [N/A]

Not integrable

Time = 27.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int { \frac {{\left (d g x + c g\right )}^{2}}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A} \,d x } \]

[In]

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

[Out]

integrate((d*g*x + c*g)^2/(B*log(e*((b*x + a)/(d*x + c))^n) + A), x)

Mupad [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {(c g+d g x)^2}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx=\int \frac {{\left (c\,g+d\,g\,x\right )}^2}{A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )} \,d x \]

[In]

int((c*g + d*g*x)^2/(A + B*log(e*((a + b*x)/(c + d*x))^n)),x)

[Out]

int((c*g + d*g*x)^2/(A + B*log(e*((a + b*x)/(c + d*x))^n)), x)