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

Optimal. Leaf size=206 \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{4 b g^5 (a+b x)^4}-\frac{B d^3}{4 b g^5 (a+b x) (b c-a d)^3}+\frac{B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}-\frac{B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}+\frac{B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}-\frac{B d}{12 b g^5 (a+b x)^3 (b c-a d)}+\frac{B}{16 b g^5 (a+b x)^4} \]

[Out]

B/(16*b*g^5*(a + b*x)^4) - (B*d)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) + (B*d^2)/(8*b*(b*c - a*d)^2*g^5*(a + b*x)
^2) - (B*d^3)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) - (B*d^4*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) + (B*d^4*Log[c
+ d*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(c + d*x))/(a + b*x)])/(4*b*g^5*(a + b*x)^4)

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Rubi [A]  time = 0.14864, antiderivative size = 206, 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} \[ -\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{4 b g^5 (a+b x)^4}-\frac{B d^3}{4 b g^5 (a+b x) (b c-a d)^3}+\frac{B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}-\frac{B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}+\frac{B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}-\frac{B d}{12 b g^5 (a+b x)^3 (b c-a d)}+\frac{B}{16 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

B/(16*b*g^5*(a + b*x)^4) - (B*d)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) + (B*d^2)/(8*b*(b*c - a*d)^2*g^5*(a + b*x)
^2) - (B*d^3)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) - (B*d^4*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) + (B*d^4*Log[c
+ d*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(c + d*x))/(a + b*x)])/(4*b*g^5*(a + b*x)^4)

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)^5} \, dx &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}+\frac{B \int \frac{-b c+a d}{g^4 (a+b x)^5 (c+d x)} \, dx}{4 b g}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}-\frac{(B (b c-a d)) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{4 b g^5}\\ &=-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}-\frac{(B (b c-a d)) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b g^5}\\ &=\frac{B}{16 b g^5 (a+b x)^4}-\frac{B d}{12 b (b c-a d) g^5 (a+b x)^3}+\frac{B d^2}{8 b (b c-a d)^2 g^5 (a+b x)^2}-\frac{B d^3}{4 b (b c-a d)^3 g^5 (a+b x)}-\frac{B d^4 \log (a+b x)}{4 b (b c-a d)^4 g^5}+\frac{B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5}-\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.202056, size = 166, normalized size = 0.81 \[ \frac{\frac{B (a d-b c) \left (\frac{12 d^3 (b c-a d)}{a+b x}-\frac{6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac{4 d (b c-a d)^3}{(a+b x)^3}-\frac{3 (b c-a d)^4}{(a+b x)^4}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^5}-\frac{B \log \left (\frac{e (c+d x)}{a+b x}\right )+A}{(a+b x)^4}}{4 b g^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.053, size = 1306, normalized size = 6.3 \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)^5,x)

[Out]

-1/4/(a*d-b*c)^5/g^5*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^4*c+1/4*b^4/(a*d-b*c)^5/g^5*A/(b*x+a)^4*c^5-1/16*b^4/
(a*d-b*c)^5/g^5*B/(b*x+a)^4*c^5-5/4*b^3/(a*d-b*c)^5/g^5*A/(b*x+a)^4*a*d*c^4-5/2*b/(a*d-b*c)^5/g^5*A/(b*x+a)^4*
a^3*d^3*c^2+5/8*b/(a*d-b*c)^5/g^5*B/(b*x+a)^4*a^3*d^3*c^2+5/4/(a*d-b*c)^5/g^5*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a)
)/(b*x+a)^4*a^4*d^4*c-1/4/b/(a*d-b*c)^5/g^5*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(b*x+a)^4*a^5*d^5+5/2*b^2/(a*d-b
*c)^5/g^5*A/(b*x+a)^4*a^2*d^2*c^3-5/8*b^2/(a*d-b*c)^5/g^5*B/(b*x+a)^4*a^2*d^2*c^3+5/16*b^3/(a*d-b*c)^5/g^5*B/(
b*x+a)^4*a*d*c^4+1/2*b/(a*d-b*c)^5/g^5*B*d^3/(b*x+a)^3*a^2*c^2-1/3*b^2/(a*d-b*c)^5/g^5*B*d^2/(b*x+a)^3*a*c^3+3
/8*b/(a*d-b*c)^5/g^5*B*d^3/(b*x+a)^2*a*c^2-1/4/(a*d-b*c)^5/g^5*A*d^4*c-25/48/b/(a*d-b*c)^5/g^5*B*d^5*a+25/48/(
a*d-b*c)^5/g^5*B*d^4*c+1/4/b/(a*d-b*c)^5/g^5*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))*d^5*a+1/4*b^4/(a*d-b*c)^5/g^5*B
*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(b*x+a)^4*c^5+5/2*b^2/(a*d-b*c)^5/g^5*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(b*x+
a)^4*a^2*d^2*c^3-3/8/(a*d-b*c)^5/g^5*B*d^4/(b*x+a)^2*a^2*c-1/2/(a*d-b*c)^5/g^5*B*d^4/(b*x+a)*a*c+1/4/b/(a*d-b*
c)^5/g^5*B*d^5/(b*x+a)*a^2+1/4*b/(a*d-b*c)^5/g^5*B*d^3/(b*x+a)*c^2+1/8/b/(a*d-b*c)^5/g^5*B*d^5/(b*x+a)^2*a^3-1
/8*b^2/(a*d-b*c)^5/g^5*B*d^2/(b*x+a)^2*c^3+1/12/b/(a*d-b*c)^5/g^5*B*d^5/(b*x+a)^3*a^4+1/12*b^3/(a*d-b*c)^5/g^5
*B*d/(b*x+a)^3*c^4+5/4/(a*d-b*c)^5/g^5*A/(b*x+a)^4*a^4*d^4*c-1/4/b/(a*d-b*c)^5/g^5*A/(b*x+a)^4*a^5*d^5+1/4/b/(
a*d-b*c)^5/g^5*A*d^5*a-5/4*b^3/(a*d-b*c)^5/g^5*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(b*x+a)^4*a*d*c^4-5/2*b/(a*d-
b*c)^5/g^5*B*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))/(b*x+a)^4*a^3*d^3*c^2+1/16/b/(a*d-b*c)^5/g^5*B/(b*x+a)^4*a^5*d^5-
5/16/(a*d-b*c)^5/g^5*B/(b*x+a)^4*a^4*d^4*c-1/3/(a*d-b*c)^5/g^5*B*d^4/(b*x+a)^3*a^3*c

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Maxima [B]  time = 1.1692, size = 873, normalized size = 4.24 \begin{align*} -\frac{1}{48} \, B{\left (\frac{12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \,{\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x +{\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} + \frac{12 \, \log \left (\frac{d e x}{b x + a} + \frac{c e}{b x + a}\right )}{b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac{12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac{12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac{A}{4 \,{\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\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)^5,x, algorithm="maxima")

[Out]

-1/48*B*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d
^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^
5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*
a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 -
a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*log(d*e*x/(b*x +
a) + c*e/(b*x + a))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) + 12*d^4
*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d
*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/4*A/(b^5*g^5*x^
4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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Fricas [B]  time = 1.10845, size = 1284, normalized size = 6.23 \begin{align*} -\frac{3 \,{\left (4 \, A - B\right )} b^{4} c^{4} - 16 \,{\left (3 \, A - B\right )} a b^{3} c^{3} d + 36 \,{\left (2 \, A - B\right )} a^{2} b^{2} c^{2} d^{2} - 48 \,{\left (A - B\right )} a^{3} b c d^{3} +{\left (12 \, A - 25 \, B\right )} a^{4} d^{4} + 12 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} x^{3} - 6 \,{\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} x^{2} + 4 \,{\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 12 \,{\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\right )} \log \left (\frac{d e x + c e}{b x + a}\right )}{48 \,{\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x +{\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\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)^5,x, algorithm="fricas")

[Out]

-1/48*(3*(4*A - B)*b^4*c^4 - 16*(3*A - B)*a*b^3*c^3*d + 36*(2*A - B)*a^2*b^2*c^2*d^2 - 48*(A - B)*a^3*b*c*d^3
+ (12*A - 25*B)*a^4*d^4 + 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*x^3 - 6*(B*b^4*c^2*d^2 - 8*B*a*b^3*c*d^3 + 7*B*a^2*b^
2*d^4)*x^2 + 4*(B*b^4*c^3*d - 6*B*a*b^3*c^2*d^2 + 18*B*a^2*b^2*c*d^3 - 13*B*a^3*b*d^4)*x - 12*(B*b^4*d^4*x^4 +
4*B*a*b^3*d^4*x^3 + 6*B*a^2*b^2*d^4*x^2 + 4*B*a^3*b*d^4*x - B*b^4*c^4 + 4*B*a*b^3*c^3*d - 6*B*a^2*b^2*c^2*d^2
+ 4*B*a^3*b*c*d^3)*log((d*e*x + c*e)/(b*x + a)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*
d^3 + a^4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^
4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4
*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*g^5*x + (a^4*b^5*c^4 - 4*
a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g^5)

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Sympy [B]  time = 7.93603, size = 944, normalized size = 4.58 \begin{align*} \text{result too large to display} \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)**5,x)

[Out]

-B*log(e*(c + d*x)/(a + b*x))/(4*a**4*b*g**5 + 16*a**3*b**2*g**5*x + 24*a**2*b**3*g**5*x**2 + 16*a*b**4*g**5*x
**3 + 4*b**5*g**5*x**4) + B*d**4*log(x + (-B*a**5*d**9/(a*d - b*c)**4 + 5*B*a**4*b*c*d**8/(a*d - b*c)**4 - 10*
B*a**3*b**2*c**2*d**7/(a*d - b*c)**4 + 10*B*a**2*b**3*c**3*d**6/(a*d - b*c)**4 - 5*B*a*b**4*c**4*d**5/(a*d - b
*c)**4 + B*a*d**5 + B*b**5*c**5*d**4/(a*d - b*c)**4 + B*b*c*d**4)/(2*B*b*d**5))/(4*b*g**5*(a*d - b*c)**4) - B*
d**4*log(x + (B*a**5*d**9/(a*d - b*c)**4 - 5*B*a**4*b*c*d**8/(a*d - b*c)**4 + 10*B*a**3*b**2*c**2*d**7/(a*d -
b*c)**4 - 10*B*a**2*b**3*c**3*d**6/(a*d - b*c)**4 + 5*B*a*b**4*c**4*d**5/(a*d - b*c)**4 + B*a*d**5 - B*b**5*c*
*5*d**4/(a*d - b*c)**4 + B*b*c*d**4)/(2*B*b*d**5))/(4*b*g**5*(a*d - b*c)**4) + (-12*A*a**3*d**3 + 36*A*a**2*b*
c*d**2 - 36*A*a*b**2*c**2*d + 12*A*b**3*c**3 + 25*B*a**3*d**3 - 23*B*a**2*b*c*d**2 + 13*B*a*b**2*c**2*d - 3*B*
b**3*c**3 + 12*B*b**3*d**3*x**3 + x**2*(42*B*a*b**2*d**3 - 6*B*b**3*c*d**2) + x*(52*B*a**2*b*d**3 - 20*B*a*b**
2*c*d**2 + 4*B*b**3*c**2*d))/(48*a**7*b*d**3*g**5 - 144*a**6*b**2*c*d**2*g**5 + 144*a**5*b**3*c**2*d*g**5 - 48
*a**4*b**4*c**3*g**5 + x**4*(48*a**3*b**5*d**3*g**5 - 144*a**2*b**6*c*d**2*g**5 + 144*a*b**7*c**2*d*g**5 - 48*
b**8*c**3*g**5) + x**3*(192*a**4*b**4*d**3*g**5 - 576*a**3*b**5*c*d**2*g**5 + 576*a**2*b**6*c**2*d*g**5 - 192*
a*b**7*c**3*g**5) + x**2*(288*a**5*b**3*d**3*g**5 - 864*a**4*b**4*c*d**2*g**5 + 864*a**3*b**5*c**2*d*g**5 - 28
8*a**2*b**6*c**3*g**5) + x*(192*a**6*b**2*d**3*g**5 - 576*a**5*b**3*c*d**2*g**5 + 576*a**4*b**4*c**2*d*g**5 -
192*a**3*b**5*c**3*g**5))

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Giac [B]  time = 1.35681, size = 967, normalized size = 4.69 \begin{align*} -\frac{B d^{4} \log \left (b x + a\right )}{4 \,{\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} + \frac{B d^{4} \log \left (d x + c\right )}{4 \,{\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} - \frac{B \log \left (\frac{d x + c}{b x + a}\right )}{4 \,{\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} - \frac{12 \, B b^{3} d^{3} x^{3} - 6 \, B b^{3} c d^{2} x^{2} + 42 \, B a b^{2} d^{3} x^{2} + 4 \, B b^{3} c^{2} d x - 20 \, B a b^{2} c d^{2} x + 52 \, B a^{2} b d^{3} x + 12 \, A b^{3} c^{3} + 9 \, B b^{3} c^{3} - 36 \, A a b^{2} c^{2} d - 23 \, B a b^{2} c^{2} d + 36 \, A a^{2} b c d^{2} + 13 \, B a^{2} b c d^{2} - 12 \, A a^{3} d^{3} + 13 \, B a^{3} d^{3}}{48 \,{\left (b^{8} c^{3} g^{5} x^{4} - 3 \, a b^{7} c^{2} d g^{5} x^{4} + 3 \, a^{2} b^{6} c d^{2} g^{5} x^{4} - a^{3} b^{5} d^{3} g^{5} x^{4} + 4 \, a b^{7} c^{3} g^{5} x^{3} - 12 \, a^{2} b^{6} c^{2} d g^{5} x^{3} + 12 \, a^{3} b^{5} c d^{2} g^{5} x^{3} - 4 \, a^{4} b^{4} d^{3} g^{5} x^{3} + 6 \, a^{2} b^{6} c^{3} g^{5} x^{2} - 18 \, a^{3} b^{5} c^{2} d g^{5} x^{2} + 18 \, a^{4} b^{4} c d^{2} g^{5} x^{2} - 6 \, a^{5} b^{3} d^{3} g^{5} x^{2} + 4 \, a^{3} b^{5} c^{3} g^{5} x - 12 \, a^{4} b^{4} c^{2} d g^{5} x + 12 \, a^{5} b^{3} c d^{2} g^{5} x - 4 \, a^{6} b^{2} d^{3} g^{5} x + a^{4} b^{4} c^{3} g^{5} - 3 \, a^{5} b^{3} c^{2} d g^{5} + 3 \, a^{6} b^{2} c d^{2} g^{5} - a^{7} b d^{3} g^{5}\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)^5,x, algorithm="giac")

[Out]

-1/4*B*d^4*log(b*x + a)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b
*d^4*g^5) + 1/4*B*d^4*log(d*x + c)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4*a^3*b^2*c*d^3*
g^5 + a^4*b*d^4*g^5) - 1/4*B*log((d*x + c)/(b*x + a))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a
^3*b^2*g^5*x + a^4*b*g^5) - 1/48*(12*B*b^3*d^3*x^3 - 6*B*b^3*c*d^2*x^2 + 42*B*a*b^2*d^3*x^2 + 4*B*b^3*c^2*d*x
- 20*B*a*b^2*c*d^2*x + 52*B*a^2*b*d^3*x + 12*A*b^3*c^3 + 9*B*b^3*c^3 - 36*A*a*b^2*c^2*d - 23*B*a*b^2*c^2*d + 3
6*A*a^2*b*c*d^2 + 13*B*a^2*b*c*d^2 - 12*A*a^3*d^3 + 13*B*a^3*d^3)/(b^8*c^3*g^5*x^4 - 3*a*b^7*c^2*d*g^5*x^4 + 3
*a^2*b^6*c*d^2*g^5*x^4 - a^3*b^5*d^3*g^5*x^4 + 4*a*b^7*c^3*g^5*x^3 - 12*a^2*b^6*c^2*d*g^5*x^3 + 12*a^3*b^5*c*d
^2*g^5*x^3 - 4*a^4*b^4*d^3*g^5*x^3 + 6*a^2*b^6*c^3*g^5*x^2 - 18*a^3*b^5*c^2*d*g^5*x^2 + 18*a^4*b^4*c*d^2*g^5*x
^2 - 6*a^5*b^3*d^3*g^5*x^2 + 4*a^3*b^5*c^3*g^5*x - 12*a^4*b^4*c^2*d*g^5*x + 12*a^5*b^3*c*d^2*g^5*x - 4*a^6*b^2
*d^3*g^5*x + a^4*b^4*c^3*g^5 - 3*a^5*b^3*c^2*d*g^5 + 3*a^6*b^2*c*d^2*g^5 - a^7*b*d^3*g^5)