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

Optimal. Leaf size=96 \[ \frac{(a+b x) e^{-\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B g^2 n (c+d x) (b c-a d)} \]

[Out]

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

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Rubi [F]  time = 0.100225, 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 = {} \[ \int \frac{1}{(c g+d g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.119399, size = 96, normalized size = 1. \[ \frac{(a+b x) e^{-\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B g^2 n (c+d x) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.904063, size = 149, normalized size = 1.55 \begin{align*} \frac{e^{\left (-\frac{B \log \left (e\right ) + A}{B n}\right )} \logintegral \left (\frac{{\left (b x + a\right )} e^{\left (\frac{B \log \left (e\right ) + A}{B n}\right )}}{d x + c}\right )}{{\left (B b c - B a d\right )} g^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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