\(\int \frac {(a g+b g x)^2}{(A+B \log (\frac {e (c+d x)}{a+b x}))^2} \, dx\) [196]

Optimal result

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

[Out]

Unintegrable((b*g*x+a*g)^2/(A+B*ln(e*(d*x+c)/(b*x+a)))^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 = {} \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx \]

[In]

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

[Out]

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

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

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

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

[In]

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

Sympy [N/A]

Not integrable

Time = 11.03 (sec) , antiderivative size = 400, normalized size of antiderivative = 12.50 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\frac {- a^{3} c g^{2} - a^{3} d g^{2} x - 3 a^{2} b c g^{2} x - 3 a^{2} b d g^{2} x^{2} - 3 a b^{2} c g^{2} x^{2} - 3 a b^{2} d g^{2} x^{3} - b^{3} c g^{2} x^{3} - b^{3} d g^{2} x^{4}}{A B a d - A B b c + \left (B^{2} a d - B^{2} b c\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}} + \frac {g^{2} \left (\int \frac {a^{3} d}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {3 a^{2} b c}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {3 b^{3} c x^{2}}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {4 b^{3} d x^{3}}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {6 a b^{2} c x}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {9 a b^{2} d x^{2}}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx + \int \frac {6 a^{2} b d x}{A + B \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx\right )}{B \left (a d - b c\right )} \]

[In]

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

[Out]

(-a**3*c*g**2 - a**3*d*g**2*x - 3*a**2*b*c*g**2*x - 3*a**2*b*d*g**2*x**2 - 3*a*b**2*c*g**2*x**2 - 3*a*b**2*d*g
**2*x**3 - b**3*c*g**2*x**3 - b**3*d*g**2*x**4)/(A*B*a*d - A*B*b*c + (B**2*a*d - B**2*b*c)*log(e*(c + d*x)/(a
+ b*x))) + g**2*(Integral(a**3*d/(A + B*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x) + Integral(3*a**2*b*c/(A + B
*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x) + Integral(3*b**3*c*x**2/(A + B*log(c*e/(a + b*x) + d*e*x/(a + b*x)
)), x) + Integral(4*b**3*d*x**3/(A + B*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x) + Integral(6*a*b**2*c*x/(A +
B*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x) + Integral(9*a*b**2*d*x**2/(A + B*log(c*e/(a + b*x) + d*e*x/(a + b
*x))), x) + Integral(6*a**2*b*d*x/(A + B*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x))/(B*(a*d - b*c))

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 309, normalized size of antiderivative = 9.66 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2}}{{\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2}} \,d x } \]

[In]

integrate((b*g*x+a*g)^2/(A+B*log(e*(d*x+c)/(b*x+a)))^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 = 0.70 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2}}{{\left (B \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^2}{{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2} \,d x \]

[In]

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

[Out]

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