3.212 \(\int (a g+b g x)^m (c i+d i x)^{-2-m} (A+B \log (e (\frac{a+b x}{c+d x})^n))^3 \, dx\)
Optimal. Leaf size=292 \[ \frac{6 B^2 n^2 (a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{i^2 (m+1)^3 (c+d x) (b c-a d)}+\frac{(a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^3}{i^2 (m+1) (c+d x) (b c-a d)}-\frac{3 B n (a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{i^2 (m+1)^2 (c+d x) (b c-a d)}-\frac{6 B^3 n^3 (a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{i^2 (m+1)^4 (c+d x) (b c-a d)} \]
[Out]
(-6*B^3*n^3*(a + b*x)*(g*(a + b*x))^m)/((b*c - a*d)*i^2*(1 + m)^4*(c + d*x)*(i*(c + d*x))^m) + (6*B^2*n^2*(a +
b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)*i^2*(1 + m)^3*(c + d*x)*(i*(c + d*x
))^m) - (3*B*n*(a + b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)*i^2*(1 + m)^2*
(c + d*x)*(i*(c + d*x))^m) + ((a + b*x)*(g*(a + b*x))^m*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/((b*c - a*d)
*i^2*(1 + m)*(c + d*x)*(i*(c + d*x))^m)
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Rubi [F] time = 1.97695, 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 = {}
\[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^3 \, dx \]
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]
(A^3*(a*g + b*g*x)^(1 + m)*(c*i + d*i*x)^(-1 - m))/((b*c - a*d)*g*i*(1 + m)) - (3*A^2*B*n*(a*g + b*g*x)^(1 + m
)*(c*i + d*i*x)^(-1 - m))/((b*c - a*d)*g*i*(1 + m)^2) + (3*A^2*B*(a*g + b*g*x)^(1 + m)*(c*i + d*i*x)^(-1 - m)*
Log[e*((a + b*x)/(c + d*x))^n])/((b*c - a*d)*g*i*(1 + m)) + 3*A*B^2*Defer[Int][(a*g + b*g*x)^m*(c*i + d*i*x)^(
-2 - m)*Log[e*((a + b*x)/(c + d*x))^n]^2, x] + B^3*Defer[Int][(a*g + b*g*x)^m*(c*i + d*i*x)^(-2 - m)*Log[e*((a
+ b*x)/(c + d*x))^n]^3, x]
Rubi steps
\begin{align*} \int (212 c+212 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 \left (A^3 (212 c+212 d x)^{-2-m} (a g+b g x)^m+3 A^2 B (212 c+212 d x)^{-2-m} (a g+b g x)^m \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+3 A B^2 (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B^3 (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx\\ &=A^3 \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \, dx+\left (3 A^2 B\right ) \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx+\left (3 A B^2\right ) \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx\\ &=\frac{A^3 (212 c+212 d x)^{-1-m} (a g+b g x)^{1+m}}{212 (b c-a d) g (1+m)}+\frac{3 A^2 B (212 c+212 d x)^{-1-m} (a g+b g x)^{1+m} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{212 (b c-a d) g (1+m)}-\left (3 A^2 B\right ) \int \frac{212^{-2-m} n (c+d x)^{-2-m} (a g+b g x)^m}{1+m} \, dx+\left (3 A B^2\right ) \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx\\ &=\frac{A^3 (212 c+212 d x)^{-1-m} (a g+b g x)^{1+m}}{212 (b c-a d) g (1+m)}+\frac{3 A^2 B (212 c+212 d x)^{-1-m} (a g+b g x)^{1+m} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{212 (b c-a d) g (1+m)}+\left (3 A B^2\right ) \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx-\frac{\left (3\ 212^{-2-m} A^2 B n\right ) \int (c+d x)^{-2-m} (a g+b g x)^m \, dx}{1+m}\\ &=-\frac{3\ 212^{-2-m} A^2 B n (c+d x)^{-1-m} (a g+b g x)^{1+m}}{(b c-a d) g (1+m)^2}+\frac{A^3 (212 c+212 d x)^{-1-m} (a g+b g x)^{1+m}}{212 (b c-a d) g (1+m)}+\frac{3 A^2 B (212 c+212 d x)^{-1-m} (a g+b g x)^{1+m} \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{212 (b c-a d) g (1+m)}+\left (3 A B^2\right ) \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (212 c+212 d x)^{-2-m} (a g+b g x)^m \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx\\ \end{align*}
Mathematica [A] time = 7.57979, size = 206, normalized size = 0.71 \[ \frac{(a+b x) (g (a+b x))^m (i (c+d x))^{-m-1} \left (3 B (m+1) \left (A^2 (m+1)^2-2 A B (m+1) n+2 B^2 n^2\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+3 B^2 (m+1)^2 (A m+A-B n) \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B^3 (m+1)^3 \log ^3\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-3 A^2 B (m+1)^2 n+A^3 (m+1)^3+6 A B^2 (m+1) n^2-6 B^3 n^3\right )}{i (m+1)^4 (b c-a d)} \]
Antiderivative was successfully verified.
[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]
((a + b*x)*(g*(a + b*x))^m*(i*(c + d*x))^(-1 - m)*(A^3*(1 + m)^3 - 3*A^2*B*(1 + m)^2*n + 6*A*B^2*(1 + m)*n^2 -
6*B^3*n^3 + 3*B*(1 + m)*(A^2*(1 + m)^2 - 2*A*B*(1 + m)*n + 2*B^2*n^2)*Log[e*((a + b*x)/(c + d*x))^n] + 3*B^2*
(1 + m)^2*(A + A*m - B*n)*Log[e*((a + b*x)/(c + d*x))^n]^2 + B^3*(1 + m)^3*Log[e*((a + b*x)/(c + d*x))^n]^3))/
((b*c - a*d)*i*(1 + m)^4)
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Maple [F] time = 5.497, 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*} \int{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{3}{\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}\,{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="maxima")
[Out]
integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^3*(b*g*x + a*g)^m*(d*i*x + c*i)^(-m - 2), x)
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Fricas [B] time = 0.82571, size = 5800, normalized size = 19.86 \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]
(A^3*a*c*m^3 - 6*B^3*a*c*n^3 + 3*A^3*a*c*m^2 + 3*A^3*a*c*m + A^3*a*c + (B^3*a*c*m^3 + 3*B^3*a*c*m^2 + 3*B^3*a*
c*m + B^3*a*c + (B^3*b*d*m^3 + 3*B^3*b*d*m^2 + 3*B^3*b*d*m + B^3*b*d)*x^2 + (B^3*b*c + B^3*a*d + (B^3*b*c + B^
3*a*d)*m^3 + 3*(B^3*b*c + B^3*a*d)*m^2 + 3*(B^3*b*c + B^3*a*d)*m)*x)*log(e)^3 + ((B^3*b*d*m^3 + 3*B^3*b*d*m^2
+ 3*B^3*b*d*m + B^3*b*d)*n^3*x^2 + (B^3*b*c + B^3*a*d + (B^3*b*c + B^3*a*d)*m^3 + 3*(B^3*b*c + B^3*a*d)*m^2 +
3*(B^3*b*c + B^3*a*d)*m)*n^3*x + (B^3*a*c*m^3 + 3*B^3*a*c*m^2 + 3*B^3*a*c*m + B^3*a*c)*n^3)*log((b*x + a)/(d*x
+ c))^3 + 6*(A*B^2*a*c*m + A*B^2*a*c)*n^2 + (A^3*b*d*m^3 - 6*B^3*b*d*n^3 + 3*A^3*b*d*m^2 + 3*A^3*b*d*m + A^3*
b*d + 6*(A*B^2*b*d*m + A*B^2*b*d)*n^2 - 3*(A^2*B*b*d*m^2 + 2*A^2*B*b*d*m + A^2*B*b*d)*n)*x^2 + 3*(A*B^2*a*c*m^
3 + 3*A*B^2*a*c*m^2 + 3*A*B^2*a*c*m + A*B^2*a*c + (A*B^2*b*d*m^3 + 3*A*B^2*b*d*m^2 + 3*A*B^2*b*d*m + A*B^2*b*d
- (B^3*b*d*m^2 + 2*B^3*b*d*m + B^3*b*d)*n)*x^2 - (B^3*a*c*m^2 + 2*B^3*a*c*m + B^3*a*c)*n + (A*B^2*b*c + A*B^2
*a*d + (A*B^2*b*c + A*B^2*a*d)*m^3 + 3*(A*B^2*b*c + A*B^2*a*d)*m^2 + 3*(A*B^2*b*c + A*B^2*a*d)*m - (B^3*b*c +
B^3*a*d + (B^3*b*c + B^3*a*d)*m^2 + 2*(B^3*b*c + B^3*a*d)*m)*n)*x + ((B^3*b*d*m^3 + 3*B^3*b*d*m^2 + 3*B^3*b*d*
m + B^3*b*d)*n*x^2 + (B^3*b*c + B^3*a*d + (B^3*b*c + B^3*a*d)*m^3 + 3*(B^3*b*c + B^3*a*d)*m^2 + 3*(B^3*b*c + B
^3*a*d)*m)*n*x + (B^3*a*c*m^3 + 3*B^3*a*c*m^2 + 3*B^3*a*c*m + B^3*a*c)*n)*log((b*x + a)/(d*x + c)))*log(e)^2 -
3*((B^3*a*c*m^2 + 2*B^3*a*c*m + B^3*a*c)*n^3 - (A*B^2*a*c*m^3 + 3*A*B^2*a*c*m^2 + 3*A*B^2*a*c*m + A*B^2*a*c)*
n^2 + ((B^3*b*d*m^2 + 2*B^3*b*d*m + B^3*b*d)*n^3 - (A*B^2*b*d*m^3 + 3*A*B^2*b*d*m^2 + 3*A*B^2*b*d*m + A*B^2*b*
d)*n^2)*x^2 + ((B^3*b*c + B^3*a*d + (B^3*b*c + B^3*a*d)*m^2 + 2*(B^3*b*c + B^3*a*d)*m)*n^3 - (A*B^2*b*c + A*B^
2*a*d + (A*B^2*b*c + A*B^2*a*d)*m^3 + 3*(A*B^2*b*c + A*B^2*a*d)*m^2 + 3*(A*B^2*b*c + A*B^2*a*d)*m)*n^2)*x)*log
((b*x + a)/(d*x + c))^2 - 3*(A^2*B*a*c*m^2 + 2*A^2*B*a*c*m + A^2*B*a*c)*n + (A^3*b*c + A^3*a*d + (A^3*b*c + A^
3*a*d)*m^3 - 6*(B^3*b*c + B^3*a*d)*n^3 + 3*(A^3*b*c + A^3*a*d)*m^2 + 6*(A*B^2*b*c + A*B^2*a*d + (A*B^2*b*c + A
*B^2*a*d)*m)*n^2 + 3*(A^3*b*c + A^3*a*d)*m - 3*(A^2*B*b*c + A^2*B*a*d + (A^2*B*b*c + A^2*B*a*d)*m^2 + 2*(A^2*B
*b*c + A^2*B*a*d)*m)*n)*x + 3*(A^2*B*a*c*m^3 + 3*A^2*B*a*c*m^2 + 3*A^2*B*a*c*m + A^2*B*a*c + 2*(B^3*a*c*m + B^
3*a*c)*n^2 + (A^2*B*b*d*m^3 + 3*A^2*B*b*d*m^2 + 3*A^2*B*b*d*m + A^2*B*b*d + 2*(B^3*b*d*m + B^3*b*d)*n^2 - 2*(A
*B^2*b*d*m^2 + 2*A*B^2*b*d*m + A*B^2*b*d)*n)*x^2 + ((B^3*b*d*m^3 + 3*B^3*b*d*m^2 + 3*B^3*b*d*m + B^3*b*d)*n^2*
x^2 + (B^3*b*c + B^3*a*d + (B^3*b*c + B^3*a*d)*m^3 + 3*(B^3*b*c + B^3*a*d)*m^2 + 3*(B^3*b*c + B^3*a*d)*m)*n^2*
x + (B^3*a*c*m^3 + 3*B^3*a*c*m^2 + 3*B^3*a*c*m + B^3*a*c)*n^2)*log((b*x + a)/(d*x + c))^2 - 2*(A*B^2*a*c*m^2 +
2*A*B^2*a*c*m + A*B^2*a*c)*n + (A^2*B*b*c + A^2*B*a*d + (A^2*B*b*c + A^2*B*a*d)*m^3 + 3*(A^2*B*b*c + A^2*B*a*
d)*m^2 + 2*(B^3*b*c + B^3*a*d + (B^3*b*c + B^3*a*d)*m)*n^2 + 3*(A^2*B*b*c + A^2*B*a*d)*m - 2*(A*B^2*b*c + A*B^
2*a*d + (A*B^2*b*c + A*B^2*a*d)*m^2 + 2*(A*B^2*b*c + A*B^2*a*d)*m)*n)*x - 2*((B^3*a*c*m^2 + 2*B^3*a*c*m + B^3*
a*c)*n^2 + ((B^3*b*d*m^2 + 2*B^3*b*d*m + B^3*b*d)*n^2 - (A*B^2*b*d*m^3 + 3*A*B^2*b*d*m^2 + 3*A*B^2*b*d*m + A*B
^2*b*d)*n)*x^2 - (A*B^2*a*c*m^3 + 3*A*B^2*a*c*m^2 + 3*A*B^2*a*c*m + A*B^2*a*c)*n + ((B^3*b*c + B^3*a*d + (B^3*
b*c + B^3*a*d)*m^2 + 2*(B^3*b*c + B^3*a*d)*m)*n^2 - (A*B^2*b*c + A*B^2*a*d + (A*B^2*b*c + A*B^2*a*d)*m^3 + 3*(
A*B^2*b*c + A*B^2*a*d)*m^2 + 3*(A*B^2*b*c + A*B^2*a*d)*m)*n)*x)*log((b*x + a)/(d*x + c)))*log(e) + 3*(2*(B^3*a
*c*m + B^3*a*c)*n^3 - 2*(A*B^2*a*c*m^2 + 2*A*B^2*a*c*m + A*B^2*a*c)*n^2 + (2*(B^3*b*d*m + B^3*b*d)*n^3 - 2*(A*
B^2*b*d*m^2 + 2*A*B^2*b*d*m + A*B^2*b*d)*n^2 + (A^2*B*b*d*m^3 + 3*A^2*B*b*d*m^2 + 3*A^2*B*b*d*m + A^2*B*b*d)*n
)*x^2 + (A^2*B*a*c*m^3 + 3*A^2*B*a*c*m^2 + 3*A^2*B*a*c*m + A^2*B*a*c)*n + (2*(B^3*b*c + B^3*a*d + (B^3*b*c + B
^3*a*d)*m)*n^3 - 2*(A*B^2*b*c + A*B^2*a*d + (A*B^2*b*c + A*B^2*a*d)*m^2 + 2*(A*B^2*b*c + A*B^2*a*d)*m)*n^2 + (
A^2*B*b*c + A^2*B*a*d + (A^2*B*b*c + A^2*B*a*d)*m^3 + 3*(A^2*B*b*c + A^2*B*a*d)*m^2 + 3*(A^2*B*b*c + A^2*B*a*d
)*m)*n)*x)*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*c - a*d)*m^4 + 4*(b*c - a*d)*m^3 + 6*(b*c - a*d)*m^2 + b*c - a*d + 4*(b*c - a*d
)*m)
________________________________________________________________________________________
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
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{3}{\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}\,{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*log(e*((b*x + a)/(d*x + c))^n) + A)^3*(b*g*x + a*g)^m*(d*i*x + c*i)^(-m - 2), x)