3.262 \(\int (f+g x)^4 (A+B \log (\frac{e (a+b x)^2}{(c+d x)^2})) \, dx\)

Optimal. Leaf size=357 \[ -\frac{B g^2 x^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (c^2 g^2-5 c d f g+10 d^2 f^2\right )\right )}{5 b^3 d^3}+\frac{2 B g x (b c-a d) \left (-a^2 b d^2 g^2 (5 d f-c g)+a^3 d^3 g^3+a b^2 d g \left (c^2 g^2-5 c d f g+10 d^2 f^2\right )+b^3 \left (-\left (5 c^2 d f g^2-c^3 g^3-10 c d^2 f^2 g+10 d^3 f^3\right )\right )\right )}{5 b^4 d^4}+\frac{(f+g x)^5 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 g}-\frac{2 B g^3 x^3 (b c-a d) (-a d g-b c g+5 b d f)}{15 b^2 d^2}-\frac{2 B (b f-a g)^5 \log (a+b x)}{5 b^5 g}-\frac{B g^4 x^4 (b c-a d)}{10 b d}+\frac{2 B (d f-c g)^5 \log (c+d x)}{5 d^5 g} \]

[Out]

(2*B*(b*c - a*d)*g*(a^3*d^3*g^3 - a^2*b*d^2*g^2*(5*d*f - c*g) + a*b^2*d*g*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2) -
b^3*(10*d^3*f^3 - 10*c*d^2*f^2*g + 5*c^2*d*f*g^2 - c^3*g^3))*x)/(5*b^4*d^4) - (B*(b*c - a*d)*g^2*(a^2*d^2*g^2
- a*b*d*g*(5*d*f - c*g) + b^2*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2))*x^2)/(5*b^3*d^3) - (2*B*(b*c - a*d)*g^3*(5*
b*d*f - b*c*g - a*d*g)*x^3)/(15*b^2*d^2) - (B*(b*c - a*d)*g^4*x^4)/(10*b*d) - (2*B*(b*f - a*g)^5*Log[a + b*x])
/(5*b^5*g) + ((f + g*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(5*g) + (2*B*(d*f - c*g)^5*Log[c + d*x])/(
5*d^5*g)

________________________________________________________________________________________

Rubi [A]  time = 0.50101, antiderivative size = 341, normalized size of antiderivative = 0.96, 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} \[ -\frac{B g^2 x^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (c^2 g^2-5 c d f g+10 d^2 f^2\right )\right )}{5 b^3 d^3}+\frac{2 B g x \left (-10 a^2 b^2 d^4 f^2 g+5 a^3 b d^4 f g^2-a^4 d^4 g^3+10 a b^3 d^4 f^3+b^4 (-c) \left (5 c^2 d f g^2-c^3 g^3-10 c d^2 f^2 g+10 d^3 f^3\right )\right )}{5 b^4 d^4}+\frac{(f+g x)^5 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{5 g}-\frac{2 B g^3 x^3 (b c-a d) (-a d g-b c g+5 b d f)}{15 b^2 d^2}-\frac{2 B (b f-a g)^5 \log (a+b x)}{5 b^5 g}-\frac{B g^4 x^4 (b c-a d)}{10 b d}+\frac{2 B (d f-c g)^5 \log (c+d x)}{5 d^5 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*B*g*(10*a*b^3*d^4*f^3 - 10*a^2*b^2*d^4*f^2*g + 5*a^3*b*d^4*f*g^2 - a^4*d^4*g^3 - b^4*c*(10*d^3*f^3 - 10*c*d
^2*f^2*g + 5*c^2*d*f*g^2 - c^3*g^3))*x)/(5*b^4*d^4) - (B*(b*c - a*d)*g^2*(a^2*d^2*g^2 - a*b*d*g*(5*d*f - c*g)
+ b^2*(10*d^2*f^2 - 5*c*d*f*g + c^2*g^2))*x^2)/(5*b^3*d^3) - (2*B*(b*c - a*d)*g^3*(5*b*d*f - b*c*g - a*d*g)*x^
3)/(15*b^2*d^2) - (B*(b*c - a*d)*g^4*x^4)/(10*b*d) - (2*B*(b*f - a*g)^5*Log[a + b*x])/(5*b^5*g) + ((f + g*x)^5
*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(5*g) + (2*B*(d*f - c*g)^5*Log[c + d*x])/(5*d^5*g)

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]

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]

Rubi steps

\begin{align*} \int (f+g x)^4 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx &=\frac{(f+g x)^5 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 g}-\frac{B \int \frac{2 (b c-a d) (f+g x)^5}{(a+b x) (c+d x)} \, dx}{5 g}\\ &=\frac{(f+g x)^5 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 g}-\frac{(2 B (b c-a d)) \int \frac{(f+g x)^5}{(a+b x) (c+d x)} \, dx}{5 g}\\ &=\frac{(f+g x)^5 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 g}-\frac{(2 B (b c-a d)) \int \left (\frac{g^2 \left (-a^3 d^3 g^3+a^2 b d^2 g^2 (5 d f-c g)-a b^2 d g \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )+b^3 \left (10 d^3 f^3-10 c d^2 f^2 g+5 c^2 d f g^2-c^3 g^3\right )\right )}{b^4 d^4}+\frac{g^3 \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )\right ) x}{b^3 d^3}+\frac{g^4 (5 b d f-b c g-a d g) x^2}{b^2 d^2}+\frac{g^5 x^3}{b d}+\frac{(b f-a g)^5}{b^4 (b c-a d) (a+b x)}+\frac{(d f-c g)^5}{d^4 (-b c+a d) (c+d x)}\right ) \, dx}{5 g}\\ &=\frac{2 B (b c-a d) g \left (a^3 d^3 g^3-a^2 b d^2 g^2 (5 d f-c g)+a b^2 d g \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )-b^3 \left (10 d^3 f^3-10 c d^2 f^2 g+5 c^2 d f g^2-c^3 g^3\right )\right ) x}{5 b^4 d^4}-\frac{B (b c-a d) g^2 \left (a^2 d^2 g^2-a b d g (5 d f-c g)+b^2 \left (10 d^2 f^2-5 c d f g+c^2 g^2\right )\right ) x^2}{5 b^3 d^3}-\frac{2 B (b c-a d) g^3 (5 b d f-b c g-a d g) x^3}{15 b^2 d^2}-\frac{B (b c-a d) g^4 x^4}{10 b d}-\frac{2 B (b f-a g)^5 \log (a+b x)}{5 b^5 g}+\frac{(f+g x)^5 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{5 g}+\frac{2 B (d f-c g)^5 \log (c+d x)}{5 d^5 g}\\ \end{align*}

Mathematica [A]  time = 0.587325, size = 282, normalized size = 0.79 \[ \frac{\frac{B g^2 x (a d-b c) \left (6 a^2 b d^2 g^2 (-2 c g+10 d f+d g x)-12 a^3 d^3 g^3-2 a b^2 d g \left (6 c^2 g^2-3 c d g (10 f+g x)+d^2 \left (60 f^2+15 f g x+2 g^2 x^2\right )\right )+b^3 \left (6 c^2 d g^2 (10 f+g x)-12 c^3 g^3-2 c d^2 g \left (60 f^2+15 f g x+2 g^2 x^2\right )+d^3 \left (60 f^2 g x+120 f^3+20 f g^2 x^2+3 g^3 x^3\right )\right )\right )}{6 b^4 d^4}+(f+g x)^5 \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )-\frac{2 B (b f-a g)^5 \log (a+b x)}{b^5}+\frac{2 B (d f-c g)^5 \log (c+d x)}{d^5}}{5 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((B*(-(b*c) + a*d)*g^2*x*(-12*a^3*d^3*g^3 + 6*a^2*b*d^2*g^2*(10*d*f - 2*c*g + d*g*x) - 2*a*b^2*d*g*(6*c^2*g^2
- 3*c*d*g*(10*f + g*x) + d^2*(60*f^2 + 15*f*g*x + 2*g^2*x^2)) + b^3*(-12*c^3*g^3 + 6*c^2*d*g^2*(10*f + g*x) -
2*c*d^2*g*(60*f^2 + 15*f*g*x + 2*g^2*x^2) + d^3*(120*f^3 + 60*f^2*g*x + 20*f*g^2*x^2 + 3*g^3*x^3))))/(6*b^4*d^
4) - (2*B*(b*f - a*g)^5*Log[a + b*x])/b^5 + (f + g*x)^5*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + (2*B*(d*f -
c*g)^5*Log[c + d*x])/d^5)/(5*g)

________________________________________________________________________________________

Maple [B]  time = 0.318, size = 2438, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^4*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2)),x)

[Out]

-2/d^4*B*g^3*c^4*ln(1/(d*x+c))*f+4*B*g^2*a^3/b^3*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*f^2-4*B*g^2*a^3/b^3*ln(1/(d*x
+c))*f^2+4*B*g/b^2*ln(1/(d*x+c))*a^2*f^3-4*B*g/b^2*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^2*f^3-4*B/(a*d-b*c)*ln(1/
(d*x+c)*a*d-b*c/(d*x+c)+b)*a*c*f^4+4/d^2*B*c^2*f^2*g^2*x-2/d*B*g^2*c*f^2*x^2+2*B*g^3*a^3/b^3*f*x+4*B*g/b*a*f^3
*x-2*B/b*ln(1/(d*x+c))*a*f^4+2/5*B*g^4*a^5/b^5*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)-2/5*B*g^4*a^5/b^5*ln(1/(d*x+c))
+1/5*B*g^4*a^3/b^3*x^2+2/15/d^2*B*g^4*c^2*x^3-1/5/d^3*B*g^4*c^3*x^2+2/5/d^4*B*g^4*c^4*x-1/10/d*B*g^4*c*x^4+2*B
*g^2*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*f^2*x^3-2/5*B*g^4*a^4/b^4*x+1/10*B*g^4*a/b*x^4-2/15*B*g^4*a^2/b
^2*x^3+2*B*g*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*f^3*x^2+1/5/d^5*B*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/
d^2)*c^5*g^4-1/d^4*B*g^3*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*f*c^4-2/d^2*B*g*ln(e*(1/(d*x+c)*a*d-b*c/(d*
x+c)+b)^2/d^2)*f^3*c^2+8/d^3*B*g^2*c^3*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*f^2-4/d^2*B*g*ln(1/(d*x+c)*a*d-b*c/(d*x
+c)+b)*c^2*f^3-4/d^2*B*ln(1/(d*x+c))*c^2*f^3*g-6/d^4*B*g^3*c^4*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*f-1/5/d^2*B*g^4
*a^3/b^3*c^2+B*g^3*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*f*x^4-1/10/d^4*B*g^4*a/b*c^4-1/d^4*A*c^4*f*g^3+2/
d^3*A*c^3*f^2*g^2-2/d^2*A*c^2*f^3*g+6/d^3*B*g^2*c^3*f^2-4/d^2*B*g*c^2*f^3-11/3/d^4*B*g^3*c^4*f-2*B*g^3*a^4/b^4
*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*f+2*B*g^3*a^4/b^4*ln(1/(d*x+c))*f-4/d*B*c*f^3*g*x-2/3/d*B*g^3*c*f*x^3+1/d^2*B
*g^3*c^2*f*x^2-2/d^3*B*g^3*c^3*f*x+4/d^3*B*g^2*c^3*ln(1/(d*x+c))*f^2+2/d^3*B*g^2*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+
c)+b)^2/d^2)*f^2*c^3-2/5/d*B*g^4*a^4/b^4*c+12/d*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^2*c^2*f^2*g^2-
8/d^2*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^2*c^3*f*g^3+2/5/d^5*B*g^4*c^5*ln(1/(d*x+c))+8/5/d^5*B*g^
4*c^5*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)+1/d*B*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*c*f^4+2/d*B*ln(1/(d*x+c)
)*c*f^4+5/6/d^5*B*g^4*c^5+1/5/d^5*A*c^5*g^4+1/d*A*c*f^4+A*x^4*f*g^3+2*A*x^3*f^2*g^2+2*A*x^2*f^3*g+1/5*B*g^4*ln
(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*x^5+B*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)*x*f^4+2/d^3*B/b/(a*d-b
*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^2*c^4*g^4-8/d^4*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^5*b*f*g^3-
8*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^2*c*f^3*g+12/d^3*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)
*c^4*b*f^2*g^2+16/d^3*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a*c^4*f*g^3+8/d*B*g/b*ln(1/(d*x+c)*a*d-b*c/(
d*x+c)+b)*a*c*f^3+16/d*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a*c^2*f^3*g-12/d^2*B*g^2*a/b*ln(1/(d*x+c)*a
*d-b*c/(d*x+c)+b)*c^2*f^2-8/d^2*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^3*b*f^3*g+8/d^3*B*g^3*a/b*ln(1/(
d*x+c)*a*d-b*c/(d*x+c)+b)*c^3*f-24/d^2*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a*c^3*f^2*g^2+A*x*f^4+1/5*A
*x^5*g^4-2/15/d^3*B*g^4*a^2/b^2*c^3+2/3*B*g^3*a/b*f*x^3-B*g^3*a^2/b^2*f*x^2+2*B*g^2*a/b*f^2*x^2-4*B*g^2*a^2/b^
2*f^2*x+2/d^5*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^6*b*g^4-2/d^4*B*g^4*a/b*ln(1/(d*x+c)*a*d-b*c/(d*x+
c)+b)*c^4+2*d*B/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*a^2*f^4-4/d^4*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*
x+c)+b)*a*c^5*g^4+2/d*B/(a*d-b*c)*ln(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*c^2*b*f^4+2/d*B*g^3*a^3/b^3*f*c+4/d*B*g/b*a*
f^3*c-4/d*B*g^2*a^2/b^2*f^2*c+1/d^2*B*g^3*a^2/b^2*c^2*f+2/3/d^3*B*g^3*a/b*c^3*f-2/d^2*B*g^2*a/b*c^2*f^2

________________________________________________________________________________________

Maxima [B]  time = 1.42897, size = 1154, normalized size = 3.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="maxima")

[Out]

1/5*A*g^4*x^5 + A*f*g^3*x^4 + 2*A*f^2*g^2*x^3 + 2*A*f^3*g*x^2 + (x*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)) + 2*a*log(b*x + a)/b - 2*c*log(d*x + c)/
d)*B*f^4 + 2*(x^2*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)) - 2*a^2*log(b*x + a)/b^2 + 2*c^2*log(d*x + c)/d^2 - 2*(b*c - a*d)*x/(b*d))*B*f^3*g + 2*(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)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/
(b^2*d^2))*B*f^2*g^2 + 1/3*(3*x^4*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)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2
*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*f*g^3 + 1/30*(6*x^5*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)) + 12*
a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4
)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4))*B*g^4 + A*f^4*x

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Fricas [A]  time = 3.49111, size = 1328, normalized size = 3.72 \begin{align*} \frac{6 \, A b^{5} d^{5} g^{4} x^{5} + 3 \,{\left (10 \, A b^{5} d^{5} f g^{3} -{\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{4}\right )} x^{4} + 4 \,{\left (15 \, A b^{5} d^{5} f^{2} g^{2} - 5 \,{\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} f g^{3} +{\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} g^{4}\right )} x^{3} + 6 \,{\left (10 \, A b^{5} d^{5} f^{3} g - 10 \,{\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} f^{2} g^{2} + 5 \,{\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} f g^{3} -{\left (B b^{5} c^{3} d^{2} - B a^{3} b^{2} d^{5}\right )} g^{4}\right )} x^{2} + 6 \,{\left (5 \, A b^{5} d^{5} f^{4} - 20 \,{\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} f^{3} g + 20 \,{\left (B b^{5} c^{2} d^{3} - B a^{2} b^{3} d^{5}\right )} f^{2} g^{2} - 10 \,{\left (B b^{5} c^{3} d^{2} - B a^{3} b^{2} d^{5}\right )} f g^{3} + 2 \,{\left (B b^{5} c^{4} d - B a^{4} b d^{5}\right )} g^{4}\right )} x + 12 \,{\left (5 \, B a b^{4} d^{5} f^{4} - 10 \, B a^{2} b^{3} d^{5} f^{3} g + 10 \, B a^{3} b^{2} d^{5} f^{2} g^{2} - 5 \, B a^{4} b d^{5} f g^{3} + B a^{5} d^{5} g^{4}\right )} \log \left (b x + a\right ) - 12 \,{\left (5 \, B b^{5} c d^{4} f^{4} - 10 \, B b^{5} c^{2} d^{3} f^{3} g + 10 \, B b^{5} c^{3} d^{2} f^{2} g^{2} - 5 \, B b^{5} c^{4} d f g^{3} + B b^{5} c^{5} g^{4}\right )} \log \left (d x + c\right ) + 6 \,{\left (B b^{5} d^{5} g^{4} x^{5} + 5 \, B b^{5} d^{5} f g^{3} x^{4} + 10 \, B b^{5} d^{5} f^{2} g^{2} x^{3} + 10 \, B b^{5} d^{5} f^{3} g x^{2} + 5 \, B b^{5} d^{5} f^{4} x\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{30 \, b^{5} d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="fricas")

[Out]

1/30*(6*A*b^5*d^5*g^4*x^5 + 3*(10*A*b^5*d^5*f*g^3 - (B*b^5*c*d^4 - B*a*b^4*d^5)*g^4)*x^4 + 4*(15*A*b^5*d^5*f^2
*g^2 - 5*(B*b^5*c*d^4 - B*a*b^4*d^5)*f*g^3 + (B*b^5*c^2*d^3 - B*a^2*b^3*d^5)*g^4)*x^3 + 6*(10*A*b^5*d^5*f^3*g
- 10*(B*b^5*c*d^4 - B*a*b^4*d^5)*f^2*g^2 + 5*(B*b^5*c^2*d^3 - B*a^2*b^3*d^5)*f*g^3 - (B*b^5*c^3*d^2 - B*a^3*b^
2*d^5)*g^4)*x^2 + 6*(5*A*b^5*d^5*f^4 - 20*(B*b^5*c*d^4 - B*a*b^4*d^5)*f^3*g + 20*(B*b^5*c^2*d^3 - B*a^2*b^3*d^
5)*f^2*g^2 - 10*(B*b^5*c^3*d^2 - B*a^3*b^2*d^5)*f*g^3 + 2*(B*b^5*c^4*d - B*a^4*b*d^5)*g^4)*x + 12*(5*B*a*b^4*d
^5*f^4 - 10*B*a^2*b^3*d^5*f^3*g + 10*B*a^3*b^2*d^5*f^2*g^2 - 5*B*a^4*b*d^5*f*g^3 + B*a^5*d^5*g^4)*log(b*x + a)
- 12*(5*B*b^5*c*d^4*f^4 - 10*B*b^5*c^2*d^3*f^3*g + 10*B*b^5*c^3*d^2*f^2*g^2 - 5*B*b^5*c^4*d*f*g^3 + B*b^5*c^5
*g^4)*log(d*x + c) + 6*(B*b^5*d^5*g^4*x^5 + 5*B*b^5*d^5*f*g^3*x^4 + 10*B*b^5*d^5*f^2*g^2*x^3 + 10*B*b^5*d^5*f^
3*g*x^2 + 5*B*b^5*d^5*f^4*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/(b^5*d^5)

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Sympy [B]  time = 35.7938, size = 1562, normalized size = 4.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**4*(A+B*ln(e*(b*x+a)**2/(d*x+c)**2)),x)

[Out]

A*g**4*x**5/5 + 2*B*a*(a**4*g**4 - 5*a**3*b*f*g**3 + 10*a**2*b**2*f**2*g**2 - 10*a*b**3*f**3*g + 5*b**4*f**4)*
log(x + (2*B*a**5*c*d**4*g**4 - 10*B*a**4*b*c*d**4*f*g**3 + 20*B*a**3*b**2*c*d**4*f**2*g**2 - 20*B*a**2*b**3*c
*d**4*f**3*g + 2*B*a**2*d**5*(a**4*g**4 - 5*a**3*b*f*g**3 + 10*a**2*b**2*f**2*g**2 - 10*a*b**3*f**3*g + 5*b**4
*f**4)/b + 2*B*a*b**4*c**5*g**4 - 10*B*a*b**4*c**4*d*f*g**3 + 20*B*a*b**4*c**3*d**2*f**2*g**2 - 20*B*a*b**4*c*
*2*d**3*f**3*g + 20*B*a*b**4*c*d**4*f**4 - 2*B*a*c*d**4*(a**4*g**4 - 5*a**3*b*f*g**3 + 10*a**2*b**2*f**2*g**2
- 10*a*b**3*f**3*g + 5*b**4*f**4))/(2*B*a**5*d**5*g**4 - 10*B*a**4*b*d**5*f*g**3 + 20*B*a**3*b**2*d**5*f**2*g*
*2 - 20*B*a**2*b**3*d**5*f**3*g + 10*B*a*b**4*d**5*f**4 + 2*B*b**5*c**5*g**4 - 10*B*b**5*c**4*d*f*g**3 + 20*B*
b**5*c**3*d**2*f**2*g**2 - 20*B*b**5*c**2*d**3*f**3*g + 10*B*b**5*c*d**4*f**4))/(5*b**5) - 2*B*c*(c**4*g**4 -
5*c**3*d*f*g**3 + 10*c**2*d**2*f**2*g**2 - 10*c*d**3*f**3*g + 5*d**4*f**4)*log(x + (2*B*a**5*c*d**4*g**4 - 10*
B*a**4*b*c*d**4*f*g**3 + 20*B*a**3*b**2*c*d**4*f**2*g**2 - 20*B*a**2*b**3*c*d**4*f**3*g + 2*B*a*b**4*c**5*g**4
- 10*B*a*b**4*c**4*d*f*g**3 + 20*B*a*b**4*c**3*d**2*f**2*g**2 - 20*B*a*b**4*c**2*d**3*f**3*g + 20*B*a*b**4*c*
d**4*f**4 - 2*B*a*b**4*c*(c**4*g**4 - 5*c**3*d*f*g**3 + 10*c**2*d**2*f**2*g**2 - 10*c*d**3*f**3*g + 5*d**4*f**
4) + 2*B*b**5*c**2*(c**4*g**4 - 5*c**3*d*f*g**3 + 10*c**2*d**2*f**2*g**2 - 10*c*d**3*f**3*g + 5*d**4*f**4)/d)/
(2*B*a**5*d**5*g**4 - 10*B*a**4*b*d**5*f*g**3 + 20*B*a**3*b**2*d**5*f**2*g**2 - 20*B*a**2*b**3*d**5*f**3*g + 1
0*B*a*b**4*d**5*f**4 + 2*B*b**5*c**5*g**4 - 10*B*b**5*c**4*d*f*g**3 + 20*B*b**5*c**3*d**2*f**2*g**2 - 20*B*b**
5*c**2*d**3*f**3*g + 10*B*b**5*c*d**4*f**4))/(5*d**5) + (B*f**4*x + 2*B*f**3*g*x**2 + 2*B*f**2*g**2*x**3 + B*f
*g**3*x**4 + B*g**4*x**5/5)*log(e*(a + b*x)**2/(c + d*x)**2) + x**4*(10*A*b*d*f*g**3 + B*a*d*g**4 - B*b*c*g**4
)/(10*b*d) - x**3*(-30*A*b**2*d**2*f**2*g**2 + 2*B*a**2*d**2*g**4 - 10*B*a*b*d**2*f*g**3 - 2*B*b**2*c**2*g**4
+ 10*B*b**2*c*d*f*g**3)/(15*b**2*d**2) + x**2*(10*A*b**3*d**3*f**3*g + B*a**3*d**3*g**4 - 5*B*a**2*b*d**3*f*g*
*3 + 10*B*a*b**2*d**3*f**2*g**2 - B*b**3*c**3*g**4 + 5*B*b**3*c**2*d*f*g**3 - 10*B*b**3*c*d**2*f**2*g**2)/(5*b
**3*d**3) - x*(-5*A*b**4*d**4*f**4 + 2*B*a**4*d**4*g**4 - 10*B*a**3*b*d**4*f*g**3 + 20*B*a**2*b**2*d**4*f**2*g
**2 - 20*B*a*b**3*d**4*f**3*g - 2*B*b**4*c**4*g**4 + 10*B*b**4*c**3*d*f*g**3 - 20*B*b**4*c**2*d**2*f**2*g**2 +
20*B*b**4*c*d**3*f**3*g)/(5*b**4*d**4)

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Giac [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((g*x+f)^4*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="giac")

[Out]

Timed out