\(\int \frac {(a g+b g x)^2}{(A+B \log (\frac {e (a+b x)}{c+d x}))^2} \, dx\) [114]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\text {Int}\left (\frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 32, 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 = {} \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.47 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2}}{{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 305, normalized size of antiderivative = 9.53 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2}}{{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(b^3*d*g^2*x^4 + a^3*c*g^2 + (b^3*c*g^2 + 3*a*b^2*d*g^2)*x^3 + 3*(a*b^2*c*g^2 + a^2*b*d*g^2)*x^2 + (3*a^2*b*c
*g^2 + a^3*d*g^2)*x)/((b*c - a*d)*B^2*log(b*x + a) - (b*c - a*d)*B^2*log(d*x + c) + (b*c - a*d)*A*B + (b*c*log
(e) - a*d*log(e))*B^2) + integrate((4*b^3*d*g^2*x^3 + 3*a^2*b*c*g^2 + a^3*d*g^2 + 3*(b^3*c*g^2 + 3*a*b^2*d*g^2
)*x^2 + 6*(a*b^2*c*g^2 + a^2*b*d*g^2)*x)/((b*c - a*d)*B^2*log(b*x + a) - (b*c - a*d)*B^2*log(d*x + c) + (b*c -
a*d)*A*B + (b*c*log(e) - a*d*log(e))*B^2), x)

Giac [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2}}{{\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 4.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^2}{{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2} \,d x \]

[In]

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

[Out]

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