3.3.0 ∫ A+B log(e(a+bx)2 (c+dx)2 ) ________________________

Integrand size = 29, antiderivative size = 381 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=-\frac {B (b c-a d)}{6 (b f-a g) (d f-c g) (f+g x)^3}-\frac {B (b c-a d) (2 b d f-b c g-a d g)}{4 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{2 (b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 B \log (a+b x)}{2 g (b f-a g)^4}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{4 g (f+g x)^4}-\frac {B d^4 \log (c+d x)}{2 g (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{2 (b f-a g)^4 (d f-c g)^4} \]

-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] (veriﬁed)

Time = 0.26 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2548, 84} \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=-\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{2 (f+g x) (b f-a g)^3 (d f-c g)^3}-\frac {B (b c-a d) \log (f+g x) (-a d g-b c g+2 b d f) \left (-a^2 d^2 g^2+2 a b d^2 f g-\left (b^2 \left (c^2 g^2-2 c d f g+2 d^2 f^2\right )\right )\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{4 g (f+g x)^4}+\frac {b^4 B \log (a+b x)}{2 g (b f-a g)^4}-\frac {B (b c-a d) (-a d g-b c g+2 b d f)}{4 (f+g x)^2 (b f-a g)^2 (d f-c g)^2}-\frac {B (b c-a d)}{6 (f+g x)^3 (b f-a g) (d f-c g)}-\frac {B d^4 \log (c+d x)}{2 g (d f-c g)^4} \]

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

-1/6*(B*(b*c - a*d))/((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

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]

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.

), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(

(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A

, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && !(EqQ[m, -2] && IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = -\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] (veriﬁed)

Time = 0.51 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\frac {-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^4}+2 B (b c-a d) \left (-\frac {g}{3 (b f-a g) (d f-c g) (f+g x)^3}+\frac {g (-2 b d f+b c g+a d g)}{2 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac {g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)}+\frac {b^4 \log (a+b x)}{(b c-a d) (b f-a g)^4}-\frac {d^4 \log (c+d x)}{(b c-a d) (d f-c g)^4}-\frac {g (-2 b d f+b c g+a d g) \left (-2 a b d^2 f g+a^2 d^2 g^2+b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{(b f-a g)^4 (d f-c g)^4}\right )}{4 g} \]

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

(-((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)

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(2298\) vs. \(2(370)=740\).

Time = 5.50 (sec) , antiderivative size = 2299, normalized size of antiderivative = 6.03


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(2299\)
default	\(\text {Expression too large to display}\)	\(2299\)
risch	\(\text {Expression too large to display}\)	\(4452\)
parallelrisch	\(\text {Expression too large to display}\)	\(5619\)

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

-1/d*(-d^5*A*(-1/4*g^3/(c*g-d*f)^4/(c*g/(d*x+c)-f/(d*x+c)*d-g)^4-g^2/(c*g-d*f)^4/(c*g/(d*x+c)-f/(d*x+c)*d-g)^3

-3/2*g/(c*g-d*f)^4/(c*g/(d*x+c)-f/(d*x+c)*d-g)^2-1/(c*g-d*f)^4/(c*g/(d*x+c)-f/(d*x+c)*d-g))+((c*g-d*f)*b^4*g^2

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

2/d^2)+(c*g-d*f)*(c^2*g^2-2*c*d*f*g+d^2*f^2)*b^4*B*d/(a^4*g^4-4*a^3*b*f*g^3+6*a^2*b^2*f^2*g^2-4*a*b^3*f^3*g+b^

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

f^2*g^4-B*b^3*c^3*d^2*g^6+3*B*b^3*c^2*d^3*f*g^5-3*B*b^3*c*d^4*f^2*g^4)/g/(a^3*c^3*g^6-3*a^3*c^2*d*f*g^5+3*a^3*

c*d^2*f^2*g^4-a^3*d^3*f^3*g^3-3*a^2*b*c^3*f*g^5+9*a^2*b*c^2*d*f^2*g^4-9*a^2*b*c*d^2*f^3*g^3+3*a^2*b*d^3*f^4*g^

2+3*a*b^2*c^3*f^2*g^4-9*a*b^2*c^2*d*f^3*g^3+9*a*b^2*c*d^2*f^4*g^2-3*a*b^2*d^3*f^5*g-b^3*c^3*f^3*g^3+3*b^3*c^2*

d*f^4*g^2-3*b^3*c*d^2*f^5*g+b^3*d^3*f^6)/(d*x+c)+1/12*(11*B*a^3*d^5*g^6-2*B*a^2*b*c*d^4*g^6-31*B*a^2*b*d^5*f*g

^5-3*B*a*b^2*c^2*d^3*g^6+10*B*a*b^2*c*d^4*f*g^5+26*B*a*b^2*d^5*f^2*g^4-6*B*b^3*c^3*d^2*g^6+21*B*b^3*c^2*d^3*f*

g^5-26*B*b^3*c*d^4*f^2*g^4)/(a^3*g^3-3*a^2*b*f*g^2+3*a*b^2*f^2*g-b^3*f^3)/g^4/(d*x+c)^4-1/6*(13*B*a^3*d^5*g^6-

B*a^2*b*c*d^4*g^6-38*B*a^2*b*d^5*f*g^5-3*B*a*b^2*c^2*d^3*g^6+8*B*a*b^2*c*d^4*f*g^5+34*B*a*b^2*d^5*f^2*g^4-9*B*

b^3*c^3*d^2*g^6+30*B*b^3*c^2*d^3*f*g^5-34*B*b^3*c*d^4*f^2*g^4)/g^3/(a^3*c*g^4-a^3*d*f*g^3-3*a^2*b*c*f*g^3+3*a^

2*b*d*f^2*g^2+3*a*b^2*c*f^2*g^2-3*a*b^2*d*f^3*g-b^3*c*f^3*g+b^3*d*f^4)/(d*x+c)^3+1/4*(7*B*a^3*d^5*g^6-21*B*a^2

*b*d^5*f*g^5-B*a*b^2*c^2*d^3*g^6+2*B*a*b^2*c*d^4*f*g^5+20*B*a*b^2*d^5*f^2*g^4-6*B*b^3*c^3*d^2*g^6+19*B*b^3*c^2

*d^3*f*g^5-20*B*b^3*c*d^4*f^2*g^4)/(c^2*g^2-2*c*d*f*g+d^2*f^2)/g^2/(a^3*g^3-3*a^2*b*f*g^2+3*a*b^2*f^2*g-b^3*f^

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

d*f*g^2-6*b^4*c^2*d^2*f^2*g+4*b^4*c*d^3*f^3)/(a^4*g^4-4*a^3*b*f*g^3+6*a^2*b^2*f^2*g^2-4*a*b^3*f^3*g+b^4*f^4)/(

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

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

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

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

^3+4*b^4*c^3*d*f*g^2-6*b^4*c^2*d^2*f^2*g+4*b^4*c*d^3*f^3)/(a^4*c^4*g^8-4*a^4*c^3*d*f*g^7+6*a^4*c^2*d^2*f^2*g^6

-4*a^4*c*d^3*f^3*g^5+a^4*d^4*f^4*g^4-4*a^3*b*c^4*f*g^7+16*a^3*b*c^3*d*f^2*g^6-24*a^3*b*c^2*d^2*f^3*g^5+16*a^3*

b*c*d^3*f^4*g^4-4*a^3*b*d^4*f^5*g^3+6*a^2*b^2*c^4*f^2*g^6-24*a^2*b^2*c^3*d*f^3*g^5+36*a^2*b^2*c^2*d^2*f^4*g^4-

24*a^2*b^2*c*d^3*f^5*g^3+6*a^2*b^2*d^4*f^6*g^2-4*a*b^3*c^4*f^3*g^5+16*a*b^3*c^3*d*f^4*g^4-24*a*b^3*c^2*d^2*f^5

*g^3+16*a*b^3*c*d^3*f^6*g^2-4*a*b^3*d^4*f^7*g+b^4*c^4*f^4*g^4-4*b^4*c^3*d*f^5*g^3+6*b^4*c^2*d^2*f^6*g^2-4*b^4*

c*d^3*f^7*g+b^4*d^4*f^8)*ln(c*g/(d*x+c)-f/(d*x+c)*d-g))

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Timed out} \]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Timed out} \]

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

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1809 vs. \(2 (367) = 734\).

Time = 0.32 (sec) , antiderivative size = 1809, normalized size of antiderivative = 4.75 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \]

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

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)

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2159 vs. \(2 (367) = 734\).

Time = 3.18 (sec) , antiderivative size = 2159, normalized size of antiderivative = 5.67 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \]

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

-1/4*(4*B*b^4*c*d^3*f^3 - 4*B*a*b^3*d^4*f^3 - 6*B*b^4*c^2*d^2*f^2*g + 6*B*a^2*b^2*d^4*f^2*g + 4*B*b^4*c^3*d*f*

g^2 - 4*B*a^3*b*d^4*f*g^2 - B*b^4*c^4*g^3 + B*a^4*d^4*g^3)*log(abs(b*d - 2*b*d*f/(g*x + f) + b*d*f^2/(g*x + f)

^2 + b*c*g/(g*x + f) + a*d*g/(g*x + f) - b*c*f*g/(g*x + f)^2 - a*d*f*g/(g*x + f)^2 + a*c*g^2/(g*x + f)^2))/(b^

4*d^4*f^8 - 4*b^4*c*d^3*f^7*g - 4*a*b^3*d^4*f^7*g + 6*b^4*c^2*d^2*f^6*g^2 + 16*a*b^3*c*d^3*f^6*g^2 + 6*a^2*b^2

*d^4*f^6*g^2 - 4*b^4*c^3*d*f^5*g^3 - 24*a*b^3*c^2*d^2*f^5*g^3 - 24*a^2*b^2*c*d^3*f^5*g^3 - 4*a^3*b*d^4*f^5*g^3

+ b^4*c^4*f^4*g^4 + 16*a*b^3*c^3*d*f^4*g^4 + 36*a^2*b^2*c^2*d^2*f^4*g^4 + 16*a^3*b*c*d^3*f^4*g^4 + a^4*d^4*f^

4*g^4 - 4*a*b^3*c^4*f^3*g^5 - 24*a^2*b^2*c^3*d*f^3*g^5 - 24*a^3*b*c^2*d^2*f^3*g^5 - 4*a^4*c*d^3*f^3*g^5 + 6*a^

2*b^2*c^4*f^2*g^6 + 16*a^3*b*c^3*d*f^2*g^6 + 6*a^4*c^2*d^2*f^2*g^6 - 4*a^3*b*c^4*f*g^7 - 4*a^4*c^3*d*f*g^7 + a

^4*c^4*g^8) + 1/4*(2*B*b^5*c*d^4*f^4*g - 2*B*a*b^4*d^5*f^4*g - 4*B*b^5*c^2*d^3*f^3*g^2 + 4*B*a^2*b^3*d^5*f^3*g

^2 + 6*B*b^5*c^3*d^2*f^2*g^3 - 6*B*a*b^4*c^2*d^3*f^2*g^3 + 6*B*a^2*b^3*c*d^4*f^2*g^3 - 6*B*a^3*b^2*d^5*f^2*g^3

- 4*B*b^5*c^4*d*f*g^4 + 4*B*a*b^4*c^3*d^2*f*g^4 - 4*B*a^3*b^2*c*d^4*f*g^4 + 4*B*a^4*b*d^5*f*g^4 + B*b^5*c^5*g

^5 - B*a*b^4*c^4*d*g^5 + B*a^4*b*c*d^4*g^5 - B*a^5*d^5*g^5)*log(abs(2*b*d*f*g - 2*b*d*f^2*g/(g*x + f) - b*c*g^

2 - a*d*g^2 + 2*b*c*f*g^2/(g*x + f) + 2*a*d*f*g^2/(g*x + f) - 2*a*c*g^3/(g*x + f) - abs(-b*c*g^2 + a*d*g^2))/a

bs(2*b*d*f*g - 2*b*d*f^2*g/(g*x + f) - b*c*g^2 - a*d*g^2 + 2*b*c*f*g^2/(g*x + f) + 2*a*d*f*g^2/(g*x + f) - 2*a

*c*g^3/(g*x + f) + abs(-b*c*g^2 + a*d*g^2)))/((b^4*d^4*f^8 - 4*b^4*c*d^3*f^7*g - 4*a*b^3*d^4*f^7*g + 6*b^4*c^2

*d^2*f^6*g^2 + 16*a*b^3*c*d^3*f^6*g^2 + 6*a^2*b^2*d^4*f^6*g^2 - 4*b^4*c^3*d*f^5*g^3 - 24*a*b^3*c^2*d^2*f^5*g^3

- 24*a^2*b^2*c*d^3*f^5*g^3 - 4*a^3*b*d^4*f^5*g^3 + b^4*c^4*f^4*g^4 + 16*a*b^3*c^3*d*f^4*g^4 + 36*a^2*b^2*c^2*

d^2*f^4*g^4 + 16*a^3*b*c*d^3*f^4*g^4 + a^4*d^4*f^4*g^4 - 4*a*b^3*c^4*f^3*g^5 - 24*a^2*b^2*c^3*d*f^3*g^5 - 24*a

^3*b*c^2*d^2*f^3*g^5 - 4*a^4*c*d^3*f^3*g^5 + 6*a^2*b^2*c^4*f^2*g^6 + 16*a^3*b*c^3*d*f^2*g^6 + 6*a^4*c^2*d^2*f^

2*g^6 - 4*a^3*b*c^4*f*g^7 - 4*a^4*c^3*d*f*g^7 + a^4*c^4*g^8)*abs(-b*c*g^2 + a*d*g^2)) - 1/2*(3*B*b^3*c*d^2*f^2

*g - 3*B*a*b^2*d^3*f^2*g - 3*B*b^3*c^2*d*f*g^2 + 3*B*a^2*b*d^3*f*g^2 + B*b^3*c^3*g^3 - B*a^3*d^3*g^3)/((b^3*d^

3*f^6 - 3*b^3*c*d^2*f^5*g - 3*a*b^2*d^3*f^5*g + 3*b^3*c^2*d*f^4*g^2 + 9*a*b^2*c*d^2*f^4*g^2 + 3*a^2*b*d^3*f^4*

g^2 - b^3*c^3*f^3*g^3 - 9*a*b^2*c^2*d*f^3*g^3 - 9*a^2*b*c*d^2*f^3*g^3 - a^3*d^3*f^3*g^3 + 3*a*b^2*c^3*f^2*g^4

+ 9*a^2*b*c^2*d*f^2*g^4 + 3*a^3*c*d^2*f^2*g^4 - 3*a^2*b*c^3*f*g^5 - 3*a^3*c^2*d*f*g^5 + a^3*c^3*g^6)*(g*x + f)

*g) - 1/4*B*log((b^2*e - 2*b^2*e*f/(g*x + f) + b^2*e*f^2/(g*x + f)^2 + 2*a*b*e*g/(g*x + f) - 2*a*b*e*f*g/(g*x

+ f)^2 + a^2*e*g^2/(g*x + f)^2)/(d^2 - 2*d^2*f/(g*x + f) + d^2*f^2/(g*x + f)^2 + 2*c*d*g/(g*x + f) - 2*c*d*f*g

/(g*x + f)^2 + c^2*g^2/(g*x + f)^2))/((g*x + f)^4*g) - 1/4*(2*B*b^2*c*d*f*g^2 - 2*B*a*b*d^2*f*g^2 - B*b^2*c^2*

g^3 + B*a^2*d^2*g^3)/((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)*(g*x + f)^2*g^2) - 1/4*A/((g*x + f)^4*g) -

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

Mupad [B] (veriﬁcation not implemented)

Time = 13.52 (sec) , antiderivative size = 2520, normalized size of antiderivative = 6.61 \[ \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(f+g x)^5} \, dx=\text {Too large to display} \]

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

(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

\(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{(f+g x)^5} \, dx\) [271]

Optimal result