3.215 \(\int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log (e (\frac{a+b x}{c+d x})^n)} \, dx\)

Optimal. Leaf size=125 \[ \frac{(a+b x) e^{-\frac{A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B i^2 n (c+d x) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.736634, 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 = {} \[ \int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{(215 c+215 d x)^{-2-m} (a g+b g x)^m}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\int \frac{(215 c+215 d x)^{-2-m} (a g+b g x)^m}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx\\ \end{align*}

Mathematica [F]  time = 0.231827, size = 0, normalized size = 0. \[ \int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 2.954, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bgx+ag \right ) ^{m} \left ( dix+ci \right ) ^{-2-m} \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((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)/(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.496147, size = 227, normalized size = 1.82 \begin{align*} \frac{{\rm Ei}\left (\frac{{\left (B m + B\right )} n \log \left (\frac{b x + a}{d x + c}\right ) + A m +{\left (B m + B\right )} \log \left (e\right ) + A}{B n}\right ) e^{\left (-\frac{{\left (B m + 2 \, B\right )} n \log \left (\frac{i}{g}\right ) + A m +{\left (B m + B\right )} \log \left (e\right ) + A}{B n}\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((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)/(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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