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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 \[ \frac {b^3 g^3 i^3 (c+d x)^7 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{7 d^4}-\frac {b^2 g^3 i^3 (c+d x)^6 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d^4}-\frac {g^3 i^3 (c+d x)^4 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4}+\frac {3 b g^3 i^3 (c+d x)^5 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4}+\frac {B g^3 i^3 n (b c-a d)^7 \log \left (\frac {a+b x}{c+d x}\right )}{140 b^4 d^4}+\frac {B g^3 i^3 n (b c-a d)^7 \log (c+d x)}{140 b^4 d^4}+\frac {B g^3 i^3 n x (b c-a d)^6}{140 b^3 d^3}+\frac {B g^3 i^3 n (c+d x)^2 (b c-a d)^5}{280 b^2 d^4}-\frac {b^2 B g^3 i^3 n (c+d x)^6 (b c-a d)}{42 d^4}+\frac {B g^3 i^3 n (c+d x)^3 (b c-a d)^4}{420 b d^4}-\frac {17 B g^3 i^3 n (c+d x)^4 (b c-a d)^3}{280 d^4}+\frac {b B g^3 i^3 n (c+d x)^5 (b c-a d)^2}{14 d^4} \]

[Out]

1/140*B*(-a*d+b*c)^6*g^3*i^3*n*x/b^3/d^3+1/280*B*(-a*d+b*c)^5*g^3*i^3*n*(d*x+c)^2/b^2/d^4+1/420*B*(-a*d+b*c)^4
*g^3*i^3*n*(d*x+c)^3/b/d^4-17/280*B*(-a*d+b*c)^3*g^3*i^3*n*(d*x+c)^4/d^4+1/14*b*B*(-a*d+b*c)^2*g^3*i^3*n*(d*x+
c)^5/d^4-1/42*b^2*B*(-a*d+b*c)*g^3*i^3*n*(d*x+c)^6/d^4-1/4*(-a*d+b*c)^3*g^3*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(
d*x+c))^n))/d^4+3/5*b*(-a*d+b*c)^2*g^3*i^3*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4-1/2*b^2*(-a*d+b*c)*g^
3*i^3*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+1/7*b^3*g^3*i^3*(d*x+c)^7*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/
d^4+1/140*B*(-a*d+b*c)^7*g^3*i^3*n*ln((b*x+a)/(d*x+c))/b^4/d^4+1/140*B*(-a*d+b*c)^7*g^3*i^3*n*ln(d*x+c)/b^4/d^
4

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Rubi [A]  time = 0.99, 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 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])

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 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]

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*}

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Mathematica [A]  time = 0.67, size = 631, normalized size = 1.32 \[ \frac {g^3 i^3 \left (120 d^7 (a+b x)^7 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+420 d^6 (a+b x)^6 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+504 d^5 (a+b x)^5 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+210 d^4 (a+b x)^4 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-35 B n (b c-a d)^4 \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+42 B n (b c-a d)^3 \left (4 d^3 (a+b x)^3 (b c-a d)-6 d^2 (a+b x)^2 (b c-a d)^2+12 b d x (b c-a d)^3-12 (b c-a d)^4 \log (c+d x)-3 d^4 (a+b x)^4\right )-7 B n (b c-a d)^2 \left (15 d^4 (a+b x)^4 (a d-b c)+20 d^3 (a+b x)^3 (b c-a d)^2+30 d^2 (a+b x)^2 (a d-b c)^3+60 b d x (b c-a d)^4-60 (b c-a d)^5 \log (c+d x)+12 d^5 (a+b x)^5\right )+2 B n (b c-a d) \left (12 d^5 (a+b x)^5 (b c-a d)-15 d^4 (a+b x)^4 (b c-a d)^2+20 d^3 (a+b x)^3 (b c-a d)^3-30 d^2 (a+b x)^2 (b c-a d)^4+60 b d x (b c-a d)^5-60 (b c-a d)^6 \log (c+d x)-10 d^6 (a+b x)^6\right )\right )}{840 b^4 d^4} \]

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)

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fricas [B]  time = 2.59, size = 1336, normalized size = 2.80 \[ \text {result too large to display} \]

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)

________________________________________________________________________________________

giac [B]  time = 15.02, size = 5502, normalized size = 11.53 \[ \text {result too large to display} \]

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]

1/840*(6*(B*b^11*c^8*g^3*i*n - 8*B*a*b^10*c^7*d*g^3*i*n - 7*(b*x + a)*B*b^10*c^8*d*g^3*i*n/(d*x + c) + 28*B*a^
2*b^9*c^6*d^2*g^3*i*n + 56*(b*x + a)*B*a*b^9*c^7*d^2*g^3*i*n/(d*x + c) + 21*(b*x + a)^2*B*b^9*c^8*d^2*g^3*i*n/
(d*x + c)^2 - 56*B*a^3*b^8*c^5*d^3*g^3*i*n - 196*(b*x + a)*B*a^2*b^8*c^6*d^3*g^3*i*n/(d*x + c) - 168*(b*x + a)
^2*B*a*b^8*c^7*d^3*g^3*i*n/(d*x + c)^2 - 35*(b*x + a)^3*B*b^8*c^8*d^3*g^3*i*n/(d*x + c)^3 + 70*B*a^4*b^7*c^4*d
^4*g^3*i*n + 392*(b*x + a)*B*a^3*b^7*c^5*d^4*g^3*i*n/(d*x + c) + 588*(b*x + a)^2*B*a^2*b^7*c^6*d^4*g^3*i*n/(d*
x + c)^2 + 280*(b*x + a)^3*B*a*b^7*c^7*d^4*g^3*i*n/(d*x + c)^3 - 56*B*a^5*b^6*c^3*d^5*g^3*i*n - 490*(b*x + a)*
B*a^4*b^6*c^4*d^5*g^3*i*n/(d*x + c) - 1176*(b*x + a)^2*B*a^3*b^6*c^5*d^5*g^3*i*n/(d*x + c)^2 - 980*(b*x + a)^3
*B*a^2*b^6*c^6*d^5*g^3*i*n/(d*x + c)^3 + 28*B*a^6*b^5*c^2*d^6*g^3*i*n + 392*(b*x + a)*B*a^5*b^5*c^3*d^6*g^3*i*
n/(d*x + c) + 1470*(b*x + a)^2*B*a^4*b^5*c^4*d^6*g^3*i*n/(d*x + c)^2 + 1960*(b*x + a)^3*B*a^3*b^5*c^5*d^6*g^3*
i*n/(d*x + c)^3 - 8*B*a^7*b^4*c*d^7*g^3*i*n - 196*(b*x + a)*B*a^6*b^4*c^2*d^7*g^3*i*n/(d*x + c) - 1176*(b*x +
a)^2*B*a^5*b^4*c^3*d^7*g^3*i*n/(d*x + c)^2 - 2450*(b*x + a)^3*B*a^4*b^4*c^4*d^7*g^3*i*n/(d*x + c)^3 + B*a^8*b^
3*d^8*g^3*i*n + 56*(b*x + a)*B*a^7*b^3*c*d^8*g^3*i*n/(d*x + c) + 588*(b*x + a)^2*B*a^6*b^3*c^2*d^8*g^3*i*n/(d*
x + c)^2 + 1960*(b*x + a)^3*B*a^5*b^3*c^3*d^8*g^3*i*n/(d*x + c)^3 - 7*(b*x + a)*B*a^8*b^2*d^9*g^3*i*n/(d*x + c
) - 168*(b*x + a)^2*B*a^7*b^2*c*d^9*g^3*i*n/(d*x + c)^2 - 980*(b*x + a)^3*B*a^6*b^2*c^2*d^9*g^3*i*n/(d*x + c)^
3 + 21*(b*x + a)^2*B*a^8*b*d^10*g^3*i*n/(d*x + c)^2 + 280*(b*x + a)^3*B*a^7*b*c*d^10*g^3*i*n/(d*x + c)^3 - 35*
(b*x + a)^3*B*a^8*d^11*g^3*i*n/(d*x + c)^3)*log((b*x + a)/(d*x + c))/(b^7*d^4 - 7*(b*x + a)*b^6*d^5/(d*x + c)
+ 21*(b*x + a)^2*b^5*d^6/(d*x + c)^2 - 35*(b*x + a)^3*b^4*d^7/(d*x + c)^3 + 35*(b*x + a)^4*b^3*d^8/(d*x + c)^4
 - 21*(b*x + a)^5*b^2*d^9/(d*x + c)^5 + 7*(b*x + a)^6*b*d^10/(d*x + c)^6 - (b*x + a)^7*d^11/(d*x + c)^7) + (6*
(b*x + a)*B*b^13*c^8*d*g^3*i*n/(d*x + c) - 48*(b*x + a)*B*a*b^12*c^7*d^2*g^3*i*n/(d*x + c) - 39*(b*x + a)^2*B*
b^12*c^8*d^2*g^3*i*n/(d*x + c)^2 + 168*(b*x + a)*B*a^2*b^11*c^6*d^3*g^3*i*n/(d*x + c) + 312*(b*x + a)^2*B*a*b^
11*c^7*d^3*g^3*i*n/(d*x + c)^2 + 107*(b*x + a)^3*B*b^11*c^8*d^3*g^3*i*n/(d*x + c)^3 - 336*(b*x + a)*B*a^3*b^10
*c^5*d^4*g^3*i*n/(d*x + c) - 1092*(b*x + a)^2*B*a^2*b^10*c^6*d^4*g^3*i*n/(d*x + c)^2 - 856*(b*x + a)^3*B*a*b^1
0*c^7*d^4*g^3*i*n/(d*x + c)^3 - 107*(b*x + a)^4*B*b^10*c^8*d^4*g^3*i*n/(d*x + c)^4 + 420*(b*x + a)*B*a^4*b^9*c
^4*d^5*g^3*i*n/(d*x + c) + 2184*(b*x + a)^2*B*a^3*b^9*c^5*d^5*g^3*i*n/(d*x + c)^2 + 2996*(b*x + a)^3*B*a^2*b^9
*c^6*d^5*g^3*i*n/(d*x + c)^3 + 856*(b*x + a)^4*B*a*b^9*c^7*d^5*g^3*i*n/(d*x + c)^4 + 39*(b*x + a)^5*B*b^9*c^8*
d^5*g^3*i*n/(d*x + c)^5 - 336*(b*x + a)*B*a^5*b^8*c^3*d^6*g^3*i*n/(d*x + c) - 2730*(b*x + a)^2*B*a^4*b^8*c^4*d
^6*g^3*i*n/(d*x + c)^2 - 5992*(b*x + a)^3*B*a^3*b^8*c^5*d^6*g^3*i*n/(d*x + c)^3 - 2996*(b*x + a)^4*B*a^2*b^8*c
^6*d^6*g^3*i*n/(d*x + c)^4 - 312*(b*x + a)^5*B*a*b^8*c^7*d^6*g^3*i*n/(d*x + c)^5 - 6*(b*x + a)^6*B*b^8*c^8*d^6
*g^3*i*n/(d*x + c)^6 + 168*(b*x + a)*B*a^6*b^7*c^2*d^7*g^3*i*n/(d*x + c) + 2184*(b*x + a)^2*B*a^5*b^7*c^3*d^7*
g^3*i*n/(d*x + c)^2 + 7490*(b*x + a)^3*B*a^4*b^7*c^4*d^7*g^3*i*n/(d*x + c)^3 + 5992*(b*x + a)^4*B*a^3*b^7*c^5*
d^7*g^3*i*n/(d*x + c)^4 + 1092*(b*x + a)^5*B*a^2*b^7*c^6*d^7*g^3*i*n/(d*x + c)^5 + 48*(b*x + a)^6*B*a*b^7*c^7*
d^7*g^3*i*n/(d*x + c)^6 - 48*(b*x + a)*B*a^7*b^6*c*d^8*g^3*i*n/(d*x + c) - 1092*(b*x + a)^2*B*a^6*b^6*c^2*d^8*
g^3*i*n/(d*x + c)^2 - 5992*(b*x + a)^3*B*a^5*b^6*c^3*d^8*g^3*i*n/(d*x + c)^3 - 7490*(b*x + a)^4*B*a^4*b^6*c^4*
d^8*g^3*i*n/(d*x + c)^4 - 2184*(b*x + a)^5*B*a^3*b^6*c^5*d^8*g^3*i*n/(d*x + c)^5 - 168*(b*x + a)^6*B*a^2*b^6*c
^6*d^8*g^3*i*n/(d*x + c)^6 + 6*(b*x + a)*B*a^8*b^5*d^9*g^3*i*n/(d*x + c) + 312*(b*x + a)^2*B*a^7*b^5*c*d^9*g^3
*i*n/(d*x + c)^2 + 2996*(b*x + a)^3*B*a^6*b^5*c^2*d^9*g^3*i*n/(d*x + c)^3 + 5992*(b*x + a)^4*B*a^5*b^5*c^3*d^9
*g^3*i*n/(d*x + c)^4 + 2730*(b*x + a)^5*B*a^4*b^5*c^4*d^9*g^3*i*n/(d*x + c)^5 + 336*(b*x + a)^6*B*a^3*b^5*c^5*
d^9*g^3*i*n/(d*x + c)^6 - 39*(b*x + a)^2*B*a^8*b^4*d^10*g^3*i*n/(d*x + c)^2 - 856*(b*x + a)^3*B*a^7*b^4*c*d^10
*g^3*i*n/(d*x + c)^3 - 2996*(b*x + a)^4*B*a^6*b^4*c^2*d^10*g^3*i*n/(d*x + c)^4 - 2184*(b*x + a)^5*B*a^5*b^4*c^
3*d^10*g^3*i*n/(d*x + c)^5 - 420*(b*x + a)^6*B*a^4*b^4*c^4*d^10*g^3*i*n/(d*x + c)^6 + 107*(b*x + a)^3*B*a^8*b^
3*d^11*g^3*i*n/(d*x + c)^3 + 856*(b*x + a)^4*B*a^7*b^3*c*d^11*g^3*i*n/(d*x + c)^4 + 1092*(b*x + a)^5*B*a^6*b^3
*c^2*d^11*g^3*i*n/(d*x + c)^5 + 336*(b*x + a)^6*B*a^5*b^3*c^3*d^11*g^3*i*n/(d*x + c)^6 - 107*(b*x + a)^4*B*a^8
*b^2*d^12*g^3*i*n/(d*x + c)^4 - 312*(b*x + a)^5*B*a^7*b^2*c*d^12*g^3*i*n/(d*x + c)^5 - 168*(b*x + a)^6*B*a^6*b
^2*c^2*d^12*g^3*i*n/(d*x + c)^6 + 39*(b*x + a)^5*B*a^8*b*d^13*g^3*i*n/(d*x + c)^5 + 48*(b*x + a)^6*B*a^7*b*c*d
^13*g^3*i*n/(d*x + c)^6 - 6*(b*x + a)^6*B*a^8*d^14*g^3*i*n/(d*x + c)^6 + 6*A*b^14*c^8*g^3*i + 6*B*b^14*c^8*g^3
*i - 48*A*a*b^13*c^7*d*g^3*i - 48*B*a*b^13*c^7*d*g^3*i - 42*(b*x + a)*A*b^13*c^8*d*g^3*i/(d*x + c) - 42*(b*x +
 a)*B*b^13*c^8*d*g^3*i/(d*x + c) + 168*A*a^2*b^12*c^6*d^2*g^3*i + 168*B*a^2*b^12*c^6*d^2*g^3*i + 336*(b*x + a)
*A*a*b^12*c^7*d^2*g^3*i/(d*x + c) + 336*(b*x + a)*B*a*b^12*c^7*d^2*g^3*i/(d*x + c) + 126*(b*x + a)^2*A*b^12*c^
8*d^2*g^3*i/(d*x + c)^2 + 126*(b*x + a)^2*B*b^12*c^8*d^2*g^3*i/(d*x + c)^2 - 336*A*a^3*b^11*c^5*d^3*g^3*i - 33
6*B*a^3*b^11*c^5*d^3*g^3*i - 1176*(b*x + a)*A*a^2*b^11*c^6*d^3*g^3*i/(d*x + c) - 1176*(b*x + a)*B*a^2*b^11*c^6
*d^3*g^3*i/(d*x + c) - 1008*(b*x + a)^2*A*a*b^11*c^7*d^3*g^3*i/(d*x + c)^2 - 1008*(b*x + a)^2*B*a*b^11*c^7*d^3
*g^3*i/(d*x + c)^2 - 210*(b*x + a)^3*A*b^11*c^8*d^3*g^3*i/(d*x + c)^3 - 210*(b*x + a)^3*B*b^11*c^8*d^3*g^3*i/(
d*x + c)^3 + 420*A*a^4*b^10*c^4*d^4*g^3*i + 420*B*a^4*b^10*c^4*d^4*g^3*i + 2352*(b*x + a)*A*a^3*b^10*c^5*d^4*g
^3*i/(d*x + c) + 2352*(b*x + a)*B*a^3*b^10*c^5*d^4*g^3*i/(d*x + c) + 3528*(b*x + a)^2*A*a^2*b^10*c^6*d^4*g^3*i
/(d*x + c)^2 + 3528*(b*x + a)^2*B*a^2*b^10*c^6*d^4*g^3*i/(d*x + c)^2 + 1680*(b*x + a)^3*A*a*b^10*c^7*d^4*g^3*i
/(d*x + c)^3 + 1680*(b*x + a)^3*B*a*b^10*c^7*d^4*g^3*i/(d*x + c)^3 - 336*A*a^5*b^9*c^3*d^5*g^3*i - 336*B*a^5*b
^9*c^3*d^5*g^3*i - 2940*(b*x + a)*A*a^4*b^9*c^4*d^5*g^3*i/(d*x + c) - 2940*(b*x + a)*B*a^4*b^9*c^4*d^5*g^3*i/(
d*x + c) - 7056*(b*x + a)^2*A*a^3*b^9*c^5*d^5*g^3*i/(d*x + c)^2 - 7056*(b*x + a)^2*B*a^3*b^9*c^5*d^5*g^3*i/(d*
x + c)^2 - 5880*(b*x + a)^3*A*a^2*b^9*c^6*d^5*g^3*i/(d*x + c)^3 - 5880*(b*x + a)^3*B*a^2*b^9*c^6*d^5*g^3*i/(d*
x + c)^3 + 168*A*a^6*b^8*c^2*d^6*g^3*i + 168*B*a^6*b^8*c^2*d^6*g^3*i + 2352*(b*x + a)*A*a^5*b^8*c^3*d^6*g^3*i/
(d*x + c) + 2352*(b*x + a)*B*a^5*b^8*c^3*d^6*g^3*i/(d*x + c) + 8820*(b*x + a)^2*A*a^4*b^8*c^4*d^6*g^3*i/(d*x +
 c)^2 + 8820*(b*x + a)^2*B*a^4*b^8*c^4*d^6*g^3*i/(d*x + c)^2 + 11760*(b*x + a)^3*A*a^3*b^8*c^5*d^6*g^3*i/(d*x
+ c)^3 + 11760*(b*x + a)^3*B*a^3*b^8*c^5*d^6*g^3*i/(d*x + c)^3 - 48*A*a^7*b^7*c*d^7*g^3*i - 48*B*a^7*b^7*c*d^7
*g^3*i - 1176*(b*x + a)*A*a^6*b^7*c^2*d^7*g^3*i/(d*x + c) - 1176*(b*x + a)*B*a^6*b^7*c^2*d^7*g^3*i/(d*x + c) -
 7056*(b*x + a)^2*A*a^5*b^7*c^3*d^7*g^3*i/(d*x + c)^2 - 7056*(b*x + a)^2*B*a^5*b^7*c^3*d^7*g^3*i/(d*x + c)^2 -
 14700*(b*x + a)^3*A*a^4*b^7*c^4*d^7*g^3*i/(d*x + c)^3 - 14700*(b*x + a)^3*B*a^4*b^7*c^4*d^7*g^3*i/(d*x + c)^3
 + 6*A*a^8*b^6*d^8*g^3*i + 6*B*a^8*b^6*d^8*g^3*i + 336*(b*x + a)*A*a^7*b^6*c*d^8*g^3*i/(d*x + c) + 336*(b*x +
a)*B*a^7*b^6*c*d^8*g^3*i/(d*x + c) + 3528*(b*x + a)^2*A*a^6*b^6*c^2*d^8*g^3*i/(d*x + c)^2 + 3528*(b*x + a)^2*B
*a^6*b^6*c^2*d^8*g^3*i/(d*x + c)^2 + 11760*(b*x + a)^3*A*a^5*b^6*c^3*d^8*g^3*i/(d*x + c)^3 + 11760*(b*x + a)^3
*B*a^5*b^6*c^3*d^8*g^3*i/(d*x + c)^3 - 42*(b*x + a)*A*a^8*b^5*d^9*g^3*i/(d*x + c) - 42*(b*x + a)*B*a^8*b^5*d^9
*g^3*i/(d*x + c) - 1008*(b*x + a)^2*A*a^7*b^5*c*d^9*g^3*i/(d*x + c)^2 - 1008*(b*x + a)^2*B*a^7*b^5*c*d^9*g^3*i
/(d*x + c)^2 - 5880*(b*x + a)^3*A*a^6*b^5*c^2*d^9*g^3*i/(d*x + c)^3 - 5880*(b*x + a)^3*B*a^6*b^5*c^2*d^9*g^3*i
/(d*x + c)^3 + 126*(b*x + a)^2*A*a^8*b^4*d^10*g^3*i/(d*x + c)^2 + 126*(b*x + a)^2*B*a^8*b^4*d^10*g^3*i/(d*x +
c)^2 + 1680*(b*x + a)^3*A*a^7*b^4*c*d^10*g^3*i/(d*x + c)^3 + 1680*(b*x + a)^3*B*a^7*b^4*c*d^10*g^3*i/(d*x + c)
^3 - 210*(b*x + a)^3*A*a^8*b^3*d^11*g^3*i/(d*x + c)^3 - 210*(b*x + a)^3*B*a^8*b^3*d^11*g^3*i/(d*x + c)^3)/(b^1
0*d^4 - 7*(b*x + a)*b^9*d^5/(d*x + c) + 21*(b*x + a)^2*b^8*d^6/(d*x + c)^2 - 35*(b*x + a)^3*b^7*d^7/(d*x + c)^
3 + 35*(b*x + a)^4*b^6*d^8/(d*x + c)^4 - 21*(b*x + a)^5*b^5*d^9/(d*x + c)^5 + 7*(b*x + a)^6*b^4*d^10/(d*x + c)
^6 - (b*x + a)^7*b^3*d^11/(d*x + c)^7) + 6*(B*b^8*c^8*g^3*i*n - 8*B*a*b^7*c^7*d*g^3*i*n + 28*B*a^2*b^6*c^6*d^2
*g^3*i*n - 56*B*a^3*b^5*c^5*d^3*g^3*i*n + 70*B*a^4*b^4*c^4*d^4*g^3*i*n - 56*B*a^5*b^3*c^3*d^5*g^3*i*n + 28*B*a
^6*b^2*c^2*d^6*g^3*i*n - 8*B*a^7*b*c*d^7*g^3*i*n + B*a^8*d^8*g^3*i*n)*log(b - (b*x + a)*d/(d*x + c))/(b^4*d^4)
 - 6*(B*b^8*c^8*g^3*i*n - 8*B*a*b^7*c^7*d*g^3*i*n + 28*B*a^2*b^6*c^6*d^2*g^3*i*n - 56*B*a^3*b^5*c^5*d^3*g^3*i*
n + 70*B*a^4*b^4*c^4*d^4*g^3*i*n - 56*B*a^5*b^3*c^3*d^5*g^3*i*n + 28*B*a^6*b^2*c^2*d^6*g^3*i*n - 8*B*a^7*b*c*d
^7*g^3*i*n + B*a^8*d^8*g^3*i*n)*log((b*x + a)/(d*x + c))/(b^4*d^4))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.44, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^3*(d*i*x+c*i)^3*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

[Out]

int((b*g*x+a*g)^3*(d*i*x+c*i)^3*(B*ln(e*((b*x+a)/(d*x+c))^n)+A),x)

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maxima [B]  time = 1.89, size = 2901, normalized size = 6.08 \[ \text {result too large to display} \]

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

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mupad [B]  time = 6.56, size = 4476, normalized size = 9.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

x^4*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3*n - 3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c
*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d^2*n))/20 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*
b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b
*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3
))/(560*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a
*d + 140*b*c))/140))/(4*b*d)) + x^3*((g^3*i^3*(4*A*a^4*d^4 + 4*A*b^4*c^4 + B*a^4*d^4*n - B*b^4*c^4*n + 144*A*a
^2*b^2*c^2*d^2 + 64*A*a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3*d*n + 8*B*a^3*b*c*d^3*n))/(12*b*d) - ((14
0*a*d + 140*b*c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3*n - 3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d +
120*A*a^2*b*c*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d^2*n))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A
*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(1
40*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c
*d^2*g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^
3*i^3*(140*a*d + 140*b*c))/140))/(b*d)))/(420*b*d) + (a*c*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n -
 B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*
A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(3*b*d)) + x
^6*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/42 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/8
40) - x^2*(((140*a*d + 140*b*c)*((g^3*i^3*(4*A*a^4*d^4 + 4*A*b^4*c^4 + B*a^4*d^4*n - B*b^4*c^4*n + 144*A*a^2*b
^2*c^2*d^2 + 64*A*a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3*d*n + 8*B*a^3*b*c*d^3*n))/(4*b*d) - ((140*a*d
 + 140*b*c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3*n - 3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d + 120*A
*a^2*b*c*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d^2*n))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d
+ 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*
d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*
g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3
*(140*a*d + 140*b*c))/140))/(b*d)))/(140*b*d) + (a*c*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*
c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2
*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(b*d)))/(280*b*d)
 + (a*c*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3*n - 3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d + 120*A*a^2
*b*c*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d^2*n))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28
*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) -
 (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*
i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(14
0*a*d + 140*b*c))/140))/(b*d)))/(2*b*d) - (a*c*g^3*i^3*(4*A*a^3*d^3 + 4*A*b^3*c^3 + B*a^3*d^3*n - B*b^3*c^3*n
+ 24*A*a*b^2*c^2*d + 24*A*a^2*b*c*d^2 - 2*B*a*b^2*c^2*d*n + 2*B*a^2*b*c*d^2*n))/(2*b*d)) - x^5*((((b^2*d^2*g^3
*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 14
0*b*c))/(700*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/10
+ (A*a*b^2*c*d^2*g^3*i^3)/5) + x*(((140*a*d + 140*b*c)*(((140*a*d + 140*b*c)*((g^3*i^3*(4*A*a^4*d^4 + 4*A*b^4*
c^4 + B*a^4*d^4*n - B*b^4*c^4*n + 144*A*a^2*b^2*c^2*d^2 + 64*A*a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3*
d*n + 8*B*a^3*b*c*d^3*n))/(4*b*d) - ((140*a*d + 140*b*c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3*
n - 3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d^2*n))/5 + ((140*
a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d
+ 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*
b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c
 + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140))/(b*d)))/(140*b*d) + (a*c*((((b^2*d^2*
g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d +
 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/
2 + A*a*b^2*c*d^2*g^3*i^3))/(b*d)))/(140*b*d) + (a*c*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3*n -
3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d + 120*A*a^2*b*c*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d^2*n))/5 + ((140*a*d
+ 140*b*c)*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 14
0*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*
c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B
*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140))/(b*d)))/(b*d) - (a*c*g^3*i^3*(4*A*a^3*d^3
 + 4*A*b^3*c^3 + B*a^3*d^3*n - B*b^3*c^3*n + 24*A*a*b^2*c^2*d + 24*A*a^2*b*c*d^2 - 2*B*a*b^2*c^2*d*n + 2*B*a^2
*b*c*d^2*n))/(b*d)))/(140*b*d) - (a*c*((g^3*i^3*(4*A*a^4*d^4 + 4*A*b^4*c^4 + B*a^4*d^4*n - B*b^4*c^4*n + 144*A
*a^2*b^2*c^2*d^2 + 64*A*a*b^3*c^3*d + 64*A*a^3*b*c*d^3 - 8*B*a*b^3*c^3*d*n + 8*B*a^3*b*c*d^3*n))/(4*b*d) - ((1
40*a*d + 140*b*c)*((g^3*i^3*(20*A*a^3*d^3 + 20*A*b^3*c^3 + 3*B*a^3*d^3*n - 3*B*b^3*c^3*n + 120*A*a*b^2*c^2*d +
 120*A*a^2*b*c*d^2 - 6*B*a*b^2*c^2*d*n + 6*B*a^2*b*c*d^2*n))/5 + ((140*a*d + 140*b*c)*((((b^2*d^2*g^3*i^3*(28*
A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(
140*b*d) - (b*d*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*
c*d^2*g^3*i^3))/(140*b*d) - (a*c*((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n - B*b*c*n))/7 - (A*b^2*d^2*g
^3*i^3*(140*a*d + 140*b*c))/140))/(b*d)))/(140*b*d) + (a*c*((((b^2*d^2*g^3*i^3*(28*A*a*d + 28*A*b*c + B*a*d*n
- B*b*c*n))/7 - (A*b^2*d^2*g^3*i^3*(140*a*d + 140*b*c))/140)*(140*a*d + 140*b*c))/(140*b*d) - (b*d*g^3*i^3*(12
*A*a^2*d^2 + 12*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d))/2 + A*a*b^2*c*d^2*g^3*i^3))/(b*d)))/(b*
d) + (a^2*c^2*g^3*i^3*(12*A*a^2*d^2 + 12*A*b^2*c^2 + 3*B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d))/(2*b*d)) +
 log(e*((a + b*x)/(c + d*x))^n)*((B*g^3*i^3*x^4*(a^3*d^3 + b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2))/4 + B*a^3
*c^3*g^3*i^3*x + (B*b^3*d^3*g^3*i^3*x^7)/7 + (3*B*a^2*c^2*g^3*i^3*x^2*(a*d + b*c))/2 + (B*b^2*d^2*g^3*i^3*x^6*
(a*d + b*c))/2 + B*a*c*g^3*i^3*x^3*(a^2*d^2 + b^2*c^2 + 3*a*b*c*d) + (3*B*b*d*g^3*i^3*x^5*(a^2*d^2 + b^2*c^2 +
 3*a*b*c*d))/5) - (log(a + b*x)*(B*a^7*d^3*g^3*i^3*n - 35*B*a^4*b^3*c^3*g^3*i^3*n + 21*B*a^5*b^2*c^2*d*g^3*i^3
*n - 7*B*a^6*b*c*d^2*g^3*i^3*n))/(140*b^4) + (log(c + d*x)*(B*b^3*c^7*g^3*i^3*n - 35*B*a^3*c^4*d^3*g^3*i^3*n +
 21*B*a^2*b*c^5*d^2*g^3*i^3*n - 7*B*a*b^2*c^6*d*g^3*i^3*n))/(140*d^4) + (A*b^3*d^3*g^3*i^3*x^7)/7

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

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