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

Optimal. Leaf size=104 \[ \frac {c+d x}{B g^2 (a+b x) (b c-a d) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}-\frac {e^{-\frac {A}{B}} \text {Ei}\left (\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{B}\right )}{B^2 e g^2 (b c-a d)} \]

[Out]

-Ei((A+B*ln(e*(d*x+c)/(b*x+a)))/B)/B^2/(-a*d+b*c)/e/exp(A/B)/g^2+(d*x+c)/B/(-a*d+b*c)/g^2/(b*x+a)/(A+B*ln(e*(d
*x+c)/(b*x+a)))

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Rubi [F]  time = 0.09, 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 = {} \[ \int \frac {1}{(a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {1}{(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 {1}{(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*}

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Mathematica [A]  time = 0.12, size = 88, normalized size = 0.85 \[ \frac {\frac {e^{-\frac {A}{B}} \text {Ei}\left (\frac {A}{B}+\log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{e}-\frac {B (c+d x)}{(a+b x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}}{B^2 g^2 (a d-b c)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.57, size = 208, normalized size = 2.00 \[ \frac {{\left (B d e x + B c e\right )} e^{\frac {A}{B}} - {\left (A b x + A a + {\left (B b x + B a\right )} \log \left (\frac {d e x + c e}{b x + a}\right )\right )} \operatorname {log\_integral}\left (\frac {{\left (d e x + c e\right )} e^{\frac {A}{B}}}{b x + a}\right )}{{\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} e g^{2} x + {\left (B^{3} a b c - B^{3} a^{2} d\right )} e g^{2}\right )} e^{\frac {A}{B}} \log \left (\frac {d e x + c e}{b x + a}\right ) + {\left ({\left (A B^{2} b^{2} c - A B^{2} a b d\right )} e g^{2} x + {\left (A B^{2} a b c - A B^{2} a^{2} d\right )} e g^{2}\right )} e^{\frac {A}{B}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.34, size = 152, normalized size = 1.46 \[ {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} {\left (\frac {d x e + c e}{{\left (B^{2} g^{2} \log \left (\frac {d x e + c e}{b x + a}\right ) + A B g^{2}\right )} {\left (b x + a\right )}} - \frac {{\rm Ei}\left (\frac {A}{B} + \log \left (\frac {d x e + c e}{b x + a}\right )\right ) e^{\left (-\frac {A}{B}\right )}}{B^{2} g^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.45, size = 258, normalized size = 2.48 \[ \frac {a d}{\left (a d -b c \right ) \left (\ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )+\frac {A}{B}\right ) \left (b x +a \right ) B^{2} b \,g^{2}}-\frac {c}{\left (a d -b c \right ) \left (\ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )+\frac {A}{B}\right ) \left (b x +a \right ) B^{2} g^{2}}-\frac {\Ei \left (1, -\ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )-\frac {A}{B}\right ) {\mathrm e}^{-\frac {A}{B}}}{\left (a d -b c \right ) B^{2} e \,g^{2}}-\frac {d}{\left (a d -b c \right ) \left (\ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )+\frac {A}{B}\right ) B^{2} b \,g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a*d-b*c)/g^2/B^2/(ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)+A/B)/b*d+1/(a*d-b*c)/g^2/B^2/(ln(1/b*d*e-(a*d-b*c)/(b*
x+a)/b*e)+A/B)/b/(b*x+a)*a*d-1/(a*d-b*c)/g^2/B^2/(ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)+A/B)/(b*x+a)*c-1/e/(a*d-b*
c)/g^2/B^2*exp(-A/B)*Ei(1,-ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)-A/B)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {d x + c}{{\left (a b c g^{2} - a^{2} d g^{2}\right )} A B + {\left (a b c g^{2} \log \relax (e) - a^{2} d g^{2} \log \relax (e)\right )} B^{2} + {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} A B + {\left (b^{2} c g^{2} \log \relax (e) - a b d g^{2} \log \relax (e)\right )} B^{2}\right )} x - {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x + {\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (b x + a\right ) + {\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x + {\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (d x + c\right )} + \int \frac {1}{B^{2} a^{2} g^{2} \log \relax (e) + A B a^{2} g^{2} + {\left (B^{2} b^{2} g^{2} \log \relax (e) + A B b^{2} g^{2}\right )} x^{2} + 2 \, {\left (B^{2} a b g^{2} \log \relax (e) + A B a b g^{2}\right )} x - {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (b x + a\right ) + {\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (d x + c\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {- c - d x}{A B a^{2} d g^{2} - A B a b c g^{2} + A B a b d g^{2} x - A B b^{2} c g^{2} x + \left (B^{2} a^{2} d g^{2} - B^{2} a b c g^{2} + B^{2} a b d g^{2} x - B^{2} b^{2} c g^{2} x\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}} + \frac {\int \frac {1}{A a^{2} + 2 A a b x + A b^{2} x^{2} + B a^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + 2 B a b x \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )} + B b^{2} x^{2} \log {\left (\frac {c e}{a + b x} + \frac {d e x}{a + b x} \right )}}\, dx}{B g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-c - d*x)/(A*B*a**2*d*g**2 - A*B*a*b*c*g**2 + A*B*a*b*d*g**2*x - A*B*b**2*c*g**2*x + (B**2*a**2*d*g**2 - B**2
*a*b*c*g**2 + B**2*a*b*d*g**2*x - B**2*b**2*c*g**2*x)*log(e*(c + d*x)/(a + b*x))) + Integral(1/(A*a**2 + 2*A*a
*b*x + A*b**2*x**2 + B*a**2*log(c*e/(a + b*x) + d*e*x/(a + b*x)) + 2*B*a*b*x*log(c*e/(a + b*x) + d*e*x/(a + b*
x)) + B*b**2*x**2*log(c*e/(a + b*x) + d*e*x/(a + b*x))), x)/(B*g**2)

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