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

Optimal. Leaf size=269 \[ -\frac{b^2 i (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^3}-\frac{d^2 i (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^3}+\frac{2 b d i (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^3}-\frac{b^2 B i (c+d x)^4}{16 g^5 (a+b x)^4 (b c-a d)^3}-\frac{B d^2 i (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^3}+\frac{2 b B d i (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^3} \]

[Out]

-(B*d^2*i*(c + d*x)^2)/(4*(b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*B*d*i*(c + d*x)^3)/(9*(b*c - a*d)^3*g^5*(a + b
*x)^3) - (b^2*B*i*(c + d*x)^4)/(16*(b*c - a*d)^3*g^5*(a + b*x)^4) - (d^2*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x)
)/(c + d*x)]))/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (2*b*d*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/
(3*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*i*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^3*
g^5*(a + b*x)^4)

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Rubi [A]  time = 0.389584, antiderivative size = 257, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2528, 2525, 12, 44} \[ -\frac{d i \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^5 (a+b x)^3}-\frac{i (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^2 g^5 (a+b x)^4}-\frac{B d^3 i}{12 b^2 g^5 (a+b x) (b c-a d)^2}+\frac{B d^2 i}{24 b^2 g^5 (a+b x)^2 (b c-a d)}-\frac{B d^4 i \log (a+b x)}{12 b^2 g^5 (b c-a d)^3}+\frac{B d^4 i \log (c+d x)}{12 b^2 g^5 (b c-a d)^3}-\frac{B i (b c-a d)}{16 b^2 g^5 (a+b x)^4}-\frac{B d i}{36 b^2 g^5 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)*i)/(16*b^2*g^5*(a + b*x)^4) - (B*d*i)/(36*b^2*g^5*(a + b*x)^3) + (B*d^2*i)/(24*b^2*(b*c - a*d)
*g^5*(a + b*x)^2) - (B*d^3*i)/(12*b^2*(b*c - a*d)^2*g^5*(a + b*x)) - (B*d^4*i*Log[a + b*x])/(12*b^2*(b*c - a*d
)^3*g^5) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*b^2*g^5*(a + b*x)^4) - (d*i*(A + B*Log[(e*(
a + b*x))/(c + d*x)]))/(3*b^2*g^5*(a + b*x)^3) + (B*d^4*i*Log[c + d*x])/(12*b^2*(b*c - a*d)^3*g^5)

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

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{(9 c+9 d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^5} \, dx &=\int \left (\frac{9 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g^5 (a+b x)^5}+\frac{9 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b g^5 (a+b x)^4}\right ) \, dx\\ &=\frac{(9 d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b g^5}+\frac{(9 (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b g^5}\\ &=-\frac{9 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac{3 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac{(3 B d) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}+\frac{(9 B (b c-a d)) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^2 g^5}\\ &=-\frac{9 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac{3 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac{(3 B d (b c-a d)) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^2 g^5}+\frac{\left (9 B (b c-a d)^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{4 b^2 g^5}\\ &=-\frac{9 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac{3 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac{(3 B d (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}{b^2 g^5}+\frac{\left (9 B (b c-a d)^2\right ) \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^2 g^5}\\ &=-\frac{9 B (b c-a d)}{16 b^2 g^5 (a+b x)^4}-\frac{B d}{4 b^2 g^5 (a+b x)^3}+\frac{3 B d^2}{8 b^2 (b c-a d) g^5 (a+b x)^2}-\frac{3 B d^3}{4 b^2 (b c-a d)^2 g^5 (a+b x)}-\frac{3 B d^4 \log (a+b x)}{4 b^2 (b c-a d)^3 g^5}-\frac{9 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^2 g^5 (a+b x)^4}-\frac{3 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^2 g^5 (a+b x)^3}+\frac{3 B d^4 \log (c+d x)}{4 b^2 (b c-a d)^3 g^5}\\ \end{align*}

Mathematica [A]  time = 0.458856, size = 210, normalized size = 0.78 \[ -\frac{i \left (\frac{36 A b c}{(a+b x)^4}+\frac{48 A d}{(a+b x)^3}-\frac{36 a A d}{(a+b x)^4}+\frac{12 B d^3}{(a+b x) (b c-a d)^2}-\frac{6 B d^2}{(a+b x)^2 (b c-a d)}+\frac{12 B d^4 \log (a+b x)}{(b c-a d)^3}-\frac{12 B d^4 \log (c+d x)}{(b c-a d)^3}+\frac{12 B (a d+3 b c+4 b d x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4}+\frac{9 b B c}{(a+b x)^4}+\frac{4 B d}{(a+b x)^3}-\frac{9 a B d}{(a+b x)^4}\right )}{144 b^2 g^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(i*((36*A*b*c)/(a + b*x)^4 + (9*b*B*c)/(a + b*x)^4 - (36*a*A*d)/(a + b*x)^4 - (9*a*B*d)/(a + b*x)^4 + (48*A*d
)/(a + b*x)^3 + (4*B*d)/(a + b*x)^3 - (6*B*d^2)/((b*c - a*d)*(a + b*x)^2) + (12*B*d^3)/((b*c - a*d)^2*(a + b*x
)) + (12*B*d^4*Log[a + b*x])/(b*c - a*d)^3 + (12*B*(3*b*c + a*d + 4*b*d*x)*Log[(e*(a + b*x))/(c + d*x)])/(a +
b*x)^4 - (12*B*d^4*Log[c + d*x])/(b*c - a*d)^3))/(144*b^2*g^5)

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Maple [B]  time = 0.051, size = 1226, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e^2*d^3*i/(a*d-b*c)^4/g^5*A/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a-1/2*e^2*d^2*i/(a*d-b*c)^4/g^5*A/(b*e/d
+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*b*c-2/3*e^3*d^2*i/(a*d-b*c)^4/g^5*A*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a+
2/3*e^3*d*i/(a*d-b*c)^4/g^5*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c+1/4*e^4*d*i/(a*d-b*c)^4/g^5*A*b^2/(b
*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/4*e^4*i/(a*d-b*c)^4/g^5*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c+
1/2*e^2*d^3*i/(a*d-b*c)^4/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-1/2*e^
2*d^2*i/(a*d-b*c)^4/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b*c+1/4*e^2*d^
3*i/(a*d-b*c)^4/g^5*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a-1/4*e^2*d^2*i/(a*d-b*c)^4/g^5*B/(b*e/d+e/(d*x+c)
*a-e/d/(d*x+c)*b*c)^2*b*c-2/3*e^3*d^2*i/(a*d-b*c)^4/g^5*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*
d-b*c)*e/d/(d*x+c))*a+2/3*e^3*d*i/(a*d-b*c)^4/g^5*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*
c)*e/d/(d*x+c))*c-2/9*e^3*d^2*i/(a*d-b*c)^4/g^5*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a+2/9*e^3*d*i/(a*d-b
*c)^4/g^5*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c+1/4*e^4*d*i/(a*d-b*c)^4/g^5*B*b^2/(b*e/d+e/(d*x+c)*a-e
/d/(d*x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-1/4*e^4*i/(a*d-b*c)^4/g^5*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*
x+c)*b*c)^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+1/16*e^4*d*i/(a*d-b*c)^4/g^5*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c
)*b*c)^4*a-1/16*e^4*i/(a*d-b*c)^4/g^5*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*c

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Maxima [B]  time = 1.80358, size = 1871, normalized size = 6.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*B*d*i*(12*(4*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g
^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) + (7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(
4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*
c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4
+ 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d
+ 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g
^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x +
a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^
4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) + 1/48*B*
c*i*((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(b*e*x/(d*x + c) +
a*e/(d*x + c))/(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)) - 1/12*(4*b*x + a)*A*d*
i/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/4*A*c*i/(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 = 1.08386, size = 1230, normalized size = 4.57 \begin{align*} -\frac{12 \,{\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} i 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 )} i x^{2} + 4 \,{\left ({\left (12 \, A + B\right )} b^{4} c^{3} d - 6 \,{\left (6 \, A + B\right )} a b^{3} c^{2} d^{2} + 18 \,{\left (2 \, A + B\right )} a^{2} b^{2} c d^{3} -{\left (12 \, A + 13 \, B\right )} a^{3} b d^{4}\right )} i x +{\left (9 \,{\left (4 \, A + B\right )} b^{4} c^{4} - 32 \,{\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} -{\left (12 \, A + 13 \, B\right )} a^{4} d^{4}\right )} i + 12 \,{\left (B b^{4} d^{4} i x^{4} + 4 \, B a b^{3} d^{4} i x^{3} + 6 \, B a^{2} b^{2} d^{4} i x^{2} + 4 \,{\left (B b^{4} c^{3} d - 3 \, B a b^{3} c^{2} d^{2} + 3 \, B a^{2} b^{2} c d^{3}\right )} i x +{\left (3 \, B b^{4} c^{4} - 8 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} i\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{144 \,{\left ({\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \,{\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \,{\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \,{\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x +{\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(12*(B*b^4*c*d^3 - B*a*b^3*d^4)*i*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)*i*x^2 + 4
*((12*A + B)*b^4*c^3*d - 6*(6*A + B)*a*b^3*c^2*d^2 + 18*(2*A + B)*a^2*b^2*c*d^3 - (12*A + 13*B)*a^3*b*d^4)*i*x
+ (9*(4*A + B)*b^4*c^4 - 32*(3*A + B)*a*b^3*c^3*d + 36*(2*A + B)*a^2*b^2*c^2*d^2 - (12*A + 13*B)*a^4*d^4)*i +
12*(B*b^4*d^4*i*x^4 + 4*B*a*b^3*d^4*i*x^3 + 6*B*a^2*b^2*d^4*i*x^2 + 4*(B*b^4*c^3*d - 3*B*a*b^3*c^2*d^2 + 3*B*
a^2*b^2*c*d^3)*i*x + (3*B*b^4*c^4 - 8*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2)*i)*log((b*e*x + a*e)/(d*x + c)))/((
b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*
c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(
a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*
b^3*c*d^2 - a^7*b^2*d^3)*g^5)

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Sympy [B]  time = 18.9915, size = 928, normalized size = 3.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*d**4*i*log(x + (-B*a**4*d**8*i/(a*d - b*c)**3 + 4*B*a**3*b*c*d**7*i/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**
6*i/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**5*i/(a*d - b*c)**3 + B*a*d**5*i - B*b**4*c**4*d**4*i/(a*d - b*c)**3 +
B*b*c*d**4*i)/(2*B*b*d**5*i))/(12*b**2*g**5*(a*d - b*c)**3) + B*d**4*i*log(x + (B*a**4*d**8*i/(a*d - b*c)**3 -
4*B*a**3*b*c*d**7*i/(a*d - b*c)**3 + 6*B*a**2*b**2*c**2*d**6*i/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**5*i/(a*d -
b*c)**3 + B*a*d**5*i + B*b**4*c**4*d**4*i/(a*d - b*c)**3 + B*b*c*d**4*i)/(2*B*b*d**5*i))/(12*b**2*g**5*(a*d -
b*c)**3) + (-B*a*d*i - 3*B*b*c*i - 4*B*b*d*i*x)*log(e*(a + b*x)/(c + d*x))/(12*a**4*b**2*g**5 + 48*a**3*b**3*
g**5*x + 72*a**2*b**4*g**5*x**2 + 48*a*b**5*g**5*x**3 + 12*b**6*g**5*x**4) - (12*A*a**3*d**3*i + 12*A*a**2*b*c
*d**2*i - 60*A*a*b**2*c**2*d*i + 36*A*b**3*c**3*i + 13*B*a**3*d**3*i + 13*B*a**2*b*c*d**2*i - 23*B*a*b**2*c**2
*d*i + 9*B*b**3*c**3*i + 12*B*b**3*d**3*i*x**3 + x**2*(42*B*a*b**2*d**3*i - 6*B*b**3*c*d**2*i) + x*(48*A*a**2*
b*d**3*i - 96*A*a*b**2*c*d**2*i + 48*A*b**3*c**2*d*i + 52*B*a**2*b*d**3*i - 20*B*a*b**2*c*d**2*i + 4*B*b**3*c*
*2*d*i))/(144*a**6*b**2*d**2*g**5 - 288*a**5*b**3*c*d*g**5 + 144*a**4*b**4*c**2*g**5 + x**4*(144*a**2*b**6*d**
2*g**5 - 288*a*b**7*c*d*g**5 + 144*b**8*c**2*g**5) + x**3*(576*a**3*b**5*d**2*g**5 - 1152*a**2*b**6*c*d*g**5 +
576*a*b**7*c**2*g**5) + x**2*(864*a**4*b**4*d**2*g**5 - 1728*a**3*b**5*c*d*g**5 + 864*a**2*b**6*c**2*g**5) +
x*(576*a**5*b**3*d**2*g**5 - 1152*a**4*b**4*c*d*g**5 + 576*a**3*b**5*c**2*g**5))

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Giac [B]  time = 1.40809, size = 905, normalized size = 3.36 \begin{align*} \frac{B d^{4} \log \left (b x + a\right )}{12 \,{\left (b^{5} c^{3} g^{5} i - 3 \, a b^{4} c^{2} d g^{5} i + 3 \, a^{2} b^{3} c d^{2} g^{5} i - a^{3} b^{2} d^{3} g^{5} i\right )}} - \frac{B d^{4} \log \left (d x + c\right )}{12 \,{\left (b^{5} c^{3} g^{5} i - 3 \, a b^{4} c^{2} d g^{5} i + 3 \, a^{2} b^{3} c d^{2} g^{5} i - a^{3} b^{2} d^{3} g^{5} i\right )}} - \frac{{\left (4 \, B b d i x + 3 \, B b c i + B a d i\right )} \log \left (\frac{b x + a}{d x + c}\right )}{12 \,{\left (b^{6} g^{5} x^{4} + 4 \, a b^{5} g^{5} x^{3} + 6 \, a^{2} b^{4} g^{5} x^{2} + 4 \, a^{3} b^{3} g^{5} x + a^{4} b^{2} 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} + 48 \, A b^{3} c^{2} d x + 52 \, B b^{3} c^{2} d x - 96 \, A a b^{2} c d^{2} x - 116 \, B a b^{2} c d^{2} x + 48 \, A a^{2} b d^{3} x + 100 \, B a^{2} b d^{3} x + 36 \, A b^{3} c^{3} + 45 \, B b^{3} c^{3} - 60 \, A a b^{2} c^{2} d - 83 \, B a b^{2} c^{2} d + 12 \, A a^{2} b c d^{2} + 25 \, B a^{2} b c d^{2} + 12 \, A a^{3} d^{3} + 25 \, B a^{3} d^{3}}{144 \,{\left (b^{8} c^{2} g^{5} i x^{4} - 2 \, a b^{7} c d g^{5} i x^{4} + a^{2} b^{6} d^{2} g^{5} i x^{4} + 4 \, a b^{7} c^{2} g^{5} i x^{3} - 8 \, a^{2} b^{6} c d g^{5} i x^{3} + 4 \, a^{3} b^{5} d^{2} g^{5} i x^{3} + 6 \, a^{2} b^{6} c^{2} g^{5} i x^{2} - 12 \, a^{3} b^{5} c d g^{5} i x^{2} + 6 \, a^{4} b^{4} d^{2} g^{5} i x^{2} + 4 \, a^{3} b^{5} c^{2} g^{5} i x - 8 \, a^{4} b^{4} c d g^{5} i x + 4 \, a^{5} b^{3} d^{2} g^{5} i x + a^{4} b^{4} c^{2} g^{5} i - 2 \, a^{5} b^{3} c d g^{5} i + a^{6} b^{2} d^{2} g^{5} i\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*B*d^4*log(b*x + a)/(b^5*c^3*g^5*i - 3*a*b^4*c^2*d*g^5*i + 3*a^2*b^3*c*d^2*g^5*i - a^3*b^2*d^3*g^5*i) - 1/
12*B*d^4*log(d*x + c)/(b^5*c^3*g^5*i - 3*a*b^4*c^2*d*g^5*i + 3*a^2*b^3*c*d^2*g^5*i - a^3*b^2*d^3*g^5*i) - 1/12
*(4*B*b*d*i*x + 3*B*b*c*i + B*a*d*i)*log((b*x + a)/(d*x + c))/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x
^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) + 1/144*(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 + 48*A*
b^3*c^2*d*x + 52*B*b^3*c^2*d*x - 96*A*a*b^2*c*d^2*x - 116*B*a*b^2*c*d^2*x + 48*A*a^2*b*d^3*x + 100*B*a^2*b*d^3
*x + 36*A*b^3*c^3 + 45*B*b^3*c^3 - 60*A*a*b^2*c^2*d - 83*B*a*b^2*c^2*d + 12*A*a^2*b*c*d^2 + 25*B*a^2*b*c*d^2 +
12*A*a^3*d^3 + 25*B*a^3*d^3)/(b^8*c^2*g^5*i*x^4 - 2*a*b^7*c*d*g^5*i*x^4 + a^2*b^6*d^2*g^5*i*x^4 + 4*a*b^7*c^2
*g^5*i*x^3 - 8*a^2*b^6*c*d*g^5*i*x^3 + 4*a^3*b^5*d^2*g^5*i*x^3 + 6*a^2*b^6*c^2*g^5*i*x^2 - 12*a^3*b^5*c*d*g^5*
i*x^2 + 6*a^4*b^4*d^2*g^5*i*x^2 + 4*a^3*b^5*c^2*g^5*i*x - 8*a^4*b^4*c*d*g^5*i*x + 4*a^5*b^3*d^2*g^5*i*x + a^4*
b^4*c^2*g^5*i - 2*a^5*b^3*c*d*g^5*i + a^6*b^2*d^2*g^5*i)