3.127 \(\int (a g+b g x)^3 (c i+d i x)^3 (A+B \log (e (\frac{a+b x}{c+d x})^n)) \, dx\)
Optimal. Leaf size=477 \[ -\frac{b^2 g^3 i^3 (c+d x)^6 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^4}+\frac{b^3 g^3 i^3 (c+d x)^7 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{7 d^4}-\frac{g^3 i^3 (c+d x)^4 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4}+\frac{3 b g^3 i^3 (c+d x)^5 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4}+\frac{B g^3 i^3 n x (b c-a d)^6}{140 b^3 d^3}+\frac{B g^3 i^3 n (c+d x)^2 (b c-a d)^5}{280 b^2 d^4}-\frac{b^2 B g^3 i^3 n (c+d x)^6 (b c-a d)}{42 d^4}+\frac{B g^3 i^3 n (b c-a d)^7 \log \left (\frac{a+b x}{c+d x}\right )}{140 b^4 d^4}+\frac{B g^3 i^3 n (b c-a d)^7 \log (c+d x)}{140 b^4 d^4}+\frac{B g^3 i^3 n (c+d x)^3 (b c-a d)^4}{420 b d^4}-\frac{17 B g^3 i^3 n (c+d x)^4 (b c-a d)^3}{280 d^4}+\frac{b B g^3 i^3 n (c+d x)^5 (b c-a d)^2}{14 d^4} \]
[Out]
(B*(b*c - a*d)^6*g^3*i^3*n*x)/(140*b^3*d^3) + (B*(b*c - a*d)^5*g^3*i^3*n*(c + d*x)^2)/(280*b^2*d^4) + (B*(b*c
- a*d)^4*g^3*i^3*n*(c + d*x)^3)/(420*b*d^4) - (17*B*(b*c - a*d)^3*g^3*i^3*n*(c + d*x)^4)/(280*d^4) + (b*B*(b*c
- a*d)^2*g^3*i^3*n*(c + d*x)^5)/(14*d^4) - (b^2*B*(b*c - a*d)*g^3*i^3*n*(c + d*x)^6)/(42*d^4) - ((b*c - a*d)^
3*g^3*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^4) + (3*b*(b*c - a*d)^2*g^3*i^3*(c + d*x)^5
*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^4) - (b^2*(b*c - a*d)*g^3*i^3*(c + d*x)^6*(A + B*Log[e*((a + b*x
)/(c + d*x))^n]))/(2*d^4) + (b^3*g^3*i^3*(c + d*x)^7*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(7*d^4) + (B*(b*c
- a*d)^7*g^3*i^3*n*Log[(a + b*x)/(c + d*x)])/(140*b^4*d^4) + (B*(b*c - a*d)^7*g^3*i^3*n*Log[c + d*x])/(140*b^
4*d^4)
________________________________________________________________________________________
Rubi [A] time = 0.991502, antiderivative size = 435, normalized size of antiderivative =
0.91, number of steps used = 18, number of rules used = 4, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.093, Rules
used = {2528, 2525, 12, 43} \[ \frac{d^2 g^3 i^3 (a+b x)^6 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4}+\frac{d^3 g^3 i^3 (a+b x)^7 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{7 b^4}+\frac{g^3 i^3 (a+b x)^4 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^4}+\frac{3 d g^3 i^3 (a+b x)^5 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^4}-\frac{B g^3 i^3 n x (b c-a d)^6}{140 b^3 d^3}+\frac{B g^3 i^3 n (a+b x)^2 (b c-a d)^5}{280 b^4 d^2}-\frac{B d^2 g^3 i^3 n (a+b x)^6 (b c-a d)}{42 b^4}+\frac{B g^3 i^3 n (b c-a d)^7 \log (c+d x)}{140 b^4 d^4}-\frac{B g^3 i^3 n (a+b x)^3 (b c-a d)^4}{420 b^4 d}-\frac{17 B g^3 i^3 n (a+b x)^4 (b c-a d)^3}{280 b^4}-\frac{B d g^3 i^3 n (a+b x)^5 (b c-a d)^2}{14 b^4} \]
Antiderivative was successfully verified.
[In]
Int[(a*g + b*g*x)^3*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]
[Out]
-(B*(b*c - a*d)^6*g^3*i^3*n*x)/(140*b^3*d^3) + (B*(b*c - a*d)^5*g^3*i^3*n*(a + b*x)^2)/(280*b^4*d^2) - (B*(b*c
- a*d)^4*g^3*i^3*n*(a + b*x)^3)/(420*b^4*d) - (17*B*(b*c - a*d)^3*g^3*i^3*n*(a + b*x)^4)/(280*b^4) - (B*d*(b*
c - a*d)^2*g^3*i^3*n*(a + b*x)^5)/(14*b^4) - (B*d^2*(b*c - a*d)*g^3*i^3*n*(a + b*x)^6)/(42*b^4) + ((b*c - a*d)
^3*g^3*i^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b^4) + (3*d*(b*c - a*d)^2*g^3*i^3*(a + b*x)^
5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b^4) + (d^2*(b*c - a*d)*g^3*i^3*(a + b*x)^6*(A + B*Log[e*((a + b*
x)/(c + d*x))^n]))/(2*b^4) + (d^3*g^3*i^3*(a + b*x)^7*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(7*b^4) + (B*(b*
c - a*d)^7*g^3*i^3*n*Log[c + d*x])/(140*b^4*d^4)
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 43
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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rubi steps
\begin{align*} \int (127 c+127 d x)^3 (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (\frac{(-b c+a d)^3 g^3 (127 c+127 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}+\frac{3 b (b c-a d)^2 g^3 (127 c+127 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{127 d^3}-\frac{3 b^2 (b c-a d) g^3 (127 c+127 d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{16129 d^3}+\frac{b^3 g^3 (127 c+127 d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2048383 d^3}\right ) \, dx\\ &=\frac{\left (b^3 g^3\right ) \int (127 c+127 d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2048383 d^3}-\frac{\left (3 b^2 (b c-a d) g^3\right ) \int (127 c+127 d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{16129 d^3}+\frac{\left (3 b (b c-a d)^2 g^3\right ) \int (127 c+127 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{127 d^3}-\frac{\left ((b c-a d)^3 g^3\right ) \int (127 c+127 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^3}\\ &=-\frac{2048383 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^4}+\frac{6145149 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^4}-\frac{2048383 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 d^4}+\frac{2048383 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac{\left (b^3 B g^3 n\right ) \int \frac{532875860165503 (b c-a d) (c+d x)^6}{a+b x} \, dx}{1821012487 d^4}+\frac{\left (b^2 B (b c-a d) g^3 n\right ) \int \frac{4195872914689 (b c-a d) (c+d x)^5}{a+b x} \, dx}{4096766 d^4}-\frac{\left (3 b B (b c-a d)^2 g^3 n\right ) \int \frac{33038369407 (b c-a d) (c+d x)^4}{a+b x} \, dx}{80645 d^4}+\frac{\left (B (b c-a d)^3 g^3 n\right ) \int \frac{260144641 (b c-a d) (c+d x)^3}{a+b x} \, dx}{508 d^4}\\ &=-\frac{2048383 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^4}+\frac{6145149 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^4}-\frac{2048383 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 d^4}+\frac{2048383 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac{\left (2048383 b^3 B (b c-a d) g^3 n\right ) \int \frac{(c+d x)^6}{a+b x} \, dx}{7 d^4}+\frac{\left (2048383 b^2 B (b c-a d)^2 g^3 n\right ) \int \frac{(c+d x)^5}{a+b x} \, dx}{2 d^4}-\frac{\left (6145149 b B (b c-a d)^3 g^3 n\right ) \int \frac{(c+d x)^4}{a+b x} \, dx}{5 d^4}+\frac{\left (2048383 B (b c-a d)^4 g^3 n\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{4 d^4}\\ &=-\frac{2048383 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^4}+\frac{6145149 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^4}-\frac{2048383 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 d^4}+\frac{2048383 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}-\frac{\left (2048383 b^3 B (b c-a d) g^3 n\right ) \int \left (\frac{d (b c-a d)^5}{b^6}+\frac{(b c-a d)^6}{b^6 (a+b x)}+\frac{d (b c-a d)^4 (c+d x)}{b^5}+\frac{d (b c-a d)^3 (c+d x)^2}{b^4}+\frac{d (b c-a d)^2 (c+d x)^3}{b^3}+\frac{d (b c-a d) (c+d x)^4}{b^2}+\frac{d (c+d x)^5}{b}\right ) \, dx}{7 d^4}+\frac{\left (2048383 b^2 B (b c-a d)^2 g^3 n\right ) \int \left (\frac{d (b c-a d)^4}{b^5}+\frac{(b c-a d)^5}{b^5 (a+b x)}+\frac{d (b c-a d)^3 (c+d x)}{b^4}+\frac{d (b c-a d)^2 (c+d x)^2}{b^3}+\frac{d (b c-a d) (c+d x)^3}{b^2}+\frac{d (c+d x)^4}{b}\right ) \, dx}{2 d^4}-\frac{\left (6145149 b B (b c-a d)^3 g^3 n\right ) \int \left (\frac{d (b c-a d)^3}{b^4}+\frac{(b c-a d)^4}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x)}{b^3}+\frac{d (b c-a d) (c+d x)^2}{b^2}+\frac{d (c+d x)^3}{b}\right ) \, dx}{5 d^4}+\frac{\left (2048383 B (b c-a d)^4 g^3 n\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{4 d^4}\\ &=\frac{2048383 B (b c-a d)^6 g^3 n x}{140 b^3 d^3}+\frac{2048383 B (b c-a d)^5 g^3 n (c+d x)^2}{280 b^2 d^4}+\frac{2048383 B (b c-a d)^4 g^3 n (c+d x)^3}{420 b d^4}-\frac{34822511 B (b c-a d)^3 g^3 n (c+d x)^4}{280 d^4}+\frac{2048383 b B (b c-a d)^2 g^3 n (c+d x)^5}{14 d^4}-\frac{2048383 b^2 B (b c-a d) g^3 n (c+d x)^6}{42 d^4}+\frac{2048383 B (b c-a d)^7 g^3 n \log (a+b x)}{140 b^4 d^4}-\frac{2048383 (b c-a d)^3 g^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 d^4}+\frac{6145149 b (b c-a d)^2 g^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 d^4}-\frac{2048383 b^2 (b c-a d) g^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 d^4}+\frac{2048383 b^3 g^3 (c+d x)^7 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7 d^4}\\ \end{align*}
Mathematica [A] time = 0.587502, size = 631, normalized size = 1.32 \[ \frac{g^3 i^3 \left (120 d^7 (a+b x)^7 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+420 d^6 (a+b x)^6 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+504 d^5 (a+b x)^5 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+210 d^4 (a+b x)^4 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-35 B n (b c-a d)^4 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+42 B n (b c-a d)^3 \left (-6 d^2 (a+b x)^2 (b c-a d)^2+4 d^3 (a+b x)^3 (b c-a d)+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )-7 B n (b c-a d)^2 \left (20 d^3 (a+b x)^3 (b c-a d)^2+15 d^4 (a+b x)^4 (a d-b c)+30 d^2 (a+b x)^2 (a d-b c)^3+60 b d x (b c-a d)^4-60 (b c-a d)^5 \log (c+d x)+12 d^5 (a+b x)^5\right )+2 B n (b c-a d) \left (-30 d^2 (a+b x)^2 (b c-a d)^4+20 d^3 (a+b x)^3 (b c-a d)^3-15 d^4 (a+b x)^4 (b c-a d)^2+12 d^5 (a+b x)^5 (b c-a d)+60 b d x (b c-a d)^5-60 (b c-a d)^6 \log (c+d x)-10 d^6 (a+b x)^6\right )\right )}{840 b^4 d^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(a*g + b*g*x)^3*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]
[Out]
(g^3*i^3*(210*d^4*(b*c - a*d)^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 504*d^5*(b*c - a*d)^2*(a
+ b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 420*d^6*(b*c - a*d)*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]) + 120*d^7*(a + b*x)^7*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 35*B*(b*c - a*d)^4*n*(6*b*d*(b*c - a
*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + 42*B*(b*c - a
*d)^3*n*(12*b*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a + b*x)^3 - 3*d^4*(a +
b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]) - 7*B*(b*c - a*d)^2*n*(60*b*d*(b*c - a*d)^4*x + 30*d^2*(-(b*c) + a*d)
^3*(a + b*x)^2 + 20*d^3*(b*c - a*d)^2*(a + b*x)^3 + 15*d^4*(-(b*c) + a*d)*(a + b*x)^4 + 12*d^5*(a + b*x)^5 - 6
0*(b*c - a*d)^5*Log[c + d*x]) + 2*B*(b*c - a*d)*n*(60*b*d*(b*c - a*d)^5*x - 30*d^2*(b*c - a*d)^4*(a + b*x)^2 +
20*d^3*(b*c - a*d)^3*(a + b*x)^3 - 15*d^4*(b*c - a*d)^2*(a + b*x)^4 + 12*d^5*(b*c - a*d)*(a + b*x)^5 - 10*d^6
*(a + b*x)^6 - 60*(b*c - a*d)^6*Log[c + d*x])))/(840*b^4*d^4)
________________________________________________________________________________________
Maple [F] time = 0.678, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) ^{3} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)
[Out]
int((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)
________________________________________________________________________________________
Maxima [B] time = 1.60214, size = 3916, normalized size = 8.21 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")
[Out]
1/7*B*b^3*d^3*g^3*i^3*x^7*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/7*A*b^3*d^3*g^3*i^3*x^7 + 1/2*B*b^3*c*d^2
*g^3*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*B*a*b^2*d^3*g^3*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*
x + c))^n) + 1/2*A*b^3*c*d^2*g^3*i^3*x^6 + 1/2*A*a*b^2*d^3*g^3*i^3*x^6 + 3/5*B*b^3*c^2*d*g^3*i^3*x^5*log(e*(b*
x/(d*x + c) + a/(d*x + c))^n) + 9/5*B*a*b^2*c*d^2*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/5*B*a
^2*b*d^3*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/5*A*b^3*c^2*d*g^3*i^3*x^5 + 9/5*A*a*b^2*c*d^2*
g^3*i^3*x^5 + 3/5*A*a^2*b*d^3*g^3*i^3*x^5 + 1/4*B*b^3*c^3*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) +
9/4*B*a*b^2*c^2*d*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 9/4*B*a^2*b*c*d^2*g^3*i^3*x^4*log(e*(b
*x/(d*x + c) + a/(d*x + c))^n) + 1/4*B*a^3*d^3*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b^3*
c^3*g^3*i^3*x^4 + 9/4*A*a*b^2*c^2*d*g^3*i^3*x^4 + 9/4*A*a^2*b*c*d^2*g^3*i^3*x^4 + 1/4*A*a^3*d^3*g^3*i^3*x^4 +
B*a*b^2*c^3*g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*B*a^2*b*c^2*d*g^3*i^3*x^3*log(e*(b*x/(d*x +
c) + a/(d*x + c))^n) + B*a^3*c*d^2*g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a*b^2*c^3*g^3*i^3*x
^3 + 3*A*a^2*b*c^2*d*g^3*i^3*x^3 + A*a^3*c*d^2*g^3*i^3*x^3 + 3/2*B*a^2*b*c^3*g^3*i^3*x^2*log(e*(b*x/(d*x + c)
+ a/(d*x + c))^n) + 3/2*B*a^3*c^2*d*g^3*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A*a^2*b*c^3*g^3*i
^3*x^2 + 3/2*A*a^3*c^2*d*g^3*i^3*x^2 + 1/420*B*b^3*d^3*g^3*i^3*n*(60*a^7*log(b*x + a)/b^7 - 60*c^7*log(d*x + c
)/d^7 - (10*(b^6*c*d^5 - a*b^5*d^6)*x^6 - 12*(b^6*c^2*d^4 - a^2*b^4*d^6)*x^5 + 15*(b^6*c^3*d^3 - a^3*b^3*d^6)*
x^4 - 20*(b^6*c^4*d^2 - a^4*b^2*d^6)*x^3 + 30*(b^6*c^5*d - a^5*b*d^6)*x^2 - 60*(b^6*c^6 - a^6*d^6)*x)/(b^6*d^6
)) - 1/120*B*b^3*c*d^2*g^3*i^3*n*(60*a^6*log(b*x + a)/b^6 - 60*c^6*log(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d
^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)
*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) - 1/120*B*a*b^2*d^3*g^3*i^3*n*(60*a^6*log(b*x + a)/b^6 - 60*c^6*lo
g(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*
b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) + 1/20*B*b^3*c^2*d*g^3*i^
3*n*(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)) + 3/20*B*a*b^2*c*d^2*g^3*
i^3*n*(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)) + 1/20*B*a^2*b*d^3*g^3*
i^3*n*(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)) - 1/24*B*b^3*c^3*g^3*i^
3*n*(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)) - 3/8*B*a*b^2*c^2*d*g^3*i^3*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*lo
g(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)) - 3/8*B*a^2*b*c*d^2*g^3*i^3*n*(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)) - 1/24*B*a^3*d^3*g^3*i^3*n*(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)) + 1/2*B*a*b^2*c^3*g^3*i^3*n*(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)) + 3/2*B*a^2*b*c^2*d*g^3*i^3*n*(2*a^3*lo
g(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)) + 1/2
*B*a^3*c*d^2*g^3*i^3*n*(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)) - 3/2*B*a^2*b*c^3*g^3*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a
*d)*x/(b*d)) - 3/2*B*a^3*c^2*d*g^3*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) +
B*a^3*c^3*g^3*i^3*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + B*a^3*c^3*g^3*i^3*x*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + A*a^3*c^3*g^3*i^3*x
________________________________________________________________________________________
Fricas [B] time = 1.83033, size = 2716, normalized size = 5.69 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")
[Out]
1/840*(120*A*b^7*d^7*g^3*i^3*x^7 + 6*(35*B*a^4*b^3*c^3*d^4 - 21*B*a^5*b^2*c^2*d^5 + 7*B*a^6*b*c*d^6 - B*a^7*d^
7)*g^3*i^3*n*log(b*x + a) + 6*(B*b^7*c^7 - 7*B*a*b^6*c^6*d + 21*B*a^2*b^5*c^5*d^2 - 35*B*a^3*b^4*c^4*d^3)*g^3*
i^3*n*log(d*x + c) - 20*((B*b^7*c*d^6 - B*a*b^6*d^7)*g^3*i^3*n - 21*(A*b^7*c*d^6 + A*a*b^6*d^7)*g^3*i^3)*x^6 -
12*(5*(B*b^7*c^2*d^5 - B*a^2*b^5*d^7)*g^3*i^3*n - 42*(A*b^7*c^2*d^5 + 3*A*a*b^6*c*d^6 + A*a^2*b^5*d^7)*g^3*i^
3)*x^5 - 3*((17*B*b^7*c^3*d^4 + 49*B*a*b^6*c^2*d^5 - 49*B*a^2*b^5*c*d^6 - 17*B*a^3*b^4*d^7)*g^3*i^3*n - 70*(A*
b^7*c^3*d^4 + 9*A*a*b^6*c^2*d^5 + 9*A*a^2*b^5*c*d^6 + A*a^3*b^4*d^7)*g^3*i^3)*x^4 - 2*((B*b^7*c^4*d^3 + 98*B*a
*b^6*c^3*d^4 - 98*B*a^3*b^4*c*d^6 - B*a^4*b^3*d^7)*g^3*i^3*n - 420*(A*a*b^6*c^3*d^4 + 3*A*a^2*b^5*c^2*d^5 + A*
a^3*b^4*c*d^6)*g^3*i^3)*x^3 + 3*((B*b^7*c^5*d^2 - 7*B*a*b^6*c^4*d^3 - 84*B*a^2*b^5*c^3*d^4 + 84*B*a^3*b^4*c^2*
d^5 + 7*B*a^4*b^3*c*d^6 - B*a^5*b^2*d^7)*g^3*i^3*n + 420*(A*a^2*b^5*c^3*d^4 + A*a^3*b^4*c^2*d^5)*g^3*i^3)*x^2
+ 6*(140*A*a^3*b^4*c^3*d^4*g^3*i^3 - (B*b^7*c^6*d - 7*B*a*b^6*c^5*d^2 + 21*B*a^2*b^5*c^4*d^3 - 21*B*a^4*b^3*c^
2*d^5 + 7*B*a^5*b^2*c*d^6 - B*a^6*b*d^7)*g^3*i^3*n)*x + 6*(20*B*b^7*d^7*g^3*i^3*x^7 + 140*B*a^3*b^4*c^3*d^4*g^
3*i^3*x + 70*(B*b^7*c*d^6 + B*a*b^6*d^7)*g^3*i^3*x^6 + 84*(B*b^7*c^2*d^5 + 3*B*a*b^6*c*d^6 + B*a^2*b^5*d^7)*g^
3*i^3*x^5 + 35*(B*b^7*c^3*d^4 + 9*B*a*b^6*c^2*d^5 + 9*B*a^2*b^5*c*d^6 + B*a^3*b^4*d^7)*g^3*i^3*x^4 + 140*(B*a*
b^6*c^3*d^4 + 3*B*a^2*b^5*c^2*d^5 + B*a^3*b^4*c*d^6)*g^3*i^3*x^3 + 210*(B*a^2*b^5*c^3*d^4 + B*a^3*b^4*c^2*d^5)
*g^3*i^3*x^2)*log(e) + 6*(20*B*b^7*d^7*g^3*i^3*n*x^7 + 140*B*a^3*b^4*c^3*d^4*g^3*i^3*n*x + 70*(B*b^7*c*d^6 + B
*a*b^6*d^7)*g^3*i^3*n*x^6 + 84*(B*b^7*c^2*d^5 + 3*B*a*b^6*c*d^6 + B*a^2*b^5*d^7)*g^3*i^3*n*x^5 + 35*(B*b^7*c^3
*d^4 + 9*B*a*b^6*c^2*d^5 + 9*B*a^2*b^5*c*d^6 + B*a^3*b^4*d^7)*g^3*i^3*n*x^4 + 140*(B*a*b^6*c^3*d^4 + 3*B*a^2*b
^5*c^2*d^5 + B*a^3*b^4*c*d^6)*g^3*i^3*n*x^3 + 210*(B*a^2*b^5*c^3*d^4 + B*a^3*b^4*c^2*d^5)*g^3*i^3*n*x^2)*log((
b*x + a)/(d*x + c)))/(b^4*d^4)
________________________________________________________________________________________
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((b*g*x+a*g)**3*(d*i*x+c*i)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n)),x)
[Out]
Timed out
________________________________________________________________________________________
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((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")
[Out]
Timed out