3.217 \(\int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{(A+B \log (e (\frac{a+b x}{c+d x})^n))^3} \, dx\)
Optimal. Leaf size=295 \[ \frac{(m+1)^2 (a+b x) e^{-\frac{A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{2 B^3 i^2 n^3 (c+d x) (b c-a d)}-\frac{(m+1) (a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{2 B^2 i^2 n^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}-\frac{(a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{2 B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2} \]
[Out]
((1 + m)^2*(a + b*x)*(g*(a + b*x))^m*ExpIntegralEi[((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)])/(2
*B^3*(b*c - a*d)*E^((A*(1 + m))/(B*n))*i^2*n^3*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(c + d*x)*(i*(c + d*x))
^m) - ((a + b*x)*(g*(a + b*x))^m)/(2*B*(b*c - a*d)*i^2*n*(c + d*x)*(i*(c + d*x))^m*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n])^2) - ((1 + m)*(a + b*x)*(g*(a + b*x))^m)/(2*B^2*(b*c - a*d)*i^2*n^2*(c + d*x)*(i*(c + d*x))^m*(A +
B*Log[e*((a + b*x)/(c + d*x))^n]))
________________________________________________________________________________________
Rubi [F] time = 0.814608, 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 = {}
\[ \int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]
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])^3,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])^3, x]
Rubi steps
\begin{align*} \int \frac{(217 c+217 d x)^{-2-m} (a g+b g x)^m}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^3} \, dx &=\int \frac{(217 c+217 d x)^{-2-m} (a g+b g x)^m}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^3} \, dx\\ \end{align*}
Mathematica [F] time = 0.324248, size = 0, normalized size = 0. \[ \int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]
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])^3,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])^3, x]
________________________________________________________________________________________
Maple [F] time = 23.922, 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 ) ^{-3}}\, 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))^3,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))^3,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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))^3,x, algorithm="maxima")
[Out]
-(m^2 + 2*m + 1)*g^m*integrate(-1/2*(b*x + a)^m/((B^3*d^2*i^(m + 2)*n^2*x^2 + 2*B^3*c*d*i^(m + 2)*n^2*x + B^3*
c^2*i^(m + 2)*n^2)*(d*x + c)^m*log((b*x + a)^n) - (B^3*d^2*i^(m + 2)*n^2*x^2 + 2*B^3*c*d*i^(m + 2)*n^2*x + B^3
*c^2*i^(m + 2)*n^2)*(d*x + c)^m*log((d*x + c)^n) + (B^3*c^2*i^(m + 2)*n^2*log(e) + A*B^2*c^2*i^(m + 2)*n^2 + (
B^3*d^2*i^(m + 2)*n^2*log(e) + A*B^2*d^2*i^(m + 2)*n^2)*x^2 + 2*(B^3*c*d*i^(m + 2)*n^2*log(e) + A*B^2*c*d*i^(m
+ 2)*n^2)*x)*(d*x + c)^m), x) - 1/2*((B*b*g^m*(m + 1)*x + B*a*g^m*(m + 1))*(b*x + a)^m*log((b*x + a)^n) - (B*
b*g^m*(m + 1)*x + B*a*g^m*(m + 1))*(b*x + a)^m*log((d*x + c)^n) + (A*a*g^m*(m + 1) + (g^m*(m + 1)*log(e) + g^m
*n)*B*a + (A*b*g^m*(m + 1) + (g^m*(m + 1)*log(e) + g^m*n)*B*b)*x)*(b*x + a)^m)/(((b*c*d*i^(m + 2)*n^2 - a*d^2*
i^(m + 2)*n^2)*B^4*x + (b*c^2*i^(m + 2)*n^2 - a*c*d*i^(m + 2)*n^2)*B^4)*(d*x + c)^m*log((b*x + a)^n)^2 + ((b*c
*d*i^(m + 2)*n^2 - a*d^2*i^(m + 2)*n^2)*B^4*x + (b*c^2*i^(m + 2)*n^2 - a*c*d*i^(m + 2)*n^2)*B^4)*(d*x + c)^m*l
og((d*x + c)^n)^2 + 2*((b*c^2*i^(m + 2)*n^2 - a*c*d*i^(m + 2)*n^2)*A*B^3 + (b*c^2*i^(m + 2)*n^2*log(e) - a*c*d
*i^(m + 2)*n^2*log(e))*B^4 + ((b*c*d*i^(m + 2)*n^2 - a*d^2*i^(m + 2)*n^2)*A*B^3 + (b*c*d*i^(m + 2)*n^2*log(e)
- a*d^2*i^(m + 2)*n^2*log(e))*B^4)*x)*(d*x + c)^m*log((b*x + a)^n) + ((b*c^2*i^(m + 2)*n^2 - a*c*d*i^(m + 2)*n
^2)*A^2*B^2 + 2*(b*c^2*i^(m + 2)*n^2*log(e) - a*c*d*i^(m + 2)*n^2*log(e))*A*B^3 + (b*c^2*i^(m + 2)*n^2*log(e)^
2 - a*c*d*i^(m + 2)*n^2*log(e)^2)*B^4 + ((b*c*d*i^(m + 2)*n^2 - a*d^2*i^(m + 2)*n^2)*A^2*B^2 + 2*(b*c*d*i^(m +
2)*n^2*log(e) - a*d^2*i^(m + 2)*n^2*log(e))*A*B^3 + (b*c*d*i^(m + 2)*n^2*log(e)^2 - a*d^2*i^(m + 2)*n^2*log(e
)^2)*B^4)*x)*(d*x + c)^m - 2*(((b*c*d*i^(m + 2)*n^2 - a*d^2*i^(m + 2)*n^2)*B^4*x + (b*c^2*i^(m + 2)*n^2 - a*c*
d*i^(m + 2)*n^2)*B^4)*(d*x + c)^m*log((b*x + a)^n) + ((b*c^2*i^(m + 2)*n^2 - a*c*d*i^(m + 2)*n^2)*A*B^3 + (b*c
^2*i^(m + 2)*n^2*log(e) - a*c*d*i^(m + 2)*n^2*log(e))*B^4 + ((b*c*d*i^(m + 2)*n^2 - a*d^2*i^(m + 2)*n^2)*A*B^3
+ (b*c*d*i^(m + 2)*n^2*log(e) - a*d^2*i^(m + 2)*n^2*log(e))*B^4)*x)*(d*x + c)^m)*log((d*x + c)^n))
________________________________________________________________________________________
Fricas [B] time = 0.582135, size = 1791, normalized size = 6.07 \begin{align*} \text{result too large to display} \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))^3,x, algorithm="fricas")
[Out]
-1/2*((B^2*a*c*g^2*n^2 + (B^2*b*d*g^2*n^2 + (A*B*b*d*g^2*m + A*B*b*d*g^2)*n)*x^2 + (A*B*a*c*g^2*m + A*B*a*c*g^
2)*n + ((B^2*b*c + B^2*a*d)*g^2*n^2 + ((A*B*b*c + A*B*a*d)*g^2*m + (A*B*b*c + A*B*a*d)*g^2)*n)*x + ((B^2*b*d*g
^2*m + B^2*b*d*g^2)*n*x^2 + ((B^2*b*c + B^2*a*d)*g^2*m + (B^2*b*c + B^2*a*d)*g^2)*n*x + (B^2*a*c*g^2*m + B^2*a
*c*g^2)*n)*log(e) + ((B^2*b*d*g^2*m + B^2*b*d*g^2)*n^2*x^2 + ((B^2*b*c + B^2*a*d)*g^2*m + (B^2*b*c + B^2*a*d)*
g^2)*n^2*x + (B^2*a*c*g^2*m + B^2*a*c*g^2)*n^2)*log((b*x + a)/(d*x + c)))*(b*g*x + a*g)^m*e^(-(m + 2)*log(b*g*
x + a*g) + (m + 2)*log((b*x + a)/(d*x + c)) - (m + 2)*log(i/g)) - ((B^2*m^2 + 2*B^2*m + B^2)*n^2*log((b*x + a)
/(d*x + c))^2 + A^2*m^2 + 2*A^2*m + (B^2*m^2 + 2*B^2*m + B^2)*log(e)^2 + 2*(A*B*m^2 + 2*A*B*m + A*B)*n*log((b*
x + a)/(d*x + c)) + A^2 + 2*(A*B*m^2 + 2*A*B*m + (B^2*m^2 + 2*B^2*m + B^2)*n*log((b*x + a)/(d*x + c)) + A*B)*l
og(e))*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^5*b*c - B^5*a*d)*g^2*n^5*log((b*x + a)/(d*x + c))^2 + (B^5*b*c -
B^5*a*d)*g^2*n^3*log(e)^2 + 2*(A*B^4*b*c - A*B^4*a*d)*g^2*n^4*log((b*x + a)/(d*x + c)) + (A^2*B^3*b*c - A^2*B^
3*a*d)*g^2*n^3 + 2*((B^5*b*c - B^5*a*d)*g^2*n^4*log((b*x + a)/(d*x + c)) + (A*B^4*b*c - A*B^4*a*d)*g^2*n^3)*lo
g(e))
________________________________________________________________________________________
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))**3,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}}{{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{3}}\,{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))^3,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)^3, x)