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

Optimal. Leaf size=35 \[ \text {Int}\left (\frac {(a g+b g x)^2}{\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \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 \left (\frac {a^2 g^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}+\frac {2 a b g^2 x}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}+\frac {b^2 g^2 x^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}\right ) \, dx\\ &=\left (a^2 g^2\right ) \int \frac {1}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx+\left (2 a b g^2\right ) \int \frac {x}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx+\left (b^2 g^2\right ) \int \frac {x^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.35, size = 0, normalized size = 0.00 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[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]

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}{B^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, A B \log \left (\frac {b e x + a e}{d x + c}\right ) + A^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{2}}{\left (B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{3} d g^{2} x^{4} + a^{3} c g^{2} + {\left (b^{3} c g^{2} + 3 \, a b^{2} d g^{2}\right )} x^{3} + 3 \, {\left (a b^{2} c g^{2} + a^{2} b d g^{2}\right )} x^{2} + {\left (3 \, a^{2} b c g^{2} + a^{3} d g^{2}\right )} x}{{\left (b c - a d\right )} B^{2} \log \left (b x + a\right ) - {\left (b c - a d\right )} B^{2} \log \left (d x + c\right ) + {\left (b c - a d\right )} A B + {\left (b c \log \relax (e) - a d \log \relax (e)\right )} B^{2}} + \int \frac {4 \, b^{3} d g^{2} x^{3} + 3 \, a^{2} b c g^{2} + a^{3} d g^{2} + 3 \, {\left (b^{3} c g^{2} + 3 \, a b^{2} d g^{2}\right )} x^{2} + 6 \, {\left (a b^{2} c g^{2} + a^{2} b d g^{2}\right )} x}{{\left (b c - a d\right )} B^{2} \log \left (b x + a\right ) - {\left (b c - a d\right )} B^{2} \log \left (d x + c\right ) + {\left (b c - a d\right )} A B + {\left (b c \log \relax (e) - a d \log \relax (e)\right )} B^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^2}{{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{3} c g^{2} + a^{3} d g^{2} x + 3 a^{2} b c g^{2} x + 3 a^{2} b d g^{2} x^{2} + 3 a b^{2} c g^{2} x^{2} + 3 a b^{2} d g^{2} x^{3} + b^{3} c g^{2} x^{3} + b^{3} d g^{2} x^{4}}{A B a d - A B b c + \left (B^{2} a d - B^{2} b c\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}} - \frac {g^{2} \left (\int \frac {a^{3} d}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {3 a^{2} b c}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {3 b^{3} c x^{2}}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {4 b^{3} d x^{3}}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {6 a b^{2} c x}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {9 a b^{2} d x^{2}}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {6 a^{2} b d x}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx\right )}{B \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________