3.29 \(\int \frac{(c i+d i x)^3 (A+B \log (\frac{e (a+b x)}{c+d x}))}{(a g+b g x)^6} \, dx\)

Optimal. Leaf size=181 \[ -\frac{b i^3 (c+d x)^5 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^2}+\frac{d i^3 (c+d x)^4 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 g^6 (a+b x)^4 (b c-a d)^2}-\frac{b B i^3 (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^2}+\frac{B d i^3 (c+d x)^4}{16 g^6 (a+b x)^4 (b c-a d)^2} \]

[Out]

(B*d*i^3*(c + d*x)^4)/(16*(b*c - a*d)^2*g^6*(a + b*x)^4) - (b*B*i^3*(c + d*x)^5)/(25*(b*c - a*d)^2*g^6*(a + b*
x)^5) + (d*i^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(b*c - a*d)^2*g^6*(a + b*x)^4) - (b*i^3*(c
+ d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(5*(b*c - a*d)^2*g^6*(a + b*x)^5)

________________________________________________________________________________________

Rubi [B]  time = 0.865167, antiderivative size = 409, normalized size of antiderivative = 2.26, number of steps used = 18, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2528, 2525, 12, 44} \[ -\frac{d^3 i^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 b^4 g^6 (a+b x)^2}-\frac{d^2 i^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{b^4 g^6 (a+b x)^3}-\frac{3 d i^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{4 b^4 g^6 (a+b x)^4}-\frac{i^3 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{5 b^4 g^6 (a+b x)^5}+\frac{B d^4 i^3}{20 b^4 g^6 (a+b x) (b c-a d)}-\frac{3 B d^2 i^3 (b c-a d)}{20 b^4 g^6 (a+b x)^3}+\frac{B d^5 i^3 \log (a+b x)}{20 b^4 g^6 (b c-a d)^2}-\frac{B d^5 i^3 \log (c+d x)}{20 b^4 g^6 (b c-a d)^2}-\frac{11 B d i^3 (b c-a d)^2}{80 b^4 g^6 (a+b x)^4}-\frac{B i^3 (b c-a d)^3}{25 b^4 g^6 (a+b x)^5}-\frac{B d^3 i^3}{40 b^4 g^6 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^6,x]

[Out]

-(B*(b*c - a*d)^3*i^3)/(25*b^4*g^6*(a + b*x)^5) - (11*B*d*(b*c - a*d)^2*i^3)/(80*b^4*g^6*(a + b*x)^4) - (3*B*d
^2*(b*c - a*d)*i^3)/(20*b^4*g^6*(a + b*x)^3) - (B*d^3*i^3)/(40*b^4*g^6*(a + b*x)^2) + (B*d^4*i^3)/(20*b^4*(b*c
- a*d)*g^6*(a + b*x)) + (B*d^5*i^3*Log[a + b*x])/(20*b^4*(b*c - a*d)^2*g^6) - ((b*c - a*d)^3*i^3*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(5*b^4*g^6*(a + b*x)^5) - (3*d*(b*c - a*d)^2*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]
))/(4*b^4*g^6*(a + b*x)^4) - (d^2*(b*c - a*d)*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^4*g^6*(a + b*x)^3)
- (d^3*i^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^4*g^6*(a + b*x)^2) - (B*d^5*i^3*Log[c + d*x])/(20*b^4*(b
*c - a*d)^2*g^6)

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{(29 c+29 d x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^6} \, dx &=\int \left (\frac{24389 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^6}+\frac{73167 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^5}+\frac{73167 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^4}+\frac{24389 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^3 g^6 (a+b x)^3}\right ) \, dx\\ &=\frac{\left (24389 d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^3 g^6}+\frac{\left (73167 d^2 (b c-a d)\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^3 g^6}+\frac{\left (73167 d (b c-a d)^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^3 g^6}+\frac{\left (24389 (b c-a d)^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^6} \, dx}{b^3 g^6}\\ &=-\frac{24389 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac{73167 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac{24389 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac{24389 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac{\left (24389 B d^3\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^6}+\frac{\left (24389 B d^2 (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^6}+\frac{\left (73167 B d (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^4 g^6}+\frac{\left (24389 B (b c-a d)^3\right ) \int \frac{b c-a d}{(a+b x)^6 (c+d x)} \, dx}{5 b^4 g^6}\\ &=-\frac{24389 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac{73167 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac{24389 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac{24389 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac{\left (24389 B d^3 (b c-a d)\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2 b^4 g^6}+\frac{\left (24389 B d^2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{b^4 g^6}+\frac{\left (73167 B d (b c-a d)^3\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{4 b^4 g^6}+\frac{\left (24389 B (b c-a d)^4\right ) \int \frac{1}{(a+b x)^6 (c+d x)} \, dx}{5 b^4 g^6}\\ &=-\frac{24389 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac{73167 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac{24389 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac{24389 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}+\frac{\left (24389 B d^3 (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 b^4 g^6}+\frac{\left (24389 B d^2 (b c-a d)^2\right ) \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^4 g^6}+\frac{\left (73167 B d (b c-a d)^3\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^4 g^6}+\frac{\left (24389 B (b c-a d)^4\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^6}-\frac{b d}{(b c-a d)^2 (a+b x)^5}+\frac{b d^2}{(b c-a d)^3 (a+b x)^4}-\frac{b d^3}{(b c-a d)^4 (a+b x)^3}+\frac{b d^4}{(b c-a d)^5 (a+b x)^2}-\frac{b d^5}{(b c-a d)^6 (a+b x)}+\frac{d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{5 b^4 g^6}\\ &=-\frac{24389 B (b c-a d)^3}{25 b^4 g^6 (a+b x)^5}-\frac{268279 B d (b c-a d)^2}{80 b^4 g^6 (a+b x)^4}-\frac{73167 B d^2 (b c-a d)}{20 b^4 g^6 (a+b x)^3}-\frac{24389 B d^3}{40 b^4 g^6 (a+b x)^2}+\frac{24389 B d^4}{20 b^4 (b c-a d) g^6 (a+b x)}+\frac{24389 B d^5 \log (a+b x)}{20 b^4 (b c-a d)^2 g^6}-\frac{24389 (b c-a d)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{5 b^4 g^6 (a+b x)^5}-\frac{73167 d (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{4 b^4 g^6 (a+b x)^4}-\frac{24389 d^2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{b^4 g^6 (a+b x)^3}-\frac{24389 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2 b^4 g^6 (a+b x)^2}-\frac{24389 B d^5 \log (c+d x)}{20 b^4 (b c-a d)^2 g^6}\\ \end{align*}

Mathematica [B]  time = 0.621215, size = 608, normalized size = 3.36 \[ -\frac{i^3 \left (200 a^2 A b^3 d^5 x^3+200 a^3 A b^2 d^5 x^2+100 a^4 A b d^5 x+20 a^5 A d^5+20 B (b c-a d)^2 \left (a^2 b d^2 (2 c+5 d x)+a^3 d^3+a b^2 d \left (3 c^2+10 c d x+10 d^2 x^2\right )+b^3 \left (15 c^2 d x+4 c^3+20 c d^2 x^2+10 d^3 x^3\right )\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )+200 a^2 b^3 B d^5 x^3 \log (c+d x)+200 a^3 b^2 B d^5 x^2 \log (c+d x)+90 a^2 b^3 B d^5 x^3+90 a^3 b^2 B d^5 x^2+100 a^4 b B d^5 x \log (c+d x)+45 a^4 b B d^5 x+20 a^5 B d^5 \log (c+d x)+9 a^5 B d^5-600 a A b^4 c^2 d^3 x^2-400 a A b^4 c^3 d^2 x-100 a A b^4 c^4 d-400 a A b^4 c d^4 x^3-150 a b^4 B c^2 d^3 x^2-100 a b^4 B c^3 d^2 x-25 a b^4 B c^4 d-100 a b^4 B c d^4 x^3+100 a b^4 B d^5 x^4 \log (c+d x)+20 a b^4 B d^5 x^4-20 B d^5 (a+b x)^5 \log (a+b x)+200 A b^5 c^2 d^3 x^3+400 A b^5 c^3 d^2 x^2+300 A b^5 c^4 d x+80 A b^5 c^5+10 b^5 B c^2 d^3 x^3+60 b^5 B c^3 d^2 x^2+55 b^5 B c^4 d x+16 b^5 B c^5-20 b^5 B c d^4 x^4+20 b^5 B d^5 x^5 \log (c+d x)\right )}{400 b^4 g^6 (a+b x)^5 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^6,x]

[Out]

-(i^3*(80*A*b^5*c^5 + 16*b^5*B*c^5 - 100*a*A*b^4*c^4*d - 25*a*b^4*B*c^4*d + 20*a^5*A*d^5 + 9*a^5*B*d^5 + 300*A
*b^5*c^4*d*x + 55*b^5*B*c^4*d*x - 400*a*A*b^4*c^3*d^2*x - 100*a*b^4*B*c^3*d^2*x + 100*a^4*A*b*d^5*x + 45*a^4*b
*B*d^5*x + 400*A*b^5*c^3*d^2*x^2 + 60*b^5*B*c^3*d^2*x^2 - 600*a*A*b^4*c^2*d^3*x^2 - 150*a*b^4*B*c^2*d^3*x^2 +
200*a^3*A*b^2*d^5*x^2 + 90*a^3*b^2*B*d^5*x^2 + 200*A*b^5*c^2*d^3*x^3 + 10*b^5*B*c^2*d^3*x^3 - 400*a*A*b^4*c*d^
4*x^3 - 100*a*b^4*B*c*d^4*x^3 + 200*a^2*A*b^3*d^5*x^3 + 90*a^2*b^3*B*d^5*x^3 - 20*b^5*B*c*d^4*x^4 + 20*a*b^4*B
*d^5*x^4 - 20*B*d^5*(a + b*x)^5*Log[a + b*x] + 20*B*(b*c - a*d)^2*(a^3*d^3 + a^2*b*d^2*(2*c + 5*d*x) + a*b^2*d
*(3*c^2 + 10*c*d*x + 10*d^2*x^2) + b^3*(4*c^3 + 15*c^2*d*x + 20*c*d^2*x^2 + 10*d^3*x^3))*Log[(e*(a + b*x))/(c
+ d*x)] + 20*a^5*B*d^5*Log[c + d*x] + 100*a^4*b*B*d^5*x*Log[c + d*x] + 200*a^3*b^2*B*d^5*x^2*Log[c + d*x] + 20
0*a^2*b^3*B*d^5*x^3*Log[c + d*x] + 100*a*b^4*B*d^5*x^4*Log[c + d*x] + 20*b^5*B*d^5*x^5*Log[c + d*x]))/(400*b^4
*(b*c - a*d)^2*g^6*(a + b*x)^5)

________________________________________________________________________________________

Maple [B]  time = 0.049, size = 828, 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)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x)

[Out]

1/4*e^4*d^2*i^3/(a*d-b*c)^3/g^6*A/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/4*e^4*d*i^3/(a*d-b*c)^3/g^6*A/(b*e
/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c-1/5*e^5*d*i^3/(a*d-b*c)^3/g^6*A*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*
a+1/5*e^5*i^3/(a*d-b*c)^3/g^6*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*c+1/4*e^4*d^2*i^3/(a*d-b*c)^3/g^6*B/
(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*d*i^3/(a*d-b*c)^3/g^6*B/(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))*b*c+1/16*e^4*d^2*i^3/(a*d-b*c)^3/g^6*B/(b*e/d+
e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*a-1/16*e^4*d*i^3/(a*d-b*c)^3/g^6*B/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^4*b*c-1/
5*e^5*d*i^3/(a*d-b*c)^3/g^6*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+1/5*e^
5*i^3/(a*d-b*c)^3/g^6*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-1/25*e^5*d
*i^3/(a*d-b*c)^3/g^6*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^5*a+1/25*e^5*i^3/(a*d-b*c)^3/g^6*B*b^2/(b*e/d+e/(
d*x+c)*a-e/d/(d*x+c)*b*c)^5*c

________________________________________________________________________________________

Maxima [B]  time = 2.5098, size = 5694, normalized size = 31.46 \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)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="maxima")

[Out]

-1/1200*B*d^3*i^3*(60*(10*b^3*x^3 + 10*a*b^2*x^2 + 5*a^2*b*x + a^3)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^9*
g^6*x^5 + 5*a*b^8*g^6*x^4 + 10*a^2*b^7*g^6*x^3 + 10*a^3*b^6*g^6*x^2 + 5*a^4*b^5*g^6*x + a^5*b^4*g^6) + (77*a^3
*b^4*c^4 - 548*a^4*b^3*c^3*d + 352*a^5*b^2*c^2*d^2 - 148*a^6*b*c*d^3 + 27*a^7*d^4 - 60*(10*b^7*c^3*d - 10*a*b^
6*c^2*d^2 + 5*a^2*b^5*c*d^3 - a^3*b^4*d^4)*x^4 + 30*(10*b^7*c^4 - 100*a*b^6*c^3*d + 95*a^2*b^5*c^2*d^2 - 46*a^
3*b^4*c*d^3 + 9*a^4*b^3*d^4)*x^3 + 10*(50*a*b^6*c^4 - 410*a^2*b^5*c^3*d + 337*a^3*b^4*c^2*d^2 - 148*a^4*b^3*c*
d^3 + 27*a^5*b^2*d^4)*x^2 + 5*(65*a^2*b^5*c^4 - 488*a^3*b^4*c^3*d + 352*a^4*b^3*c^2*d^2 - 148*a^5*b^2*c*d^3 +
27*a^6*b*d^4)*x)/((b^13*c^4 - 4*a*b^12*c^3*d + 6*a^2*b^11*c^2*d^2 - 4*a^3*b^10*c*d^3 + a^4*b^9*d^4)*g^6*x^5 +
5*(a*b^12*c^4 - 4*a^2*b^11*c^3*d + 6*a^3*b^10*c^2*d^2 - 4*a^4*b^9*c*d^3 + a^5*b^8*d^4)*g^6*x^4 + 10*(a^2*b^11*
c^4 - 4*a^3*b^10*c^3*d + 6*a^4*b^9*c^2*d^2 - 4*a^5*b^8*c*d^3 + a^6*b^7*d^4)*g^6*x^3 + 10*(a^3*b^10*c^4 - 4*a^4
*b^9*c^3*d + 6*a^5*b^8*c^2*d^2 - 4*a^6*b^7*c*d^3 + a^7*b^6*d^4)*g^6*x^2 + 5*(a^4*b^9*c^4 - 4*a^5*b^8*c^3*d + 6
*a^6*b^7*c^2*d^2 - 4*a^7*b^6*c*d^3 + a^8*b^5*d^4)*g^6*x + (a^5*b^8*c^4 - 4*a^6*b^7*c^3*d + 6*a^7*b^6*c^2*d^2 -
4*a^8*b^5*c*d^3 + a^9*b^4*d^4)*g^6) - 60*(10*b^3*c^3*d^2 - 10*a*b^2*c^2*d^3 + 5*a^2*b*c*d^4 - a^3*d^5)*log(b*
x + a)/((b^9*c^5 - 5*a*b^8*c^4*d + 10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5)*g^
6) + 60*(10*b^3*c^3*d^2 - 10*a*b^2*c^2*d^3 + 5*a^2*b*c*d^4 - a^3*d^5)*log(d*x + c)/((b^9*c^5 - 5*a*b^8*c^4*d +
10*a^2*b^7*c^3*d^2 - 10*a^3*b^6*c^2*d^3 + 5*a^4*b^5*c*d^4 - a^5*b^4*d^5)*g^6)) - 1/600*B*c*d^2*i^3*(60*(10*b^
2*x^2 + 5*a*b*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*b^6*g^6*x^
3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) + (47*a^2*b^4*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^
2*d^2 - 278*a^5*b*c*d^3 + 47*a^6*d^4 + 60*(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 - 30*(10*b^6*c^3*
d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4*c*d^3 - 9*a^3*b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*
c^2*d^2 - 248*a^3*b^3*c*d^3 + 47*a^4*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^4*c^3*d + 702*a^3*b^3*c^2*d^2
- 278*a^4*b^2*c*d^3 + 47*a^5*b*d^4)*x)/((b^12*c^4 - 4*a*b^11*c^3*d + 6*a^2*b^10*c^2*d^2 - 4*a^3*b^9*c*d^3 + a^
4*b^8*d^4)*g^6*x^5 + 5*(a*b^11*c^4 - 4*a^2*b^10*c^3*d + 6*a^3*b^9*c^2*d^2 - 4*a^4*b^8*c*d^3 + a^5*b^7*d^4)*g^6
*x^4 + 10*(a^2*b^10*c^4 - 4*a^3*b^9*c^3*d + 6*a^4*b^8*c^2*d^2 - 4*a^5*b^7*c*d^3 + a^6*b^6*d^4)*g^6*x^3 + 10*(a
^3*b^9*c^4 - 4*a^4*b^8*c^3*d + 6*a^5*b^7*c^2*d^2 - 4*a^6*b^6*c*d^3 + a^7*b^5*d^4)*g^6*x^2 + 5*(a^4*b^8*c^4 - 4
*a^5*b^7*c^3*d + 6*a^6*b^6*c^2*d^2 - 4*a^7*b^5*c*d^3 + a^8*b^4*d^4)*g^6*x + (a^5*b^7*c^4 - 4*a^6*b^6*c^3*d + 6
*a^7*b^5*c^2*d^2 - 4*a^8*b^4*c*d^3 + a^9*b^3*d^4)*g^6) + 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(b*x +
a)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6)
- 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(d*x + c)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10
*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6)) - 1/400*B*c^2*d*i^3*(60*(5*b*x + a)*log(b*e*x/(d*x + c
) + a*e/(d*x + c))/(b^7*g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x
+ a^5*b^2*g^6) + (27*a*b^4*c^4 - 148*a^2*b^3*c^3*d + 352*a^3*b^2*c^2*d^2 - 548*a^4*b*c*d^3 + 77*a^5*d^4 - 60*(
5*b^5*c*d^3 - a*b^4*d^4)*x^4 + 30*(5*b^5*c^2*d^2 - 46*a*b^4*c*d^3 + 9*a^2*b^3*d^4)*x^3 - 10*(10*b^5*c^3*d - 67
*a*b^4*c^2*d^2 + 248*a^2*b^3*c*d^3 - 47*a^3*b^2*d^4)*x^2 + 5*(15*b^5*c^4 - 88*a*b^4*c^3*d + 232*a^2*b^3*c^2*d^
2 - 428*a^3*b^2*c*d^3 + 77*a^4*b*d^4)*x)/((b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d^2 - 4*a^3*b^8*c*d^3 + a
^4*b^7*d^4)*g^6*x^5 + 5*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^7*c*d^3 + a^5*b^6*d^4)*g^6
*x^4 + 10*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b^5*d^4)*g^6*x^3 + 10*(a^
3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*g^6*x^2 + 5*(a^4*b^7*c^4 - 4*
a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*g^6*x + (a^5*b^6*c^4 - 4*a^6*b^5*c^3*d + 6*
a^7*b^4*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*g^6) - 60*(5*b*c*d^4 - a*d^5)*log(b*x + a)/((b^7*c^5 - 5*a*b^
6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6) + 60*(5*b*c*d^4 - a*d^
5)*log(d*x + c)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^
2*d^5)*g^6)) - 1/300*B*c^3*i^3*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 163*a^3*
b*c*d^3 + 137*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4
)*x^2 - 5*(3*b^4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3*d + 6
*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2
- 4*a^4*b^6*c*d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*
c*d^3 + a^6*b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b
^3*d^4)*g^6*x^2 + 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4)*g^6*x
+ (a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*g^6) + 60*log(b*e*x/(d*x +
c) + a*e/(d*x + c))/(b^6*g^6*x^5 + 5*a*b^5*g^6*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^2 + 5*a^4*b^2*g^6*
x + a^5*b*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a
^4*b^2*c*d^4 - a^5*b*d^5)*g^6) - 60*d^5*log(d*x + c)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b
^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6)) - 3/20*(5*b*x + a)*A*c^2*d*i^3/(b^7*g^6*x^5 + 5*a*b^6*g^6*x^4
+ 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/10*(10*b^2*x^2 + 5*a*b*x + a^2)
*A*c*d^2*i^3/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*
b^3*g^6) - 1/20*(10*b^3*x^3 + 10*a*b^2*x^2 + 5*a^2*b*x + a^3)*A*d^3*i^3/(b^9*g^6*x^5 + 5*a*b^8*g^6*x^4 + 10*a^
2*b^7*g^6*x^3 + 10*a^3*b^6*g^6*x^2 + 5*a^4*b^5*g^6*x + a^5*b^4*g^6) - 1/5*A*c^3*i^3/(b^6*g^6*x^5 + 5*a*b^5*g^6
*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^2 + 5*a^4*b^2*g^6*x + a^5*b*g^6)

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Fricas [B]  time = 0.52562, size = 1337, normalized size = 7.39 \begin{align*} \frac{20 \,{\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} i^{3} x^{4} - 10 \,{\left ({\left (20 \, A + B\right )} b^{5} c^{2} d^{3} - 10 \,{\left (4 \, A + B\right )} a b^{4} c d^{4} +{\left (20 \, A + 9 \, B\right )} a^{2} b^{3} d^{5}\right )} i^{3} x^{3} - 10 \,{\left (2 \,{\left (20 \, A + 3 \, B\right )} b^{5} c^{3} d^{2} - 15 \,{\left (4 \, A + B\right )} a b^{4} c^{2} d^{3} +{\left (20 \, A + 9 \, B\right )} a^{3} b^{2} d^{5}\right )} i^{3} x^{2} - 5 \,{\left ({\left (60 \, A + 11 \, B\right )} b^{5} c^{4} d - 20 \,{\left (4 \, A + B\right )} a b^{4} c^{3} d^{2} +{\left (20 \, A + 9 \, B\right )} a^{4} b d^{5}\right )} i^{3} x -{\left (16 \,{\left (5 \, A + B\right )} b^{5} c^{5} - 25 \,{\left (4 \, A + B\right )} a b^{4} c^{4} d +{\left (20 \, A + 9 \, B\right )} a^{5} d^{5}\right )} i^{3} + 20 \,{\left (B b^{5} d^{5} i^{3} x^{5} + 5 \, B a b^{4} d^{5} i^{3} x^{4} - 10 \,{\left (B b^{5} c^{2} d^{3} - 2 \, B a b^{4} c d^{4}\right )} i^{3} x^{3} - 10 \,{\left (2 \, B b^{5} c^{3} d^{2} - 3 \, B a b^{4} c^{2} d^{3}\right )} i^{3} x^{2} - 5 \,{\left (3 \, B b^{5} c^{4} d - 4 \, B a b^{4} c^{3} d^{2}\right )} i^{3} x -{\left (4 \, B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d\right )} i^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{400 \,{\left ({\left (b^{11} c^{2} - 2 \, a b^{10} c d + a^{2} b^{9} d^{2}\right )} g^{6} x^{5} + 5 \,{\left (a b^{10} c^{2} - 2 \, a^{2} b^{9} c d + a^{3} b^{8} d^{2}\right )} g^{6} x^{4} + 10 \,{\left (a^{2} b^{9} c^{2} - 2 \, a^{3} b^{8} c d + a^{4} b^{7} d^{2}\right )} g^{6} x^{3} + 10 \,{\left (a^{3} b^{8} c^{2} - 2 \, a^{4} b^{7} c d + a^{5} b^{6} d^{2}\right )} g^{6} x^{2} + 5 \,{\left (a^{4} b^{7} c^{2} - 2 \, a^{5} b^{6} c d + a^{6} b^{5} d^{2}\right )} g^{6} x +{\left (a^{5} b^{6} c^{2} - 2 \, a^{6} b^{5} c d + a^{7} b^{4} d^{2}\right )} g^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="fricas")

[Out]

1/400*(20*(B*b^5*c*d^4 - B*a*b^4*d^5)*i^3*x^4 - 10*((20*A + B)*b^5*c^2*d^3 - 10*(4*A + B)*a*b^4*c*d^4 + (20*A
+ 9*B)*a^2*b^3*d^5)*i^3*x^3 - 10*(2*(20*A + 3*B)*b^5*c^3*d^2 - 15*(4*A + B)*a*b^4*c^2*d^3 + (20*A + 9*B)*a^3*b
^2*d^5)*i^3*x^2 - 5*((60*A + 11*B)*b^5*c^4*d - 20*(4*A + B)*a*b^4*c^3*d^2 + (20*A + 9*B)*a^4*b*d^5)*i^3*x - (1
6*(5*A + B)*b^5*c^5 - 25*(4*A + B)*a*b^4*c^4*d + (20*A + 9*B)*a^5*d^5)*i^3 + 20*(B*b^5*d^5*i^3*x^5 + 5*B*a*b^4
*d^5*i^3*x^4 - 10*(B*b^5*c^2*d^3 - 2*B*a*b^4*c*d^4)*i^3*x^3 - 10*(2*B*b^5*c^3*d^2 - 3*B*a*b^4*c^2*d^3)*i^3*x^2
- 5*(3*B*b^5*c^4*d - 4*B*a*b^4*c^3*d^2)*i^3*x - (4*B*b^5*c^5 - 5*B*a*b^4*c^4*d)*i^3)*log((b*e*x + a*e)/(d*x +
c)))/((b^11*c^2 - 2*a*b^10*c*d + a^2*b^9*d^2)*g^6*x^5 + 5*(a*b^10*c^2 - 2*a^2*b^9*c*d + a^3*b^8*d^2)*g^6*x^4
+ 10*(a^2*b^9*c^2 - 2*a^3*b^8*c*d + a^4*b^7*d^2)*g^6*x^3 + 10*(a^3*b^8*c^2 - 2*a^4*b^7*c*d + a^5*b^6*d^2)*g^6*
x^2 + 5*(a^4*b^7*c^2 - 2*a^5*b^6*c*d + a^6*b^5*d^2)*g^6*x + (a^5*b^6*c^2 - 2*a^6*b^5*c*d + a^7*b^4*d^2)*g^6)

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Sympy [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((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**6,x)

[Out]

Timed out

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Giac [B]  time = 1.37452, size = 1183, normalized size = 6.54 \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)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^6,x, algorithm="giac")

[Out]

1/20*B*d^5*log(b*x + a)/(b^6*c^2*g^6*i - 2*a*b^5*c*d*g^6*i + a^2*b^4*d^2*g^6*i) - 1/20*B*d^5*log(d*x + c)/(b^6
*c^2*g^6*i - 2*a*b^5*c*d*g^6*i + a^2*b^4*d^2*g^6*i) + 1/20*(10*B*b^3*d^3*i*x^3 + 20*B*b^3*c*d^2*i*x^2 + 10*B*a
*b^2*d^3*i*x^2 + 15*B*b^3*c^2*d*i*x + 10*B*a*b^2*c*d^2*i*x + 5*B*a^2*b*d^3*i*x + 4*B*b^3*c^3*i + 3*B*a*b^2*c^2
*d*i + 2*B*a^2*b*c*d^2*i + B*a^3*d^3*i)*log((b*x + a)/(d*x + c))/(b^9*g^6*x^5 + 5*a*b^8*g^6*x^4 + 10*a^2*b^7*g
^6*x^3 + 10*a^3*b^6*g^6*x^2 + 5*a^4*b^5*g^6*x + a^5*b^4*g^6) - 1/400*(20*B*b^4*d^4*i*x^4 - 200*A*b^4*c*d^3*i*x
^3 - 210*B*b^4*c*d^3*i*x^3 + 200*A*a*b^3*d^4*i*x^3 + 290*B*a*b^3*d^4*i*x^3 - 400*A*b^4*c^2*d^2*i*x^2 - 460*B*b
^4*c^2*d^2*i*x^2 + 200*A*a*b^3*c*d^3*i*x^2 + 290*B*a*b^3*c*d^3*i*x^2 + 200*A*a^2*b^2*d^4*i*x^2 + 290*B*a^2*b^2
*d^4*i*x^2 - 300*A*b^4*c^3*d*i*x - 355*B*b^4*c^3*d*i*x + 100*A*a*b^3*c^2*d^2*i*x + 145*B*a*b^3*c^2*d^2*i*x + 1
00*A*a^2*b^2*c*d^3*i*x + 145*B*a^2*b^2*c*d^3*i*x + 100*A*a^3*b*d^4*i*x + 145*B*a^3*b*d^4*i*x - 80*A*b^4*c^4*i
- 96*B*b^4*c^4*i + 20*A*a*b^3*c^3*d*i + 29*B*a*b^3*c^3*d*i + 20*A*a^2*b^2*c^2*d^2*i + 29*B*a^2*b^2*c^2*d^2*i +
20*A*a^3*b*c*d^3*i + 29*B*a^3*b*c*d^3*i + 20*A*a^4*d^4*i + 29*B*a^4*d^4*i)/(b^10*c*g^6*x^5 - a*b^9*d*g^6*x^5
+ 5*a*b^9*c*g^6*x^4 - 5*a^2*b^8*d*g^6*x^4 + 10*a^2*b^8*c*g^6*x^3 - 10*a^3*b^7*d*g^6*x^3 + 10*a^3*b^7*c*g^6*x^2
- 10*a^4*b^6*d*g^6*x^2 + 5*a^4*b^6*c*g^6*x - 5*a^5*b^5*d*g^6*x + a^5*b^5*c*g^6 - a^6*b^4*d*g^6)