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

Optimal. Leaf size=175 \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{3 b g^4 (a+b x)^3}+\frac{B d^2}{3 b g^4 (a+b x) (b c-a d)^2}+\frac{B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}-\frac{B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}-\frac{B d}{6 b g^4 (a+b x)^2 (b c-a d)}+\frac{B}{9 b g^4 (a+b x)^3} \]

[Out]

B/(9*b*g^4*(a + b*x)^3) - (B*d)/(6*b*(b*c - a*d)*g^4*(a + b*x)^2) + (B*d^2)/(3*b*(b*c - a*d)^2*g^4*(a + b*x))
+ (B*d^3*Log[a + b*x])/(3*b*(b*c - a*d)^3*g^4) - (B*d^3*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*
(c + d*x))/(a + b*x)])/(3*b*g^4*(a + b*x)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.131136, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 44} \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{3 b g^4 (a+b x)^3}+\frac{B d^2}{3 b g^4 (a+b x) (b c-a d)^2}+\frac{B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}-\frac{B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}-\frac{B d}{6 b g^4 (a+b x)^2 (b c-a d)}+\frac{B}{9 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

B/(9*b*g^4*(a + b*x)^3) - (B*d)/(6*b*(b*c - a*d)*g^4*(a + b*x)^2) + (B*d^2)/(3*b*(b*c - a*d)^2*g^4*(a + b*x))
+ (B*d^3*Log[a + b*x])/(3*b*(b*c - a*d)^3*g^4) - (B*d^3*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*
(c + d*x))/(a + b*x)])/(3*b*g^4*(a + b*x)^3)

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{(a g+b g x)^4} \, dx &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{3 b g^4 (a+b x)^3}+\frac{B \int \frac{-b c+a d}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{3 b g^4 (a+b x)^3}-\frac{(B (b c-a d)) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{3 b g^4 (a+b x)^3}-\frac{(B (b c-a d)) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^4}\\ &=\frac{B}{9 b g^4 (a+b x)^3}-\frac{B d}{6 b (b c-a d) g^4 (a+b x)^2}+\frac{B d^2}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac{B d^3 \log (a+b x)}{3 b (b c-a d)^3 g^4}-\frac{B d^3 \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{3 b g^4 (a+b x)^3}\\ \end{align*}

Mathematica [A]  time = 0.157511, size = 141, normalized size = 0.81 \[ \frac{\frac{B \left ((b c-a d) \left (11 a^2 d^2+a b d (15 d x-7 c)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )-6 d^3 (a+b x)^3 \log (c+d x)+6 d^3 (a+b x)^3 \log (a+b x)\right )}{(b c-a d)^3}-6 \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{18 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((B*((b*c - a*d)*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*x^2)) + 6*d^3*(a + b*x)^3*
Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]))/(b*c - a*d)^3 - 6*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(18*b*
g^4*(a + b*x)^3)

________________________________________________________________________________________

Maple [B]  time = 0.056, size = 1012, normalized size = 5.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/b/(a*d-b*c)^4/g^4*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(b*x+a)^3*a^4*d^4-2*b/(a*d-b*c)^4/g^4*B*ln(d*e/b-e*(a
*d-b*c)/b/(b*x+a))/(b*x+a)^3*a^2*d^2*c^2-1/3/(a*d-b*c)^4/g^4*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^3*c+1/9*b^3/(
a*d-b*c)^4/g^4*B/(b*x+a)^3*c^4-1/3*b^3/(a*d-b*c)^4/g^4*A/(b*x+a)^3*c^4-1/2/(a*d-b*c)^4/g^4*B*d^3/(b*x+a)^2*a^2
*c+4/3*b^2/(a*d-b*c)^4/g^4*A/(b*x+a)^3*a*d*c^3-4/9*b^2/(a*d-b*c)^4/g^4*B/(b*x+a)^3*c^3*a*d-4/9/(a*d-b*c)^4/g^4
*B/(b*x+a)^3*a^3*d^3*c-2/3/(a*d-b*c)^4/g^4*B*d^3/(b*x+a)*c*a+4/3*b^2/(a*d-b*c)^4/g^4*B*ln(d*e/b-e*(a*d-b*c)/b/
(b*x+a))/(b*x+a)^3*a*d*c^3+4/3/(a*d-b*c)^4/g^4*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(b*x+a)^3*a^3*d^3*c+1/2*b/(a*
d-b*c)^4/g^4*B*d^2/(b*x+a)^2*c^2*a-2*b/(a*d-b*c)^4/g^4*A/(b*x+a)^3*a^2*d^2*c^2+2/3*b/(a*d-b*c)^4/g^4*B/(b*x+a)
^3*a^2*d^2*c^2-11/18/b/(a*d-b*c)^4/g^4*B*d^4*a+1/3/b/(a*d-b*c)^4/g^4*A*d^4*a+4/3/(a*d-b*c)^4/g^4*A/(b*x+a)^3*a
^3*d^3*c-1/3/(a*d-b*c)^4/g^4*A*d^3*c+1/6/b/(a*d-b*c)^4/g^4*B*d^4/(b*x+a)^2*a^3-1/6*b^2/(a*d-b*c)^4/g^4*B*d/(b*
x+a)^2*c^3+1/3/b/(a*d-b*c)^4/g^4*B*d^4/(b*x+a)*a^2+1/3/b/(a*d-b*c)^4/g^4*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^4
*a-1/3*b^3/(a*d-b*c)^4/g^4*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(b*x+a)^3*c^4-1/3/b/(a*d-b*c)^4/g^4*A/(b*x+a)^3*a
^4*d^4+11/18/(a*d-b*c)^4/g^4*B*d^3*c+1/9/b/(a*d-b*c)^4/g^4*B/(b*x+a)^3*a^4*d^4+1/3*b/(a*d-b*c)^4/g^4*B*d^2/(b*
x+a)*c^2

________________________________________________________________________________________

Maxima [B]  time = 1.26679, size = 578, normalized size = 3.3 \begin{align*} \frac{1}{18} \, B{\left (\frac{6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \,{\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x +{\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} - \frac{6 \, \log \left (\frac{d e x}{b x + a} + \frac{c e}{b x + a}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac{6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac{6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac{A}{3 \,{\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c
*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c
*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) - 6*log(d*e*x/(b*x + a) + c*e/(b*x +
a))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2
*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*
b*d^3)*g^4)) - 1/3*A/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

________________________________________________________________________________________

Fricas [B]  time = 1.02002, size = 826, normalized size = 4.72 \begin{align*} -\frac{2 \,{\left (3 \, A - B\right )} b^{3} c^{3} - 9 \,{\left (2 \, A - B\right )} a b^{2} c^{2} d + 18 \,{\left (A - B\right )} a^{2} b c d^{2} -{\left (6 \, A - 11 \, B\right )} a^{3} d^{3} - 6 \,{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} + 3 \,{\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} x + 6 \,{\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} \log \left (\frac{d e x + c e}{b x + a}\right )}{18 \,{\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x +{\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(2*(3*A - B)*b^3*c^3 - 9*(2*A - B)*a*b^2*c^2*d + 18*(A - B)*a^2*b*c*d^2 - (6*A - 11*B)*a^3*d^3 - 6*(B*b^
3*c*d^2 - B*a*b^2*d^3)*x^2 + 3*(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b*d^3)*x + 6*(B*b^3*d^3*x^3 + 3*B*a*b^
2*d^3*x^2 + 3*B*a^2*b*d^3*x + B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*log((d*e*x + c*e)/(b*x + a)))/((b
^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c
*d^2 - a^4*b^3*d^3)*g^4*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + (a^3*b
^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)

________________________________________________________________________________________

Sympy [B]  time = 5.22966, size = 656, normalized size = 3.75 \begin{align*} - \frac{B \log{\left (\frac{e \left (c + d x\right )}{a + b x} \right )}}{3 a^{3} b g^{4} + 9 a^{2} b^{2} g^{4} x + 9 a b^{3} g^{4} x^{2} + 3 b^{4} g^{4} x^{3}} + \frac{B d^{3} \log{\left (x + \frac{- \frac{B a^{4} d^{7}}{\left (a d - b c\right )^{3}} + \frac{4 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} - \frac{6 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} + \frac{4 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + B a d^{4} - \frac{B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + B b c d^{3}}{2 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} - \frac{B d^{3} \log{\left (x + \frac{\frac{B a^{4} d^{7}}{\left (a d - b c\right )^{3}} - \frac{4 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} + \frac{6 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac{4 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + B a d^{4} + \frac{B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + B b c d^{3}}{2 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac{- 6 A a^{2} d^{2} + 12 A a b c d - 6 A b^{2} c^{2} + 11 B a^{2} d^{2} - 7 B a b c d + 2 B b^{2} c^{2} + 6 B b^{2} d^{2} x^{2} + x \left (15 B a b d^{2} - 3 B b^{2} c d\right )}{18 a^{5} b d^{2} g^{4} - 36 a^{4} b^{2} c d g^{4} + 18 a^{3} b^{3} c^{2} g^{4} + x^{3} \left (18 a^{2} b^{4} d^{2} g^{4} - 36 a b^{5} c d g^{4} + 18 b^{6} c^{2} g^{4}\right ) + x^{2} \left (54 a^{3} b^{3} d^{2} g^{4} - 108 a^{2} b^{4} c d g^{4} + 54 a b^{5} c^{2} g^{4}\right ) + x \left (54 a^{4} b^{2} d^{2} g^{4} - 108 a^{3} b^{3} c d g^{4} + 54 a^{2} b^{4} c^{2} g^{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*log(e*(c + d*x)/(a + b*x))/(3*a**3*b*g**4 + 9*a**2*b**2*g**4*x + 9*a*b**3*g**4*x**2 + 3*b**4*g**4*x**3) + B
*d**3*log(x + (-B*a**4*d**7/(a*d - b*c)**3 + 4*B*a**3*b*c*d**6/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**5/(a*d -
b*c)**3 + 4*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + B*a*d**4 - B*b**4*c**4*d**3/(a*d - b*c)**3 + B*b*c*d**3)/(2*B
*b*d**4))/(3*b*g**4*(a*d - b*c)**3) - B*d**3*log(x + (B*a**4*d**7/(a*d - b*c)**3 - 4*B*a**3*b*c*d**6/(a*d - b*
c)**3 + 6*B*a**2*b**2*c**2*d**5/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + B*a*d**4 + B*b**4*c**4*
d**3/(a*d - b*c)**3 + B*b*c*d**3)/(2*B*b*d**4))/(3*b*g**4*(a*d - b*c)**3) + (-6*A*a**2*d**2 + 12*A*a*b*c*d - 6
*A*b**2*c**2 + 11*B*a**2*d**2 - 7*B*a*b*c*d + 2*B*b**2*c**2 + 6*B*b**2*d**2*x**2 + x*(15*B*a*b*d**2 - 3*B*b**2
*c*d))/(18*a**5*b*d**2*g**4 - 36*a**4*b**2*c*d*g**4 + 18*a**3*b**3*c**2*g**4 + x**3*(18*a**2*b**4*d**2*g**4 -
36*a*b**5*c*d*g**4 + 18*b**6*c**2*g**4) + x**2*(54*a**3*b**3*d**2*g**4 - 108*a**2*b**4*c*d*g**4 + 54*a*b**5*c*
*2*g**4) + x*(54*a**4*b**2*d**2*g**4 - 108*a**3*b**3*c*d*g**4 + 54*a**2*b**4*c**2*g**4))

________________________________________________________________________________________

Giac [B]  time = 1.39764, size = 609, normalized size = 3.48 \begin{align*} \frac{B d^{3} \log \left (b x + a\right )}{3 \,{\left (b^{4} c^{3} g^{4} - 3 \, a b^{3} c^{2} d g^{4} + 3 \, a^{2} b^{2} c d^{2} g^{4} - a^{3} b d^{3} g^{4}\right )}} - \frac{B d^{3} \log \left (d x + c\right )}{3 \,{\left (b^{4} c^{3} g^{4} - 3 \, a b^{3} c^{2} d g^{4} + 3 \, a^{2} b^{2} c d^{2} g^{4} - a^{3} b d^{3} g^{4}\right )}} - \frac{B \log \left (\frac{d x + c}{b x + a}\right )}{3 \,{\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} + \frac{6 \, B b^{2} d^{2} x^{2} - 3 \, B b^{2} c d x + 15 \, B a b d^{2} x - 6 \, A b^{2} c^{2} - 4 \, B b^{2} c^{2} + 12 \, A a b c d + 5 \, B a b c d - 6 \, A a^{2} d^{2} + 5 \, B a^{2} d^{2}}{18 \,{\left (b^{6} c^{2} g^{4} x^{3} - 2 \, a b^{5} c d g^{4} x^{3} + a^{2} b^{4} d^{2} g^{4} x^{3} + 3 \, a b^{5} c^{2} g^{4} x^{2} - 6 \, a^{2} b^{4} c d g^{4} x^{2} + 3 \, a^{3} b^{3} d^{2} g^{4} x^{2} + 3 \, a^{2} b^{4} c^{2} g^{4} x - 6 \, a^{3} b^{3} c d g^{4} x + 3 \, a^{4} b^{2} d^{2} g^{4} x + a^{3} b^{3} c^{2} g^{4} - 2 \, a^{4} b^{2} c d g^{4} + a^{5} b d^{2} g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*d^3*log(b*x + a)/(b^4*c^3*g^4 - 3*a*b^3*c^2*d*g^4 + 3*a^2*b^2*c*d^2*g^4 - a^3*b*d^3*g^4) - 1/3*B*d^3*log
(d*x + c)/(b^4*c^3*g^4 - 3*a*b^3*c^2*d*g^4 + 3*a^2*b^2*c*d^2*g^4 - a^3*b*d^3*g^4) - 1/3*B*log((d*x + c)/(b*x +
a))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) + 1/18*(6*B*b^2*d^2*x^2 - 3*B*b^2*c*d*x + 1
5*B*a*b*d^2*x - 6*A*b^2*c^2 - 4*B*b^2*c^2 + 12*A*a*b*c*d + 5*B*a*b*c*d - 6*A*a^2*d^2 + 5*B*a^2*d^2)/(b^6*c^2*g
^4*x^3 - 2*a*b^5*c*d*g^4*x^3 + a^2*b^4*d^2*g^4*x^3 + 3*a*b^5*c^2*g^4*x^2 - 6*a^2*b^4*c*d*g^4*x^2 + 3*a^3*b^3*d
^2*g^4*x^2 + 3*a^2*b^4*c^2*g^4*x - 6*a^3*b^3*c*d*g^4*x + 3*a^4*b^2*d^2*g^4*x + a^3*b^3*c^2*g^4 - 2*a^4*b^2*c*d
*g^4 + a^5*b*d^2*g^4)