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

Optimal. Leaf size=210 \[ \frac{(a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{i^2 (m+1) (c+d x) (b c-a d)}-\frac{2 B n (a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{i^2 (m+1)^2 (c+d x) (b c-a d)}+\frac{2 B^2 n^2 (a+b x) (g (a+b x))^m (i (c+d x))^{-m}}{i^2 (m+1)^3 (c+d x) (b c-a d)} \]

[Out]

(2*B^2*n^2*(a + b*x)*(g*(a + b*x))^m)/((b*c - a*d)*i^2*(1 + m)^3*(c + d*x)*(i*(c + d*x))^m) - (2*B*n*(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)^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])^2)/((b*c - a*d)*i^2*(1 + m)*(c + d*x)*(i*
(c + d*x))^m)

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Rubi [F]  time = 1.21513, 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 = {} \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx \]

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])^2,x]

[Out]

(A^2*(a*g + b*g*x)^(1 + m)*(c*i + d*i*x)^(-1 - m))/((b*c - a*d)*g*i*(1 + m)) - (2*A*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) + (2*A*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)) + 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]

Rubi steps

\begin{align*} \int (213 c+213 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 )^2 \, dx &=\int \left (A^2 (213 c+213 d x)^{-2-m} (a g+b g x)^m+2 A B (213 c+213 d x)^{-2-m} (a g+b g x)^m \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B^2 (213 c+213 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 )\right ) \, dx\\ &=A^2 \int (213 c+213 d x)^{-2-m} (a g+b g x)^m \, dx+(2 A B) \int (213 c+213 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+B^2 \int (213 c+213 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\\ &=\frac{A^2 (213 c+213 d x)^{-1-m} (a g+b g x)^{1+m}}{213 (b c-a d) g (1+m)}+\frac{2 A B (213 c+213 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 )}{213 (b c-a d) g (1+m)}-(2 A B) \int \frac{213^{-2-m} n (c+d x)^{-2-m} (a g+b g x)^m}{1+m} \, dx+B^2 \int (213 c+213 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\\ &=\frac{A^2 (213 c+213 d x)^{-1-m} (a g+b g x)^{1+m}}{213 (b c-a d) g (1+m)}+\frac{2 A B (213 c+213 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 )}{213 (b c-a d) g (1+m)}+B^2 \int (213 c+213 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-\frac{\left (2\ 213^{-2-m} A B n\right ) \int (c+d x)^{-2-m} (a g+b g x)^m \, dx}{1+m}\\ &=-\frac{2\ 213^{-2-m} A B n (c+d x)^{-1-m} (a g+b g x)^{1+m}}{(b c-a d) g (1+m)^2}+\frac{A^2 (213 c+213 d x)^{-1-m} (a g+b g x)^{1+m}}{213 (b c-a d) g (1+m)}+\frac{2 A B (213 c+213 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 )}{213 (b c-a d) g (1+m)}+B^2 \int (213 c+213 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\\ \end{align*}

Mathematica [A]  time = 2.07189, size = 134, normalized size = 0.64 \[ \frac{(a+b x) (g (a+b x))^m (i (c+d x))^{-m-1} \left (2 B (m+1) (A m+A-B n) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B^2 (m+1)^2 \log ^2\left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A^2 (m+1)^2-2 A B (m+1) n+2 B^2 n^2\right )}{i (m+1)^3 (b c-a d)} \]

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])^2,x]

[Out]

((a + b*x)*(g*(a + b*x))^m*(i*(c + d*x))^(-1 - m)*(A^2*(1 + m)^2 - 2*A*B*(1 + m)*n + 2*B^2*n^2 + 2*B*(1 + m)*(
A + A*m - B*n)*Log[e*((a + b*x)/(c + d*x))^n] + B^2*(1 + m)^2*Log[e*((a + b*x)/(c + d*x))^n]^2))/((b*c - a*d)*
i*(1 + m)^3)

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Maple [F]  time = 4.484, 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 ) ^{2}\, 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))^2,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))^2,x)

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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 )}^{2}{\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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.651262, size = 2201, normalized size = 10.48 \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))^2,x, algorithm="fricas")

[Out]

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

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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((b*g*x+a*g)**m*(d*i*x+c*i)**(-2-m)*(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{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}{\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))^2,x, algorithm="giac")

[Out]

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