3.28 \(\int \frac{1}{(a g+b g x)^3 (A+B \log (e (\frac{a+b x}{c+d x})^n))^2} \, dx\)
Optimal. Leaf size=314 \[ -\frac{2 b e^{\frac{2 A}{B n}} (c+d x)^2 \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{2/n} \text{Ei}\left (-\frac{2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B^2 g^3 n^2 (a+b x)^2 (b c-a d)^2}+\frac{d e^{\frac{A}{B n}} (c+d x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} \text{Ei}\left (-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B^2 g^3 n^2 (a+b x) (b c-a d)^2}-\frac{b (c+d x)^2}{B g^3 n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}+\frac{d (c+d x)}{B g^3 n (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )} \]
[Out]
(-2*b*E^((2*A)/(B*n))*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)^2*ExpIntegralEi[(-2*(A + B*Log[e*((a + b*x)/
(c + d*x))^n]))/(B*n)])/(B^2*(b*c - a*d)^2*g^3*n^2*(a + b*x)^2) + (d*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)^2*g^3*n^2*(a
+ b*x)) + (d*(c + d*x))/(B*(b*c - a*d)^2*g^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])) - (b*(c + d*x
)^2)/(B*(b*c - a*d)^2*g^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))
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Rubi [F] time = 0.0928482, 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 = {}
\[ \int \frac{1}{(a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
[In]
Int[1/((a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2),x]
[Out]
Defer[Int][1/((a*g + b*g*x)^3*(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)^3 \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)^3 \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 = 0.643753, size = 254, normalized size = 0.81 \[ \frac{(c+d x) \left (-2 b e^{\frac{2 A}{B n}} (c+d x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{2/n} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right ) \text{Ei}\left (-\frac{2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )+d (a+b x) e^{\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right ) \text{Ei}\left (-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )+B n (a d-b c)\right )}{B^2 g^3 n^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a*g + b*g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2),x]
[Out]
((c + d*x)*(B*(-(b*c) + a*d)*n - 2*b*E^((2*A)/(B*n))*(e*((a + b*x)/(c + d*x))^n)^(2/n)*(c + d*x)*ExpIntegralEi
[(-2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(B*n)]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + d*E^(A/(B*n))*(a
+ b*x)*(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)^2*g^3*n^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^
n]))
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Maple [F] time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3}} \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)^3/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)
[Out]
int(1/(b*g*x+a*g)^3/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)
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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(1/(b*g*x+a*g)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")
[Out]
-(d*x + c)/((a^2*b*c*g^3*n - a^3*d*g^3*n)*A*B + (a^2*b*c*g^3*n*log(e) - a^3*d*g^3*n*log(e))*B^2 + ((b^3*c*g^3*
n - a*b^2*d*g^3*n)*A*B + (b^3*c*g^3*n*log(e) - a*b^2*d*g^3*n*log(e))*B^2)*x^2 + 2*((a*b^2*c*g^3*n - a^2*b*d*g^
3*n)*A*B + (a*b^2*c*g^3*n*log(e) - a^2*b*d*g^3*n*log(e))*B^2)*x + ((b^3*c*g^3*n - a*b^2*d*g^3*n)*B^2*x^2 + 2*(
a*b^2*c*g^3*n - a^2*b*d*g^3*n)*B^2*x + (a^2*b*c*g^3*n - a^3*d*g^3*n)*B^2)*log((b*x + a)^n) - ((b^3*c*g^3*n - a
*b^2*d*g^3*n)*B^2*x^2 + 2*(a*b^2*c*g^3*n - a^2*b*d*g^3*n)*B^2*x + (a^2*b*c*g^3*n - a^3*d*g^3*n)*B^2)*log((d*x
+ c)^n)) - integrate((b*d*x + 2*b*c - a*d)/(((b^4*c*g^3*n - a*b^3*d*g^3*n)*A*B + (b^4*c*g^3*n*log(e) - a*b^3*d
*g^3*n*log(e))*B^2)*x^3 + (a^3*b*c*g^3*n - a^4*d*g^3*n)*A*B + (a^3*b*c*g^3*n*log(e) - a^4*d*g^3*n*log(e))*B^2
+ 3*((a*b^3*c*g^3*n - a^2*b^2*d*g^3*n)*A*B + (a*b^3*c*g^3*n*log(e) - a^2*b^2*d*g^3*n*log(e))*B^2)*x^2 + 3*((a^
2*b^2*c*g^3*n - a^3*b*d*g^3*n)*A*B + (a^2*b^2*c*g^3*n*log(e) - a^3*b*d*g^3*n*log(e))*B^2)*x + ((b^4*c*g^3*n -
a*b^3*d*g^3*n)*B^2*x^3 + 3*(a*b^3*c*g^3*n - a^2*b^2*d*g^3*n)*B^2*x^2 + 3*(a^2*b^2*c*g^3*n - a^3*b*d*g^3*n)*B^2
*x + (a^3*b*c*g^3*n - a^4*d*g^3*n)*B^2)*log((b*x + a)^n) - ((b^4*c*g^3*n - a*b^3*d*g^3*n)*B^2*x^3 + 3*(a*b^3*c
*g^3*n - a^2*b^2*d*g^3*n)*B^2*x^2 + 3*(a^2*b^2*c*g^3*n - a^3*b*d*g^3*n)*B^2*x + (a^3*b*c*g^3*n - a^4*d*g^3*n)*
B^2)*log((d*x + c)^n)), x)
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Fricas [B] time = 0.953429, size = 1612, normalized size = 5.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(b*g*x+a*g)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")
[Out]
-((B*b*c*d - B*a*d^2)*n*x - (A*b^2*d*x^2 + 2*A*a*b*d*x + A*a^2*d + (B*b^2*d*x^2 + 2*B*a*b*d*x + B*a^2*d)*log(e
) + (B*b^2*d*n*x^2 + 2*B*a*b*d*n*x + B*a^2*d*n)*log((b*x + a)/(d*x + c)))*e^((B*log(e) + A)/(B*n))*log_integra
l((d*x + c)*e^(-(B*log(e) + A)/(B*n))/(b*x + a)) + 2*(A*b^3*x^2 + 2*A*a*b^2*x + A*a^2*b + (B*b^3*x^2 + 2*B*a*b
^2*x + B*a^2*b)*log(e) + (B*b^3*n*x^2 + 2*B*a*b^2*n*x + B*a^2*b*n)*log((b*x + a)/(d*x + c)))*e^(2*(B*log(e) +
A)/(B*n))*log_integral((d^2*x^2 + 2*c*d*x + c^2)*e^(-2*(B*log(e) + A)/(B*n))/(b^2*x^2 + 2*a*b*x + a^2)) + (B*b
*c^2 - B*a*c*d)*n)/((A*B^2*b^4*c^2 - 2*A*B^2*a*b^3*c*d + A*B^2*a^2*b^2*d^2)*g^3*n^2*x^2 + 2*(A*B^2*a*b^3*c^2 -
2*A*B^2*a^2*b^2*c*d + A*B^2*a^3*b*d^2)*g^3*n^2*x + (A*B^2*a^2*b^2*c^2 - 2*A*B^2*a^3*b*c*d + A*B^2*a^4*d^2)*g^
3*n^2 + ((B^3*b^4*c^2 - 2*B^3*a*b^3*c*d + B^3*a^2*b^2*d^2)*g^3*n^2*x^2 + 2*(B^3*a*b^3*c^2 - 2*B^3*a^2*b^2*c*d
+ B^3*a^3*b*d^2)*g^3*n^2*x + (B^3*a^2*b^2*c^2 - 2*B^3*a^3*b*c*d + B^3*a^4*d^2)*g^3*n^2)*log(e) + ((B^3*b^4*c^2
- 2*B^3*a*b^3*c*d + B^3*a^2*b^2*d^2)*g^3*n^3*x^2 + 2*(B^3*a*b^3*c^2 - 2*B^3*a^2*b^2*c*d + B^3*a^3*b*d^2)*g^3*
n^3*x + (B^3*a^2*b^2*c^2 - 2*B^3*a^3*b*c*d + B^3*a^4*d^2)*g^3*n^3)*log((b*x + a)/(d*x + c)))
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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/(b*g*x+a*g)**3/(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b g x + a g\right )}^{3}{\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)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")
[Out]
integrate(1/((b*g*x + a*g)^3*(B*log(e*((b*x + a)/(d*x + c))^n) + A)^2), x)