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

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Rubi [F]  time = 1.22, 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 = {} \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx \]

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*}

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Mathematica [A]  time = 2.02, 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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fricas [B]  time = 1.08, size = 991, normalized size = 4.72 \[ \frac {{\left (A^{2} a c m^{2} + 2 \, B^{2} a c n^{2} + 2 \, A^{2} a c m + A^{2} a c + {\left (A^{2} b d m^{2} + 2 \, B^{2} b d n^{2} + 2 \, A^{2} b d m + A^{2} b d - 2 \, {\left (A B b d m + A B b d\right )} n\right )} x^{2} + {\left (B^{2} a c m^{2} + 2 \, B^{2} a c m + B^{2} a c + {\left (B^{2} b d m^{2} + 2 \, B^{2} b d m + B^{2} b d\right )} x^{2} + {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m^{2} + 2 \, {\left (B^{2} b c + B^{2} a d\right )} m\right )} x\right )} \log \relax (e)^{2} + {\left ({\left (B^{2} b d m^{2} + 2 \, B^{2} b d m + B^{2} b d\right )} n^{2} x^{2} + {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m^{2} + 2 \, {\left (B^{2} b c + B^{2} a d\right )} m\right )} n^{2} x + {\left (B^{2} a c m^{2} + 2 \, B^{2} a c m + B^{2} a c\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 2 \, {\left (A B a c m + A B a c\right )} n + {\left (A^{2} b c + A^{2} a d + {\left (A^{2} b c + A^{2} a d\right )} m^{2} + 2 \, {\left (B^{2} b c + B^{2} a d\right )} n^{2} + 2 \, {\left (A^{2} b c + A^{2} a d\right )} m - 2 \, {\left (A B b c + A B a d + {\left (A B b c + A B a d\right )} m\right )} n\right )} x + 2 \, {\left (A B a c m^{2} + 2 \, A B a c m + A B a c + {\left (A B b d m^{2} + 2 \, A B b d m + A B b d - {\left (B^{2} b d m + B^{2} b d\right )} n\right )} x^{2} - {\left (B^{2} a c m + B^{2} a c\right )} n + {\left (A B b c + A B a d + {\left (A B b c + A B a d\right )} m^{2} + 2 \, {\left (A B b c + A B a d\right )} m - {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m\right )} n\right )} x + {\left ({\left (B^{2} b d m^{2} + 2 \, B^{2} b d m + B^{2} b d\right )} n x^{2} + {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m^{2} + 2 \, {\left (B^{2} b c + B^{2} a d\right )} m\right )} n x + {\left (B^{2} a c m^{2} + 2 \, B^{2} a c m + B^{2} a c\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) - 2 \, {\left ({\left (B^{2} a c m + B^{2} a c\right )} n^{2} + {\left ({\left (B^{2} b d m + B^{2} b d\right )} n^{2} - {\left (A B b d m^{2} + 2 \, A B b d m + A B b d\right )} n\right )} x^{2} - {\left (A B a c m^{2} + 2 \, A B a c m + A B a c\right )} n + {\left ({\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m\right )} n^{2} - {\left (A B b c + A B a d + {\left (A B b c + A B a d\right )} m^{2} + 2 \, {\left (A B b c + A B a d\right )} m\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (b g x + a g\right )}^{m} e^{\left (-{\left (m + 2\right )} \log \left (b g x + a g\right ) + {\left (m + 2\right )} \log \left (\frac {b x + a}{d x + c}\right ) - {\left (m + 2\right )} \log \left (\frac {i}{g}\right )\right )}}{{\left (b c - a d\right )} m^{3} + 3 \, {\left (b c - a d\right )} m^{2} + b c - a d + 3 \, {\left (b c - a d\right )} m} \]

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

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)

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maple [F]  time = 7.86, size = 0, normalized size = 0.00 \[ \int \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2} \left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-m -2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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