3.223 \(\int \frac{(a g+b g x)^{-2-m} (c i+d i x)^m}{(A+B \log (e (\frac{a+b x}{c+d x})^n))^3} \, dx\)
Optimal. Leaf size=306 \[ \frac{(m+1)^2 (a+b x) e^{\frac{A (m+1)}{B n}} (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{m+1}{n}} \text{Ei}\left (-\frac{(m+1) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{2 B^3 i^2 n^3 (c+d x) (b c-a d)}+\frac{(m+1) (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{2 B^2 i^2 n^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}-\frac{(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{2 B i^2 n (c+d x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2} \]
[Out]
(E^((A*(1 + m))/(B*n))*(1 + m)^2*(a + b*x)*(g*(a + b*x))^(-2 - m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(i*(
c + d*x))^(2 + m)*ExpIntegralEi[-(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n))])/(2*B^3*(b*c - a*d)
*i^2*n^3*(c + d*x)) - ((a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))/(2*B*(b*c - a*d)*i^2*n*(c + d*x
)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2) + ((1 + m)*(a + b*x)*(g*(a + b*x))^(-2 - m)*(i*(c + d*x))^(2 + m))
/(2*B^2*(b*c - a*d)*i^2*n^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))
________________________________________________________________________________________
Rubi [F] time = 0.757231, 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 = {}
\[ \int \frac{(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Int[((a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m)/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3,x]
[Out]
Defer[Int][((a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m)/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3, x]
Rubi steps
\begin{align*} \int \frac{(223 c+223 d x)^m (a g+b g x)^{-2-m}}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^3} \, dx &=\int \frac{(223 c+223 d x)^m (a g+b g x)^{-2-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 = 0.324389, size = 0, normalized size = 0. \[ \int \frac{(a g+b g x)^{-2-m} (c i+d i x)^m}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^3} \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[((a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m)/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3,x]
[Out]
Integrate[((a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m)/(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3, x]
________________________________________________________________________________________
Maple [F] time = 25.781, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bgx+ag \right ) ^{-2-m} \left ( dix+ci \right ) ^{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)^(-2-m)*(d*i*x+c*i)^m/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^3,x)
[Out]
int((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^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)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n))^3,x, algorithm="maxima")
[Out]
-(m^2 + 2*m + 1)*i^m*integrate(-1/2*(d*x + c)^m/((B^3*b^2*g^(m + 2)*n^2*x^2 + 2*B^3*a*b*g^(m + 2)*n^2*x + B^3*
a^2*g^(m + 2)*n^2)*(b*x + a)^m*log((b*x + a)^n) - (B^3*b^2*g^(m + 2)*n^2*x^2 + 2*B^3*a*b*g^(m + 2)*n^2*x + B^3
*a^2*g^(m + 2)*n^2)*(b*x + a)^m*log((d*x + c)^n) + (B^3*a^2*g^(m + 2)*n^2*log(e) + A*B^2*a^2*g^(m + 2)*n^2 + (
B^3*b^2*g^(m + 2)*n^2*log(e) + A*B^2*b^2*g^(m + 2)*n^2)*x^2 + 2*(B^3*a*b*g^(m + 2)*n^2*log(e) + A*B^2*a*b*g^(m
+ 2)*n^2)*x)*(b*x + a)^m), x) + 1/2*((B*d*i^m*(m + 1)*x + B*c*i^m*(m + 1))*(d*x + c)^m*log((b*x + a)^n) - (B*
d*i^m*(m + 1)*x + B*c*i^m*(m + 1))*(d*x + c)^m*log((d*x + c)^n) + (A*c*i^m*(m + 1) + (i^m*(m + 1)*log(e) - i^m
*n)*B*c + (A*d*i^m*(m + 1) + (i^m*(m + 1)*log(e) - i^m*n)*B*d)*x)*(d*x + c)^m)/(((b^2*c*g^(m + 2)*n^2 - a*b*d*
g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*log((b*x + a)^n)^2 + ((b^2
*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*l
og((d*x + c)^n)^2 + 2*((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*A*B^3 + (a*b*c*g^(m + 2)*n^2*log(e) - a^2*d
*g^(m + 2)*n^2*log(e))*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A*B^3 + (b^2*c*g^(m + 2)*n^2*log(e)
- a*b*d*g^(m + 2)*n^2*log(e))*B^4)*x)*(b*x + a)^m*log((b*x + a)^n) + ((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n
^2)*A^2*B^2 + 2*(a*b*c*g^(m + 2)*n^2*log(e) - a^2*d*g^(m + 2)*n^2*log(e))*A*B^3 + (a*b*c*g^(m + 2)*n^2*log(e)^
2 - a^2*d*g^(m + 2)*n^2*log(e)^2)*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A^2*B^2 + 2*(b^2*c*g^(m +
2)*n^2*log(e) - a*b*d*g^(m + 2)*n^2*log(e))*A*B^3 + (b^2*c*g^(m + 2)*n^2*log(e)^2 - a*b*d*g^(m + 2)*n^2*log(e
)^2)*B^4)*x)*(b*x + a)^m - 2*(((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*B^4*x + (a*b*c*g^(m + 2)*n^2 - a^2*
d*g^(m + 2)*n^2)*B^4)*(b*x + a)^m*log((b*x + a)^n) + ((a*b*c*g^(m + 2)*n^2 - a^2*d*g^(m + 2)*n^2)*A*B^3 + (a*b
*c*g^(m + 2)*n^2*log(e) - a^2*d*g^(m + 2)*n^2*log(e))*B^4 + ((b^2*c*g^(m + 2)*n^2 - a*b*d*g^(m + 2)*n^2)*A*B^3
+ (b^2*c*g^(m + 2)*n^2*log(e) - a*b*d*g^(m + 2)*n^2*log(e))*B^4)*x)*(b*x + a)^m)*log((d*x + c)^n))
________________________________________________________________________________________
Fricas [B] time = 0.547344, size = 1764, normalized size = 5.76 \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)^(-2-m)*(d*i*x+c*i)^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 - 2)*e^(m*log(b*g*
x + a*g) - m*log((b*x + a)/(d*x + c)) + m*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)*log(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*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)*log(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)**(-2-m)*(d*i*x+c*i)**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 - 2}{\left (d i x + c i\right )}^{m}}{{\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)^(-2-m)*(d*i*x+c*i)^m/(A+B*log(e*((b*x+a)/(d*x+c))^n))^3,x, algorithm="giac")
[Out]
integrate((b*g*x + a*g)^(-m - 2)*(d*i*x + c*i)^m/(B*log(e*((b*x + a)/(d*x + c))^n) + A)^3, x)