\(\int \frac {1}{(f+g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2} \, dx\) [260]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\text {Int}\left (\frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2},x\right ) \]

[Out]

Unintegrable(1/(g*x+f)^2/(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 29, 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 = {} \[ \int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]

[In]

Int[1/((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2),x]

[Out]

Defer[Int][1/((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(f+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 = 1.55 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]

[In]

Integrate[1/((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2),x]

[Out]

Integrate[1/((f + g*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2), x]

Maple [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (g x +f \right )^{2} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}d x\]

[In]

int(1/(g*x+f)^2/(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

[Out]

int(1/(g*x+f)^2/(A+B*ln(e*(b*x+a)/(d*x+c)))^2,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 118, normalized size of antiderivative = 4.07 \[ \int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

integral(1/(A^2*g^2*x^2 + 2*A^2*f*g*x + A^2*f^2 + (B^2*g^2*x^2 + 2*B^2*f*g*x + B^2*f^2)*log((b*e*x + a*e)/(d*x
+ c))^2 + 2*(A*B*g^2*x^2 + 2*A*B*f*g*x + A*B*f^2)*log((b*e*x + a*e)/(d*x + c))), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\text {Timed out} \]

[In]

integrate(1/(g*x+f)**2/(A+B*ln(e*(b*x+a)/(d*x+c)))**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 688, normalized size of antiderivative = 23.72 \[ \int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 1.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}} \,d x } \]

[In]

integrate(1/(g*x+f)^2/(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="giac")

[Out]

integrate(1/((g*x + f)^2*(B*log((b*x + a)*e/(d*x + c)) + A)^2), x)

Mupad [N/A]

Not integrable

Time = 22.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2} \,d x \]

[In]

int(1/((f + g*x)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2),x)

[Out]

int(1/((f + g*x)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2), x)