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

Optimal. Leaf size=153 \[ -\frac{e^{\frac{A}{B n}} (c+d x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^2 n^2 (a+b x) (b c-a d)}-\frac{c+d x}{B g^2 n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )} \]

[Out]

-((E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*(c + d*x)*ExpIntegralEi[-((A + B*Log[e*((a + b*x)/(c + d*x))
^n])/(B*n))])/(B^2*(b*c - a*d)*g^2*n^2*(a + b*x))) - (c + d*x)/(B*(b*c - a*d)*g^2*n*(a + b*x)*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]))

________________________________________________________________________________________

Rubi [F]  time = 0.104589, 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 = {} \[ \int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.182995, size = 146, normalized size = 0.95 \[ -\frac{(c+d x) \left (e^{\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right ) \text{Ei}\left (-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )+B n\right )}{B^2 g^2 n^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c + d*x)*(B*n + E^(A/(B*n))*(e*((a + b*x)/(c + d*x))^n)^n^(-1)*ExpIntegralEi[-((A + B*Log[e*((a + b*x)/(c
+ d*x))^n])/(B*n))]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(B^2*(b*c - a*d)*g^2*n^2*(a + b*x)*(A + B*Log[e*(
(a + b*x)/(c + d*x))^n])))

________________________________________________________________________________________

Maple [F]  time = 0.441, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-2}}\, dx \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*((b*x+a)/(d*x+c))^n))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{d x + c}{{\left (a b c g^{2} n - a^{2} d g^{2} n\right )} A B +{\left (a b c g^{2} n \log \left (e\right ) - a^{2} d g^{2} n \log \left (e\right )\right )} B^{2} +{\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} A B +{\left (b^{2} c g^{2} n \log \left (e\right ) - a b d g^{2} n \log \left (e\right )\right )} B^{2}\right )} x +{\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} B^{2} x +{\left (a b c g^{2} n - a^{2} d g^{2} n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left ({\left (b^{2} c g^{2} n - a b d g^{2} n\right )} B^{2} x +{\left (a b c g^{2} n - a^{2} d g^{2} n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )} + \int -\frac{1}{B^{2} a^{2} g^{2} n \log \left (e\right ) + A B a^{2} g^{2} n +{\left (B^{2} b^{2} g^{2} n \log \left (e\right ) + A B b^{2} g^{2} n\right )} x^{2} + 2 \,{\left (B^{2} a b g^{2} n \log \left (e\right ) + A B a b g^{2} n\right )} x +{\left (B^{2} b^{2} g^{2} n x^{2} + 2 \, B^{2} a b g^{2} n x + B^{2} a^{2} g^{2} n\right )} \log \left ({\left (b x + a\right )}^{n}\right ) -{\left (B^{2} b^{2} g^{2} n x^{2} + 2 \, B^{2} a b g^{2} n x + B^{2} a^{2} g^{2} n\right )} \log \left ({\left (d x + c\right )}^{n}\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*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.930972, size = 598, normalized size = 3.91 \begin{align*} -\frac{B d n x + B c n +{\left (A b x + A a +{\left (B b x + B a\right )} \log \left (e\right ) +{\left (B b n x + B a n\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} e^{\left (\frac{B \log \left (e\right ) + A}{B n}\right )} \logintegral \left (\frac{{\left (d x + c\right )} e^{\left (-\frac{B \log \left (e\right ) + A}{B n}\right )}}{b x + a}\right )}{{\left (A B^{2} b^{2} c - A B^{2} a b d\right )} g^{2} n^{2} x +{\left (A B^{2} a b c - A B^{2} a^{2} d\right )} g^{2} n^{2} +{\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} n^{2} x +{\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2} n^{2}\right )} \log \left (e\right ) +{\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} n^{3} x +{\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2} n^{3}\right )} \log \left (\frac{b x + a}{d x + c}\right )} \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*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b g x + a g\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{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*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

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