3.218 \(\int (a g+b g x)^{-2-m} (c i+d i x)^m (A+B \log (e (\frac {a+b x}{c+d x})^n))^3 \, dx\)

Optimal. Leaf size=309 \[ -\frac {6 B^2 n^2 (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{i^2 (m+1)^3 (c+d x) (b c-a d)}-\frac {(a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{i^2 (m+1) (c+d x) (b c-a d)}-\frac {3 B n (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{i^2 (m+1)^2 (c+d x) (b c-a d)}-\frac {6 B^3 n^3 (a+b x) (g (a+b x))^{-m-2} (i (c+d x))^{m+2}}{i^2 (m+1)^4 (c+d x) (b c-a d)} \]

[Out]

-6*B^3*n^3*(b*x+a)*(g*(b*x+a))^(-2-m)*(i*(d*x+c))^(2+m)/(-a*d+b*c)/i^2/(1+m)^4/(d*x+c)-6*B^2*n^2*(b*x+a)*(g*(b
*x+a))^(-2-m)*(i*(d*x+c))^(2+m)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)/i^2/(1+m)^3/(d*x+c)-3*B*n*(b*x+a)*(
g*(b*x+a))^(-2-m)*(i*(d*x+c))^(2+m)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)/i^2/(1+m)^2/(d*x+c)-(b*x+a)*(
g*(b*x+a))^(-2-m)*(i*(d*x+c))^(2+m)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^3/(-a*d+b*c)/i^2/(1+m)/(d*x+c)

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Rubi [F]  time = 2.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 \, dx \]

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]

-((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)^(-2 - m)*(c*i
+ d*i*x)^m*Log[e*((a + b*x)/(c + d*x))^n]^2, x] + B^3*Defer[Int][(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*Log[e*
((a + b*x)/(c + d*x))^n]^3, x]

Rubi steps

\begin {align*} \int (218 c+218 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 \left (A^3 (218 c+218 d x)^m (a g+b g x)^{-2-m}+3 A^2 B (218 c+218 d x)^m (a g+b g x)^{-2-m} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 A B^2 (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B^3 (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx\\ &=A^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \, dx+\left (3 A^2 B\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+\left (3 A B^2\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx\\ &=-\frac {A^3 (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m}}{218 (b c-a d) g (1+m)}-\frac {3 A^2 B (218 c+218 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 )}{218 (b c-a d) g (1+m)}-\left (3 A^2 B\right ) \int \frac {218^m n (c+d x)^m (a g+b g x)^{-2-m}}{-1-m} \, dx+\left (3 A B^2\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx\\ &=-\frac {A^3 (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m}}{218 (b c-a d) g (1+m)}-\frac {3 A^2 B (218 c+218 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 )}{218 (b c-a d) g (1+m)}+\left (3 A B^2\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+\frac {\left (3\ 218^m A^2 B n\right ) \int (c+d x)^m (a g+b g x)^{-2-m} \, dx}{1+m}\\ &=-\frac {3\ 218^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 (218 c+218 d x)^{1+m} (a g+b g x)^{-1-m}}{218 (b c-a d) g (1+m)}-\frac {3 A^2 B (218 c+218 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 )}{218 (b c-a d) g (1+m)}+\left (3 A B^2\right ) \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx+B^3 \int (218 c+218 d x)^m (a g+b g x)^{-2-m} \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx\\ \end {align*}

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Mathematica [A]  time = 7.18, size = 206, normalized size = 0.67 \[ -\frac {(c+d x) (g (a+b x))^{-m-1} (i (c+d x))^m \left (3 B (m+1) \left (A^2 (m+1)^2+2 A B (m+1) n+2 B^2 n^2\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 B^2 (m+1)^2 (A m+A+B n) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B^3 (m+1)^3 \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A^3 (m+1)^3+3 A^2 B (m+1)^2 n+6 A B^2 (m+1) n^2+6 B^3 n^3\right )}{g (m+1)^4 (b c-a d)} \]

Antiderivative was successfully verified.

[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]

-(((g*(a + b*x))^(-1 - m)*(c + d*x)*(i*(c + d*x))^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)*g*(1 + m)^4))

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fricas [B]  time = 0.81, size = 2669, normalized size = 8.64 \[ \text {result too large to display} \]

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]

-(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)*lo
g((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 - 2)*e^(m*log(b*g*x + a*g) - m*log((b*x + a)/(d*x + c)
) + m*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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{3} {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m}\,{d x} \]

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*log(e*((b*x + a)/(d*x + c))^n) + A)^3*(b*g*x + a*g)^(-m - 2)*(d*i*x + c*i)^m, x)

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maple [F]  time = 8.71, size = 0, normalized size = 0.00 \[ \int \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{3} \left (b g x +a g \right )^{-m -2} \left (d i x +c i \right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^(-m-2)*(d*i*x+c*i)^m*(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^3,x)

[Out]

int((b*g*x+a*g)^(-m-2)*(d*i*x+c*i)^m*(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{3} {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m}\,{d x} \]

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]

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^3*(b*g*x + a*g)^(-m - 2)*(d*i*x + c*i)^m, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^m\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^3}{{\left (a\,g+b\,g\,x\right )}^{m+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)^m*(A + B*log(e*((a + b*x)/(c + d*x))^n))^3)/(a*g + b*g*x)^(m + 2),x)

[Out]

int(((c*i + d*i*x)^m*(A + B*log(e*((a + b*x)/(c + d*x))^n))^3)/(a*g + b*g*x)^(m + 2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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