3.271 \(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{(f+g x)^5} \, dx\)

Optimal. Leaf size=381 \[ -\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{2 (f+g x) (b f-a g)^3 (d f-c g)^3}-\frac {B (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f) \left (-a^2 d^2 g^2+2 a b d^2 f g-\left (b^2 \left (c^2 g^2-2 c d f g+2 d^2 f^2\right )\right )\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{4 g (f+g x)^4}+\frac {b^4 B \log (a+b x)}{2 g (b f-a g)^4}-\frac {B (b c-a d) (-a d g-b c g+2 b d f)}{4 (f+g x)^2 (b f-a g)^2 (d f-c g)^2}-\frac {B (b c-a d)}{6 (f+g x)^3 (b f-a g) (d f-c g)}-\frac {B d^4 \log (c+d x)}{2 g (d f-c g)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.55, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2525, 12, 72} \[ -\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{2 (f+g x) (b f-a g)^3 (d f-c g)^3}-\frac {B (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f) \left (-a^2 d^2 g^2+2 a b d^2 f g+b^2 \left (-\left (c^2 g^2-2 c d f g+2 d^2 f^2\right )\right )\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{4 g (f+g x)^4}+\frac {b^4 B \log (a+b x)}{2 g (b f-a g)^4}-\frac {B (b c-a d) (-a d g-b c g+2 b d f)}{4 (f+g x)^2 (b f-a g)^2 (d f-c g)^2}-\frac {B (b c-a d)}{6 (f+g x)^3 (b f-a g) (d f-c g)}-\frac {B d^4 \log (c+d x)}{2 g (d f-c g)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^5,x]

[Out]

-(B*(b*c - a*d))/(6*(b*f - a*g)*(d*f - c*g)*(f + g*x)^3) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g))/(4*(b*f -
a*g)^2*(d*f - c*g)^2*(f + g*x)^2) - (B*(b*c - a*d)*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*
c*d*f*g + c^2*g^2)))/(2*(b*f - a*g)^3*(d*f - c*g)^3*(f + g*x)) + (b^4*B*Log[a + b*x])/(2*g*(b*f - a*g)^4) - (A
+ B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(4*g*(f + g*x)^4) - (B*d^4*Log[c + d*x])/(2*g*(d*f - c*g)^4) - (B*(b*c
- a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*Log[f +
g*x])/(2*(b*f - a*g)^4*(d*f - c*g)^4)

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 (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx &=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{4 g (f+g x)^4}+\frac {B \int \frac {2 (b c-a d)}{(a+b x) (c+d x) (f+g x)^4} \, dx}{4 g}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{4 g (f+g x)^4}+\frac {(B (b c-a d)) \int \frac {1}{(a+b x) (c+d x) (f+g x)^4} \, dx}{2 g}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{4 g (f+g x)^4}+\frac {(B (b c-a d)) \int \left (\frac {b^5}{(b c-a d) (b f-a g)^4 (a+b x)}-\frac {d^5}{(b c-a d) (-d f+c g)^4 (c+d x)}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^4}-\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)^3}+\frac {g^2 \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)^2}+\frac {g^2 (2 b d f-b c g-a d g) \left (2 b^2 d^2 f^2-2 b^2 c d f g-2 a b d^2 f g+b^2 c^2 g^2+a^2 d^2 g^2\right )}{(b f-a g)^4 (d f-c g)^4 (f+g x)}\right ) \, dx}{2 g}\\ &=-\frac {B (b c-a d)}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g)}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x)}{2 g (b f-a g)^4}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{4 g (f+g x)^4}-\frac {B d^4 \log (c+d x)}{2 g (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.97, size = 358, normalized size = 0.94 \[ \frac {2 B (b c-a d) \left (-\frac {g \left (a^2 d^2 g^2+a b d g (c g-3 d f)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{(f+g x) (b f-a g)^3 (d f-c g)^3}-\frac {g \log (f+g x) (a d g+b c g-2 b d f) \left (a^2 d^2 g^2-2 a b d^2 f g+b^2 \left (c^2 g^2-2 c d f g+2 d^2 f^2\right )\right )}{(b f-a g)^4 (d f-c g)^4}+\frac {b^4 \log (a+b x)}{(b c-a d) (b f-a g)^4}-\frac {d^4 \log (c+d x)}{(b c-a d) (d f-c g)^4}+\frac {g (a d g+b c g-2 b d f)}{2 (f+g x)^2 (b f-a g)^2 (d f-c g)^2}-\frac {g}{3 (f+g x)^3 (b f-a g) (d f-c g)}\right )-\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{(f+g x)^4}}{4 g} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(f + g*x)^5,x]

[Out]

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

________________________________________________________________________________________

fricas [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*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^5,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^5,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Undef/Unsigned Inf encountered in limitU
ndef/Unsigned Inf encountered in limit(-1/4*A*g^3-1/4*B*g^3)*(-(g*x+f)^-1/g)^4+(-B*a*d*g^3+B*b*c*g^3)/(6*a*c*g
^2-6*a*d*f*g-6*b*c*f*g+6*b*d*f^2)*(-(g*x+f)^-1/g)^3+(-B*a^2*d^2*g^3+2*B*a*b*d^2*f*g^2+B*b^2*c^2*g^3-2*B*b^2*c*
d*f*g^2)/(4*a^2*c^2*g^4-8*a^2*c*d*f*g^3+4*a^2*d^2*f^2*g^2-8*a*b*c^2*f*g^3+16*a*b*c*d*f^2*g^2-8*a*b*d^2*f^3*g+4
*b^2*c^2*f^2*g^2-8*b^2*c*d*f^3*g+4*b^2*d^2*f^4)*(-(g*x+f)^-1/g)^2-(-B*a^3*d^3*g^3+3*B*a^2*b*d^3*f*g^2-3*B*a*b^
2*d^3*f^2*g+B*b^3*c^3*g^3-3*B*b^3*c^2*d*f*g^2+3*B*b^3*c*d^2*f^2*g)/(2*a^3*c^3*g^6-6*a^3*c^2*d*f*g^5+6*a^3*c*d^
2*f^2*g^4-2*a^3*d^3*f^3*g^3-6*a^2*b*c^3*f*g^5+18*a^2*b*c^2*d*f^2*g^4-18*a^2*b*c*d^2*f^3*g^3+6*a^2*b*d^3*f^4*g^
2+6*a*b^2*c^3*f^2*g^4-18*a*b^2*c^2*d*f^3*g^3+18*a*b^2*c*d^2*f^4*g^2-6*a*b^2*d^3*f^5*g-2*b^3*c^3*f^3*g^3+6*b^3*
c^2*d*f^4*g^2-6*b^3*c*d^2*f^5*g+2*b^3*d^3*f^6)*(g*x+f)^-1/g-1/4*B*g^3*(-(g*x+f)^-1/g)^4*ln((a^2*g^4*(-(g*x+f)^
-1/g)^2-2*a*b*f*g^3*(-(g*x+f)^-1/g)^2+2*a*b*g^2*(g*x+f)^-1/g+b^2*f^2*g^2*(-(g*x+f)^-1/g)^2-2*b^2*f*g*(g*x+f)^-
1/g+b^2)/(c^2*g^4*(-(g*x+f)^-1/g)^2-2*c*d*f*g^3*(-(g*x+f)^-1/g)^2+2*c*d*g^2*(g*x+f)^-1/g+d^2*f^2*g^2*(-(g*x+f)
^-1/g)^2-2*d^2*f*g*(g*x+f)^-1/g+d^2))+g/g*((-B*a^4*d^4*g^3+4*B*a^3*d^4*g^2*b*f-6*B*a^2*d^4*g*b^2*f^2+4*B*a*d^4
*b^3*f^3-4*B*d^3*b^4*c*f^3+6*B*d^2*g*b^4*c^2*f^2-4*B*d*g^2*b^4*c^3*f+B*g^3*b^4*c^4)/(4*a^4*d^4*g^4*f^4-16*a^4*
d^3*g^5*c*f^3+24*a^4*d^2*g^6*c^2*f^2-16*a^4*d*g^7*c^3*f+4*a^4*g^8*c^4-16*a^3*d^4*g^3*b*f^5+64*a^3*d^3*g^4*b*c*
f^4-96*a^3*d^2*g^5*b*c^2*f^3+64*a^3*d*g^6*b*c^3*f^2-16*a^3*g^7*b*c^4*f+24*a^2*d^4*g^2*b^2*f^6-96*a^2*d^3*g^3*b
^2*c*f^5+144*a^2*d^2*g^4*b^2*c^2*f^4-96*a^2*d*g^5*b^2*c^3*f^3+24*a^2*g^6*b^2*c^4*f^2-16*a*d^4*g*b^3*f^7+64*a*d
^3*g^2*b^3*c*f^6-96*a*d^2*g^3*b^3*c^2*f^5+64*a*d*g^4*b^3*c^3*f^4-16*a*g^5*b^3*c^4*f^3+4*d^4*b^4*f^8-16*d^3*g*b
^4*c*f^7+24*d^2*g^2*b^4*c^2*f^6-16*d*g^3*b^4*c^3*f^5+4*g^4*b^4*c^4*f^4)*ln(abs(-(-(g*x+f)^-1/g)^2*a*d*g^3*f+(-
(g*x+f)^-1/g)^2*a*g^4*c+(-(g*x+f)^-1/g)^2*d*g^2*b*f^2-(-(g*x+f)^-1/g)^2*g^3*b*c*f+(g*x+f)^-1/g*a*d*g^2-2*(g*x+
f)^-1/g*d*g*b*f+(g*x+f)^-1/g*g^2*b*c+d*b))+(-B*a^5*d^5*g^5+4*B*a^4*d^5*g^4*b*f+B*a^4*d^4*g^5*b*c-6*B*a^3*d^5*g
^3*b^2*f^2-4*B*a^3*d^4*g^4*b^2*c*f+4*B*a^2*d^5*g^2*b^3*f^3+6*B*a^2*d^4*g^3*b^3*c*f^2-2*B*a*d^5*g*b^4*f^4-6*B*a
*d^3*g^3*b^4*c^2*f^2+4*B*a*d^2*g^4*b^4*c^3*f-B*a*d*g^5*b^4*c^4+2*B*d^4*g*b^5*c*f^4-4*B*d^3*g^2*b^5*c^2*f^3+6*B
*d^2*g^3*b^5*c^3*f^2-4*B*d*g^4*b^5*c^4*f+B*g^5*b^5*c^5)/(4*a^4*d^4*g^4*f^4-16*a^4*d^3*g^5*c*f^3+24*a^4*d^2*g^6
*c^2*f^2-16*a^4*d*g^7*c^3*f+4*a^4*g^8*c^4-16*a^3*d^4*g^3*b*f^5+64*a^3*d^3*g^4*b*c*f^4-96*a^3*d^2*g^5*b*c^2*f^3
+64*a^3*d*g^6*b*c^3*f^2-16*a^3*g^7*b*c^4*f+24*a^2*d^4*g^2*b^2*f^6-96*a^2*d^3*g^3*b^2*c*f^5+144*a^2*d^2*g^4*b^2
*c^2*f^4-96*a^2*d*g^5*b^2*c^3*f^3+24*a^2*g^6*b^2*c^4*f^2-16*a*d^4*g*b^3*f^7+64*a*d^3*g^2*b^3*c*f^6-96*a*d^2*g^
3*b^3*c^2*f^5+64*a*d*g^4*b^3*c^3*f^4-16*a*g^5*b^3*c^4*f^3+4*d^4*b^4*f^8-16*d^3*g*b^4*c*f^7+24*d^2*g^2*b^4*c^2*
f^6-16*d*g^3*b^4*c^3*f^5+4*g^4*b^4*c^4*f^4)/abs(a*d*g^2-g^2*b*c)*ln(abs(2*(g*x+f)^-1/g*a*d*g^3*f-2*(g*x+f)^-1/
g*a*g^4*c-2*(g*x+f)^-1/g*d*g^2*b*f^2+2*(g*x+f)^-1/g*g^3*b*c*f-a*d*g^2+2*d*g*b*f-g^2*b*c-abs(a*d*g^2-g^2*b*c))/
abs(2*(g*x+f)^-1/g*a*d*g^3*f-2*(g*x+f)^-1/g*a*g^4*c-2*(g*x+f)^-1/g*d*g^2*b*f^2+2*(g*x+f)^-1/g*g^3*b*c*f-a*d*g^
2+2*d*g*b*f-g^2*b*c+abs(a*d*g^2-g^2*b*c))))

________________________________________________________________________________________

maple [B]  time = 0.45, size = 10401, normalized size = 27.30 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*ln((b*x+a)^2/(d*x+c)^2*e)+A)/(g*x+f)^5,x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [B]  time = 2.00, size = 1809, normalized size = 4.75 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))/(g*x+f)^5,x, algorithm="maxima")

[Out]

1/12*(6*b^4*log(b*x + a)/(b^4*f^4*g - 4*a*b^3*f^3*g^2 + 6*a^2*b^2*f^2*g^3 - 4*a^3*b*f*g^4 + a^4*g^5) - 6*d^4*l
og(d*x + c)/(d^4*f^4*g - 4*c*d^3*f^3*g^2 + 6*c^2*d^2*f^2*g^3 - 4*c^3*d*f*g^4 + c^4*g^5) + 6*(4*(b^4*c*d^3 - a*
b^3*d^4)*f^3 - 6*(b^4*c^2*d^2 - a^2*b^2*d^4)*f^2*g + 4*(b^4*c^3*d - a^3*b*d^4)*f*g^2 - (b^4*c^4 - a^4*d^4)*g^3
)*log(g*x + f)/(b^4*d^4*f^8 + a^4*c^4*g^8 - 4*(b^4*c*d^3 + a*b^3*d^4)*f^7*g + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3
+ 3*a^2*b^2*d^4)*f^6*g^2 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*f^5*g^3 + (b^4*c^4 +
16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*f^4*g^4 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*
a^3*b*c^2*d^2 + a^4*c*d^3)*f^3*g^5 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*f^2*g^6 - 4*(a^3*b*c^4
+ a^4*c^3*d)*f*g^7) - (26*(b^3*c*d^2 - a*b^2*d^3)*f^4 - 31*(b^3*c^2*d - a^2*b*d^3)*f^3*g + (11*b^3*c^3 + 15*a*
b^2*c^2*d - 15*a^2*b*c*d^2 - 11*a^3*d^3)*f^2*g^2 - 7*(a*b^2*c^3 - a^3*c*d^2)*f*g^3 + 2*(a^2*b*c^3 - a^3*c^2*d)
*g^4 + 6*(3*(b^3*c*d^2 - a*b^2*d^3)*f^2*g^2 - 3*(b^3*c^2*d - a^2*b*d^3)*f*g^3 + (b^3*c^3 - a^3*d^3)*g^4)*x^2 +
3*(14*(b^3*c*d^2 - a*b^2*d^3)*f^3*g - 15*(b^3*c^2*d - a^2*b*d^3)*f^2*g^2 + (5*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2
*b*c*d^2 - 5*a^3*d^3)*f*g^3 - (a*b^2*c^3 - a^3*c*d^2)*g^4)*x)/(b^3*d^3*f^9 + a^3*c^3*f^3*g^6 - 3*(b^3*c*d^2 +
a*b^2*d^3)*f^8*g + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^7*g^2 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^
2 + a^3*d^3)*f^6*g^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^5*g^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^4*g^5 +
(b^3*d^3*f^6*g^3 + a^3*c^3*g^9 - 3*(b^3*c*d^2 + a*b^2*d^3)*f^5*g^4 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3
)*f^4*g^5 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^3*g^6 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c
*d^2)*f^2*g^7 - 3*(a^2*b*c^3 + a^3*c^2*d)*f*g^8)*x^3 + 3*(b^3*d^3*f^7*g^2 + a^3*c^3*f*g^8 - 3*(b^3*c*d^2 + a*b
^2*d^3)*f^6*g^3 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^5*g^4 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2
+ a^3*d^3)*f^4*g^5 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^3*g^6 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^2*g^7)*x
^2 + 3*(b^3*d^3*f^8*g + a^3*c^3*f^2*g^7 - 3*(b^3*c*d^2 + a*b^2*d^3)*f^7*g^2 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a
^2*b*d^3)*f^6*g^3 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^5*g^4 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d
+ a^3*c*d^2)*f^4*g^5 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^3*g^6)*x) - 3*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*
a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(g^5*x^4 + 4*f*g^4*x^3 + 6*f^2*g^3*x^2 +
4*f^3*g^2*x + f^4*g))*B - 1/4*A/(g^5*x^4 + 4*f*g^4*x^3 + 6*f^2*g^3*x^2 + 4*f^3*g^2*x + f^4*g)

________________________________________________________________________________________

mupad [B]  time = 17.43, size = 2520, normalized size = 6.61 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))/(f + g*x)^5,x)

[Out]

(log(f + g*x)*(g*(6*B*a^2*b^2*d^4*f^2 - 6*B*b^4*c^2*d^2*f^2) - g^2*(4*B*a^3*b*d^4*f - 4*B*b^4*c^3*d*f) + g^3*(
B*a^4*d^4 - B*b^4*c^4) - 4*B*a*b^3*d^4*f^3 + 4*B*b^4*c*d^3*f^3))/(2*a^4*c^4*g^8 + 2*b^4*d^4*f^8 + 2*a^4*d^4*f^
4*g^4 + 2*b^4*c^4*f^4*g^4 + 12*a^2*b^2*c^4*f^2*g^6 + 12*a^2*b^2*d^4*f^6*g^2 + 12*a^4*c^2*d^2*f^2*g^6 + 12*b^4*
c^2*d^2*f^6*g^2 - 8*a^3*b*c^4*f*g^7 - 8*a*b^3*d^4*f^7*g - 8*a^4*c^3*d*f*g^7 - 8*b^4*c*d^3*f^7*g - 8*a*b^3*c^4*
f^3*g^5 - 8*a^3*b*d^4*f^5*g^3 - 8*a^4*c*d^3*f^3*g^5 - 8*b^4*c^3*d*f^5*g^3 + 32*a*b^3*c*d^3*f^6*g^2 + 32*a*b^3*
c^3*d*f^4*g^4 + 32*a^3*b*c*d^3*f^4*g^4 + 32*a^3*b*c^3*d*f^2*g^6 - 48*a*b^3*c^2*d^2*f^5*g^3 - 48*a^2*b^2*c*d^3*
f^5*g^3 - 48*a^2*b^2*c^3*d*f^3*g^5 - 48*a^3*b*c^2*d^2*f^3*g^5 + 72*a^2*b^2*c^2*d^2*f^4*g^4) - ((3*A*a^3*c^3*g^
6 + 3*A*b^3*d^3*f^6 - 3*A*a^3*d^3*f^3*g^3 - 3*A*b^3*c^3*f^3*g^3 - 11*B*a^3*d^3*f^3*g^3 + 11*B*b^3*c^3*f^3*g^3
+ 9*A*a*b^2*c^3*f^2*g^4 + 9*A*a^2*b*d^3*f^4*g^2 - 7*B*a*b^2*c^3*f^2*g^4 + 9*A*a^3*c*d^2*f^2*g^4 + 31*B*a^2*b*d
^3*f^4*g^2 + 9*A*b^3*c^2*d*f^4*g^2 + 7*B*a^3*c*d^2*f^2*g^4 - 31*B*b^3*c^2*d*f^4*g^2 - 9*A*a^2*b*c^3*f*g^5 - 9*
A*a*b^2*d^3*f^5*g + 2*B*a^2*b*c^3*f*g^5 - 9*A*a^3*c^2*d*f*g^5 - 26*B*a*b^2*d^3*f^5*g - 9*A*b^3*c*d^2*f^5*g - 2
*B*a^3*c^2*d*f*g^5 + 26*B*b^3*c*d^2*f^5*g + 27*A*a*b^2*c*d^2*f^4*g^2 - 27*A*a*b^2*c^2*d*f^3*g^3 - 27*A*a^2*b*c
*d^2*f^3*g^3 + 27*A*a^2*b*c^2*d*f^2*g^4 + 15*B*a*b^2*c^2*d*f^3*g^3 - 15*B*a^2*b*c*d^2*f^3*g^3)/(6*(a^3*c^3*g^6
+ b^3*d^3*f^6 - a^3*d^3*f^3*g^3 - b^3*c^3*f^3*g^3 - 3*a^2*b*c^3*f*g^5 - 3*a*b^2*d^3*f^5*g - 3*a^3*c^2*d*f*g^5
- 3*b^3*c*d^2*f^5*g + 3*a*b^2*c^3*f^2*g^4 + 3*a^2*b*d^3*f^4*g^2 + 3*a^3*c*d^2*f^2*g^4 + 3*b^3*c^2*d*f^4*g^2 +
9*a*b^2*c*d^2*f^4*g^2 - 9*a*b^2*c^2*d*f^3*g^3 - 9*a^2*b*c*d^2*f^3*g^3 + 9*a^2*b*c^2*d*f^2*g^4)) - (x^2*(B*a*b
^2*c^3*g^6 - B*a^3*c*d^2*g^6 + 7*B*a^3*d^3*f*g^5 - 7*B*b^3*c^3*f*g^5 + 20*B*a*b^2*d^3*f^3*g^3 - 21*B*a^2*b*d^3
*f^2*g^4 - 20*B*b^3*c*d^2*f^3*g^3 + 21*B*b^3*c^2*d*f^2*g^4 - 3*B*a*b^2*c^2*d*f*g^5 + 3*B*a^2*b*c*d^2*f*g^5))/(
2*(a^3*c^3*g^6 + b^3*d^3*f^6 - a^3*d^3*f^3*g^3 - b^3*c^3*f^3*g^3 - 3*a^2*b*c^3*f*g^5 - 3*a*b^2*d^3*f^5*g - 3*a
^3*c^2*d*f*g^5 - 3*b^3*c*d^2*f^5*g + 3*a*b^2*c^3*f^2*g^4 + 3*a^2*b*d^3*f^4*g^2 + 3*a^3*c*d^2*f^2*g^4 + 3*b^3*c
^2*d*f^4*g^2 + 9*a*b^2*c*d^2*f^4*g^2 - 9*a*b^2*c^2*d*f^3*g^3 - 9*a^2*b*c*d^2*f^3*g^3 + 9*a^2*b*c^2*d*f^2*g^4))
+ (x*(B*a^2*b*c^3*g^6 - B*a^3*c^2*d*g^6 - 13*B*a^3*d^3*f^2*g^4 + 13*B*b^3*c^3*f^2*g^4 - 34*B*a*b^2*d^3*f^4*g^
2 + 38*B*a^2*b*d^3*f^3*g^3 + 34*B*b^3*c*d^2*f^4*g^2 - 38*B*b^3*c^2*d*f^3*g^3 - 5*B*a*b^2*c^3*f*g^5 + 5*B*a^3*c
*d^2*f*g^5 + 12*B*a*b^2*c^2*d*f^2*g^4 - 12*B*a^2*b*c*d^2*f^2*g^4))/(3*(a^3*c^3*g^6 + b^3*d^3*f^6 - a^3*d^3*f^3
*g^3 - b^3*c^3*f^3*g^3 - 3*a^2*b*c^3*f*g^5 - 3*a*b^2*d^3*f^5*g - 3*a^3*c^2*d*f*g^5 - 3*b^3*c*d^2*f^5*g + 3*a*b
^2*c^3*f^2*g^4 + 3*a^2*b*d^3*f^4*g^2 + 3*a^3*c*d^2*f^2*g^4 + 3*b^3*c^2*d*f^4*g^2 + 9*a*b^2*c*d^2*f^4*g^2 - 9*a
*b^2*c^2*d*f^3*g^3 - 9*a^2*b*c*d^2*f^3*g^3 + 9*a^2*b*c^2*d*f^2*g^4)) - (x^3*(B*a^3*d^3*g^6 - B*b^3*c^3*g^6 + 3
*B*a*b^2*d^3*f^2*g^4 - 3*B*b^3*c*d^2*f^2*g^4 - 3*B*a^2*b*d^3*f*g^5 + 3*B*b^3*c^2*d*f*g^5))/(a^3*c^3*g^6 + b^3*
d^3*f^6 - a^3*d^3*f^3*g^3 - b^3*c^3*f^3*g^3 - 3*a^2*b*c^3*f*g^5 - 3*a*b^2*d^3*f^5*g - 3*a^3*c^2*d*f*g^5 - 3*b^
3*c*d^2*f^5*g + 3*a*b^2*c^3*f^2*g^4 + 3*a^2*b*d^3*f^4*g^2 + 3*a^3*c*d^2*f^2*g^4 + 3*b^3*c^2*d*f^4*g^2 + 9*a*b^
2*c*d^2*f^4*g^2 - 9*a*b^2*c^2*d*f^3*g^3 - 9*a^2*b*c*d^2*f^3*g^3 + 9*a^2*b*c^2*d*f^2*g^4))/(2*f^4*g + 2*g^5*x^4
+ 8*f^3*g^2*x + 8*f*g^4*x^3 + 12*f^2*g^3*x^2) + (B*b^4*log(a + b*x))/(2*a^4*g^5 + 2*b^4*f^4*g - 8*a*b^3*f^3*g
^2 + 12*a^2*b^2*f^2*g^3 - 8*a^3*b*f*g^4) - (B*log((e*(a + b*x)^2)/(c + d*x)^2))/(4*g*(f^4 + g^4*x^4 + 4*f^3*g*
x + 4*f*g^3*x^3 + 6*f^2*g^2*x^2)) - (B*d^4*log(c + d*x))/(2*c^4*g^5 + 2*d^4*f^4*g - 8*c*d^3*f^3*g^2 + 12*c^2*d
^2*f^2*g^3 - 8*c^3*d*f*g^4)

________________________________________________________________________________________

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)**2/(d*x+c)**2))/(g*x+f)**5,x)

[Out]

Timed out

________________________________________________________________________________________