3.64 \(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(f+g x)^3} \, dx\)

Optimal. Leaf size=190 \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 g (f+g x)^2}+\frac {b^2 B n \log (a+b x)}{2 g (b f-a g)^2}-\frac {B n (b c-a d)}{2 (f+g x) (b f-a g) (d f-c g)}+\frac {B n (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f)}{2 (b f-a g)^2 (d f-c g)^2}-\frac {B d^2 n \log (c+d x)}{2 g (d f-c g)^2} \]

[Out]

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

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Rubi [A]  time = 0.24, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2525, 12, 72} \[ -\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 g (f+g x)^2}+\frac {b^2 B n \log (a+b x)}{2 g (b f-a g)^2}-\frac {B n (b c-a d)}{2 (f+g x) (b f-a g) (d f-c g)}+\frac {B n (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f)}{2 (b f-a g)^2 (d f-c g)^2}-\frac {B d^2 n \log (c+d x)}{2 g (d f-c g)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^3,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.54, size = 173, normalized size = 0.91 \[ \frac {B n (b c-a d) \left (\frac {b^2 \log (a+b x)}{(b c-a d) (b f-a g)^2}+\frac {\frac {d^2 \log (c+d x)}{a d-b c}+\frac {g (c g-d f)}{(f+g x) (b f-a g)}-\frac {g \log (f+g x) (a d g+b c g-2 b d f)}{(b f-a g)^2}}{(d f-c g)^2}\right )-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(f+g x)^2}}{2 g} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^3,x]

[Out]

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

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fricas [B]  time = 159.27, size = 1175, normalized size = 6.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^3,x, algorithm="fricas")

[Out]

-1/2*(A*b^2*d^2*f^4 + A*a^2*c^2*g^4 - 2*(A*b^2*c*d + A*a*b*d^2)*f^3*g + (A*b^2*c^2 + 4*A*a*b*c*d + A*a^2*d^2)*
f^2*g^2 - 2*(A*a*b*c^2 + A*a^2*c*d)*f*g^3 + ((B*b^2*c*d - B*a*b*d^2)*f^2*g^2 - (B*b^2*c^2 - B*a^2*d^2)*f*g^3 +
(B*a*b*c^2 - B*a^2*c*d)*g^4)*n*x + (B*b^2*d^2*f^4 + B*a^2*c^2*g^4 - 2*(B*b^2*c*d + B*a*b*d^2)*f^3*g + (B*b^2*
c^2 + 4*B*a*b*c*d + B*a^2*d^2)*f^2*g^2 - 2*(B*a*b*c^2 + B*a^2*c*d)*f*g^3)*n*log((b*x + a)/(d*x + c)) + ((B*b^2
*c*d - B*a*b*d^2)*f^3*g - (B*b^2*c^2 - B*a^2*d^2)*f^2*g^2 + (B*a*b*c^2 - B*a^2*c*d)*f*g^3)*n - ((B*b^2*d^2*f^2
*g^2 - 2*B*b^2*c*d*f*g^3 + B*b^2*c^2*g^4)*n*x^2 + 2*(B*b^2*d^2*f^3*g - 2*B*b^2*c*d*f^2*g^2 + B*b^2*c^2*f*g^3)*
n*x + (B*b^2*d^2*f^4 - 2*B*b^2*c*d*f^3*g + B*b^2*c^2*f^2*g^2)*n)*log(b*x + a) + ((B*b^2*d^2*f^2*g^2 - 2*B*a*b*
d^2*f*g^3 + B*a^2*d^2*g^4)*n*x^2 + 2*(B*b^2*d^2*f^3*g - 2*B*a*b*d^2*f^2*g^2 + B*a^2*d^2*f*g^3)*n*x + (B*b^2*d^
2*f^4 - 2*B*a*b*d^2*f^3*g + B*a^2*d^2*f^2*g^2)*n)*log(d*x + c) - ((2*(B*b^2*c*d - B*a*b*d^2)*f*g^3 - (B*b^2*c^
2 - B*a^2*d^2)*g^4)*n*x^2 + 2*(2*(B*b^2*c*d - B*a*b*d^2)*f^2*g^2 - (B*b^2*c^2 - B*a^2*d^2)*f*g^3)*n*x + (2*(B*
b^2*c*d - B*a*b*d^2)*f^3*g - (B*b^2*c^2 - B*a^2*d^2)*f^2*g^2)*n)*log(g*x + f) + (B*b^2*d^2*f^4 + B*a^2*c^2*g^4
- 2*(B*b^2*c*d + B*a*b*d^2)*f^3*g + (B*b^2*c^2 + 4*B*a*b*c*d + B*a^2*d^2)*f^2*g^2 - 2*(B*a*b*c^2 + B*a^2*c*d)
*f*g^3)*log(e))/(b^2*d^2*f^6*g + a^2*c^2*f^2*g^5 - 2*(b^2*c*d + a*b*d^2)*f^5*g^2 + (b^2*c^2 + 4*a*b*c*d + a^2*
d^2)*f^4*g^3 - 2*(a*b*c^2 + a^2*c*d)*f^3*g^4 + (b^2*d^2*f^4*g^3 + a^2*c^2*g^7 - 2*(b^2*c*d + a*b*d^2)*f^3*g^4
+ (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^2*g^5 - 2*(a*b*c^2 + a^2*c*d)*f*g^6)*x^2 + 2*(b^2*d^2*f^5*g^2 + a^2*c^2*f*
g^6 - 2*(b^2*c*d + a*b*d^2)*f^4*g^3 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^3*g^4 - 2*(a*b*c^2 + a^2*c*d)*f^2*g^5)
*x)

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giac [B]  time = 6.80, size = 2952, normalized size = 15.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^3,x, algorithm="giac")

[Out]

1/2*((2*B*b^3*c^2*d*f*n - 4*B*a*b^2*c*d^2*f*n + 2*B*a^2*b*d^3*f*n - B*b^3*c^3*g*n + B*a*b^2*c^2*d*g*n + B*a^2*
b*c*d^2*g*n - B*a^3*d^3*g*n)*log(-b*f + (b*x + a)*d*f/(d*x + c) + a*g - (b*x + a)*c*g/(d*x + c))/(b^2*d^2*f^4
- 2*b^2*c*d*f^3*g - 2*a*b*d^2*f^3*g + b^2*c^2*f^2*g^2 + 4*a*b*c*d*f^2*g^2 + a^2*d^2*f^2*g^2 - 2*a*b*c^2*f*g^3
- 2*a^2*c*d*f*g^3 + a^2*c^2*g^4) + (2*B*b^3*c^2*d*f*n - 4*B*a*b^2*c*d^2*f*n - 2*(b*x + a)*B*b^2*c^2*d^2*f*n/(d
*x + c) + 2*B*a^2*b*d^3*f*n + 4*(b*x + a)*B*a*b*c*d^3*f*n/(d*x + c) - 2*(b*x + a)*B*a^2*d^4*f*n/(d*x + c) - B*
b^3*c^3*g*n + B*a*b^2*c^2*d*g*n + 2*(b*x + a)*B*b^2*c^3*d*g*n/(d*x + c) + B*a^2*b*c*d^2*g*n - 4*(b*x + a)*B*a*
b*c^2*d^2*g*n/(d*x + c) - B*a^3*d^3*g*n + 2*(b*x + a)*B*a^2*c*d^3*g*n/(d*x + c))*log((b*x + a)/(d*x + c))/(b^2
*d^2*f^4 - 2*(b*x + a)*b*d^3*f^4/(d*x + c) + (b*x + a)^2*d^4*f^4/(d*x + c)^2 - 2*b^2*c*d*f^3*g - 2*a*b*d^2*f^3
*g + 6*(b*x + a)*b*c*d^2*f^3*g/(d*x + c) + 2*(b*x + a)*a*d^3*f^3*g/(d*x + c) - 4*(b*x + a)^2*c*d^3*f^3*g/(d*x
+ c)^2 + b^2*c^2*f^2*g^2 + 4*a*b*c*d*f^2*g^2 - 6*(b*x + a)*b*c^2*d*f^2*g^2/(d*x + c) + a^2*d^2*f^2*g^2 - 6*(b*
x + a)*a*c*d^2*f^2*g^2/(d*x + c) + 6*(b*x + a)^2*c^2*d^2*f^2*g^2/(d*x + c)^2 - 2*a*b*c^2*f*g^3 + 2*(b*x + a)*b
*c^3*f*g^3/(d*x + c) - 2*a^2*c*d*f*g^3 + 6*(b*x + a)*a*c^2*d*f*g^3/(d*x + c) - 4*(b*x + a)^2*c^3*d*f*g^3/(d*x
+ c)^2 + a^2*c^2*g^4 - 2*(b*x + a)*a*c^3*g^4/(d*x + c) + (b*x + a)^2*c^4*g^4/(d*x + c)^2) - (2*B*b^3*c^2*d*f*n
- 4*B*a*b^2*c*d^2*f*n + 2*B*a^2*b*d^3*f*n - B*b^3*c^3*g*n + B*a*b^2*c^2*d*g*n + B*a^2*b*c*d^2*g*n - B*a^3*d^3
*g*n)*log((b*x + a)/(d*x + c))/(b^2*d^2*f^4 - 2*b^2*c*d*f^3*g - 2*a*b*d^2*f^3*g + b^2*c^2*f^2*g^2 + 4*a*b*c*d*
f^2*g^2 + a^2*d^2*f^2*g^2 - 2*a*b*c^2*f*g^3 - 2*a^2*c*d*f*g^3 + a^2*c^2*g^4) + (B*b^4*c^3*f*g*n - 3*B*a*b^3*c^
2*d*f*g*n - (b*x + a)*B*b^3*c^3*d*f*g*n/(d*x + c) + 3*B*a^2*b^2*c*d^2*f*g*n + 3*(b*x + a)*B*a*b^2*c^2*d^2*f*g*
n/(d*x + c) - B*a^3*b*d^3*f*g*n - 3*(b*x + a)*B*a^2*b*c*d^3*f*g*n/(d*x + c) + (b*x + a)*B*a^3*d^4*f*g*n/(d*x +
c) - B*a*b^3*c^3*g^2*n + (b*x + a)*B*b^3*c^4*g^2*n/(d*x + c) + 3*B*a^2*b^2*c^2*d*g^2*n - 3*(b*x + a)*B*a*b^2*
c^3*d*g^2*n/(d*x + c) - 3*B*a^3*b*c*d^2*g^2*n + 3*(b*x + a)*B*a^2*b*c^2*d^2*g^2*n/(d*x + c) + B*a^4*d^3*g^2*n
- (b*x + a)*B*a^3*c*d^3*g^2*n/(d*x + c) + 2*A*b^4*c^2*d*f^2 + 2*B*b^4*c^2*d*f^2 - 4*A*a*b^3*c*d^2*f^2 - 4*B*a*
b^3*c*d^2*f^2 - 2*(b*x + a)*A*b^3*c^2*d^2*f^2/(d*x + c) - 2*(b*x + a)*B*b^3*c^2*d^2*f^2/(d*x + c) + 2*A*a^2*b^
2*d^3*f^2 + 2*B*a^2*b^2*d^3*f^2 + 4*(b*x + a)*A*a*b^2*c*d^3*f^2/(d*x + c) + 4*(b*x + a)*B*a*b^2*c*d^3*f^2/(d*x
+ c) - 2*(b*x + a)*A*a^2*b*d^4*f^2/(d*x + c) - 2*(b*x + a)*B*a^2*b*d^4*f^2/(d*x + c) - A*b^4*c^3*f*g - B*b^4*
c^3*f*g - A*a*b^3*c^2*d*f*g - B*a*b^3*c^2*d*f*g + 2*(b*x + a)*A*b^3*c^3*d*f*g/(d*x + c) + 2*(b*x + a)*B*b^3*c^
3*d*f*g/(d*x + c) + 5*A*a^2*b^2*c*d^2*f*g + 5*B*a^2*b^2*c*d^2*f*g - 2*(b*x + a)*A*a*b^2*c^2*d^2*f*g/(d*x + c)
- 2*(b*x + a)*B*a*b^2*c^2*d^2*f*g/(d*x + c) - 3*A*a^3*b*d^3*f*g - 3*B*a^3*b*d^3*f*g - 2*(b*x + a)*A*a^2*b*c*d^
3*f*g/(d*x + c) - 2*(b*x + a)*B*a^2*b*c*d^3*f*g/(d*x + c) + 2*(b*x + a)*A*a^3*d^4*f*g/(d*x + c) + 2*(b*x + a)*
B*a^3*d^4*f*g/(d*x + c) + A*a*b^3*c^3*g^2 + B*a*b^3*c^3*g^2 - A*a^2*b^2*c^2*d*g^2 - B*a^2*b^2*c^2*d*g^2 - 2*(b
*x + a)*A*a*b^2*c^3*d*g^2/(d*x + c) - 2*(b*x + a)*B*a*b^2*c^3*d*g^2/(d*x + c) - A*a^3*b*c*d^2*g^2 - B*a^3*b*c*
d^2*g^2 + 4*(b*x + a)*A*a^2*b*c^2*d^2*g^2/(d*x + c) + 4*(b*x + a)*B*a^2*b*c^2*d^2*g^2/(d*x + c) + A*a^4*d^3*g^
2 + B*a^4*d^3*g^2 - 2*(b*x + a)*A*a^3*c*d^3*g^2/(d*x + c) - 2*(b*x + a)*B*a^3*c*d^3*g^2/(d*x + c))/(b^3*d^2*f^
5 - 2*(b*x + a)*b^2*d^3*f^5/(d*x + c) + (b*x + a)^2*b*d^4*f^5/(d*x + c)^2 - 2*b^3*c*d*f^4*g - 3*a*b^2*d^2*f^4*
g + 6*(b*x + a)*b^2*c*d^2*f^4*g/(d*x + c) + 4*(b*x + a)*a*b*d^3*f^4*g/(d*x + c) - 4*(b*x + a)^2*b*c*d^3*f^4*g/
(d*x + c)^2 - (b*x + a)^2*a*d^4*f^4*g/(d*x + c)^2 + b^3*c^2*f^3*g^2 + 6*a*b^2*c*d*f^3*g^2 - 6*(b*x + a)*b^2*c^
2*d*f^3*g^2/(d*x + c) + 3*a^2*b*d^2*f^3*g^2 - 12*(b*x + a)*a*b*c*d^2*f^3*g^2/(d*x + c) + 6*(b*x + a)^2*b*c^2*d
^2*f^3*g^2/(d*x + c)^2 - 2*(b*x + a)*a^2*d^3*f^3*g^2/(d*x + c) + 4*(b*x + a)^2*a*c*d^3*f^3*g^2/(d*x + c)^2 - 3
*a*b^2*c^2*f^2*g^3 + 2*(b*x + a)*b^2*c^3*f^2*g^3/(d*x + c) - 6*a^2*b*c*d*f^2*g^3 + 12*(b*x + a)*a*b*c^2*d*f^2*
g^3/(d*x + c) - 4*(b*x + a)^2*b*c^3*d*f^2*g^3/(d*x + c)^2 - a^3*d^2*f^2*g^3 + 6*(b*x + a)*a^2*c*d^2*f^2*g^3/(d
*x + c) - 6*(b*x + a)^2*a*c^2*d^2*f^2*g^3/(d*x + c)^2 + 3*a^2*b*c^2*f*g^4 - 4*(b*x + a)*a*b*c^3*f*g^4/(d*x + c
) + (b*x + a)^2*b*c^4*f*g^4/(d*x + c)^2 + 2*a^3*c*d*f*g^4 - 6*(b*x + a)*a^2*c^2*d*f*g^4/(d*x + c) + 4*(b*x + a
)^2*a*c^3*d*f*g^4/(d*x + c)^2 - a^3*c^2*g^5 + 2*(b*x + a)*a^2*c^3*g^5/(d*x + c) - (b*x + a)^2*a*c^4*g^5/(d*x +
c)^2))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (g x +f \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.00, size = 355, normalized size = 1.87 \[ \frac {1}{2} \, {\left (\frac {b^{2} \log \left (b x + a\right )}{b^{2} f^{2} g - 2 \, a b f g^{2} + a^{2} g^{3}} - \frac {d^{2} \log \left (d x + c\right )}{d^{2} f^{2} g - 2 \, c d f g^{2} + c^{2} g^{3}} + \frac {{\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} f - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} g\right )} \log \left (g x + f\right )}{b^{2} d^{2} f^{4} + a^{2} c^{2} g^{4} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3} g + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{2} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f g^{3}} - \frac {b c - a d}{b d f^{3} + a c f g^{2} - {\left (b c + a d\right )} f^{2} g + {\left (b d f^{2} g + a c g^{3} - {\left (b c + a d\right )} f g^{2}\right )} x}\right )} B n - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} - \frac {A}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^3,x, algorithm="maxima")

[Out]

1/2*(b^2*log(b*x + a)/(b^2*f^2*g - 2*a*b*f*g^2 + a^2*g^3) - d^2*log(d*x + c)/(d^2*f^2*g - 2*c*d*f*g^2 + c^2*g^
3) + (2*(b^2*c*d - a*b*d^2)*f - (b^2*c^2 - a^2*d^2)*g)*log(g*x + f)/(b^2*d^2*f^4 + a^2*c^2*g^4 - 2*(b^2*c*d +
a*b*d^2)*f^3*g + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^2*g^2 - 2*(a*b*c^2 + a^2*c*d)*f*g^3) - (b*c - a*d)/(b*d*f^3
+ a*c*f*g^2 - (b*c + a*d)*f^2*g + (b*d*f^2*g + a*c*g^3 - (b*c + a*d)*f*g^2)*x))*B*n - 1/2*B*log(e*(b*x/(d*x +
c) + a/(d*x + c))^n)/(g^3*x^2 + 2*f*g^2*x + f^2*g) - 1/2*A/(g^3*x^2 + 2*f*g^2*x + f^2*g)

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mupad [B]  time = 6.20, size = 430, normalized size = 2.26 \[ \frac {\ln \left (f+g\,x\right )\,\left (g\,\left (B\,a^2\,d^2\,n-B\,b^2\,c^2\,n\right )-2\,B\,a\,b\,d^2\,f\,n+2\,B\,b^2\,c\,d\,f\,n\right )}{2\,a^2\,c^2\,g^4-4\,a^2\,c\,d\,f\,g^3+2\,a^2\,d^2\,f^2\,g^2-4\,a\,b\,c^2\,f\,g^3+8\,a\,b\,c\,d\,f^2\,g^2-4\,a\,b\,d^2\,f^3\,g+2\,b^2\,c^2\,f^2\,g^2-4\,b^2\,c\,d\,f^3\,g+2\,b^2\,d^2\,f^4}-\frac {\frac {A\,a\,c\,g^2+A\,b\,d\,f^2-A\,a\,d\,f\,g-A\,b\,c\,f\,g-B\,a\,d\,f\,g\,n+B\,b\,c\,f\,g\,n}{a\,c\,g^2+b\,d\,f^2-a\,d\,f\,g-b\,c\,f\,g}-\frac {x\,\left (B\,a\,d\,g^2\,n-B\,b\,c\,g^2\,n\right )}{a\,c\,g^2+b\,d\,f^2-a\,d\,f\,g-b\,c\,f\,g}}{2\,f^2\,g+4\,f\,g^2\,x+2\,g^3\,x^2}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{2\,g\,\left (f^2+2\,f\,g\,x+g^2\,x^2\right )}+\frac {B\,b^2\,n\,\ln \left (a+b\,x\right )}{2\,a^2\,g^3-4\,a\,b\,f\,g^2+2\,b^2\,f^2\,g}-\frac {B\,d^2\,n\,\ln \left (c+d\,x\right )}{2\,c^2\,g^3-4\,c\,d\,f\,g^2+2\,d^2\,f^2\,g} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(f + g*x)*(g*(B*a^2*d^2*n - B*b^2*c^2*n) - 2*B*a*b*d^2*f*n + 2*B*b^2*c*d*f*n))/(2*a^2*c^2*g^4 + 2*b^2*d^2*
f^4 + 2*a^2*d^2*f^2*g^2 + 2*b^2*c^2*f^2*g^2 - 4*a*b*c^2*f*g^3 - 4*a*b*d^2*f^3*g - 4*a^2*c*d*f*g^3 - 4*b^2*c*d*
f^3*g + 8*a*b*c*d*f^2*g^2) - ((A*a*c*g^2 + A*b*d*f^2 - A*a*d*f*g - A*b*c*f*g - B*a*d*f*g*n + B*b*c*f*g*n)/(a*c
*g^2 + b*d*f^2 - a*d*f*g - b*c*f*g) - (x*(B*a*d*g^2*n - B*b*c*g^2*n))/(a*c*g^2 + b*d*f^2 - a*d*f*g - b*c*f*g))
/(2*f^2*g + 2*g^3*x^2 + 4*f*g^2*x) - (B*log(e*((a + b*x)/(c + d*x))^n))/(2*g*(f^2 + g^2*x^2 + 2*f*g*x)) + (B*b
^2*n*log(a + b*x))/(2*a^2*g^3 + 2*b^2*f^2*g - 4*a*b*f*g^2) - (B*d^2*n*log(c + d*x))/(2*c^2*g^3 + 2*d^2*f^2*g -
4*c*d*f*g^2)

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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((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(g*x+f)**3,x)

[Out]

Timed out

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