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]

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

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Rubi [F]  time = 0.0886496, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.124988, 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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Maple [B]  time = 0.16, size = 258, normalized size = 2.5 \begin{align*} -{\frac{d}{ \left ( ad-bc \right ){g}^{2}{B}^{2}b} \left ( \ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) +{\frac{A}{B}} \right ) ^{-1}}+{\frac{ad}{ \left ( ad-bc \right ){g}^{2}{B}^{2}b \left ( bx+a \right ) } \left ( \ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) +{\frac{A}{B}} \right ) ^{-1}}-{\frac{c}{ \left ( ad-bc \right ){g}^{2}{B}^{2} \left ( bx+a \right ) } \left ( \ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) +{\frac{A}{B}} \right ) ^{-1}}-{\frac{1}{e \left ( ad-bc \right ){g}^{2}{B}^{2}}{{\rm e}^{-{\frac{A}{B}}}}{\it Ei} \left ( 1,-\ln \left ({\frac{de}{b}}-{\frac{e \left ( ad-bc \right ) }{b \left ( bx+a \right ) }} \right ) -{\frac{A}{B}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \left (e\right ) - a^{2} d g^{2} \log \left (e\right )\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 \left (e\right ) - a b d g^{2} \log \left (e\right )\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 \left (e\right ) + A B a^{2} g^{2} +{\left (B^{2} b^{2} g^{2} \log \left (e\right ) + A B b^{2} g^{2}\right )} x^{2} + 2 \,{\left (B^{2} a b g^{2} \log \left (e\right ) + 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} \end{align*}

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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Fricas [B]  time = 0.970356, size = 439, normalized size = 4.22 \begin{align*} \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 )} \logintegral \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}}} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [B]  time = 1.41631, size = 344, normalized size = 3.31 \begin{align*} -\frac{{\rm Ei}\left (\frac{A}{B} + \log \left (\frac{d x + c}{b x + a}\right ) + 1\right ) e^{\left (-\frac{A}{B} - 1\right )}}{B^{2} b c g^{2} - B^{2} a d g^{2}} + \frac{d x + c}{B^{2} b^{2} c g^{2} x \log \left (\frac{d x + c}{b x + a}\right ) - B^{2} a b d g^{2} x \log \left (\frac{d x + c}{b x + a}\right ) + A B b^{2} c g^{2} x + B^{2} b^{2} c g^{2} x - A B a b d g^{2} x - B^{2} a b d g^{2} x + B^{2} a b c g^{2} \log \left (\frac{d x + c}{b x + a}\right ) - B^{2} a^{2} d g^{2} \log \left (\frac{d x + c}{b x + a}\right ) + A B a b c g^{2} + B^{2} a b c g^{2} - A B a^{2} d g^{2} - B^{2} a^{2} d g^{2}} \end{align*}

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]

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