3.2.28 \(\int (a g+b g x)^2 (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [128]
Optimal. Leaf size=387 \[ -\frac {B (b c-a d)^5 g^2 i^3 n x}{60 b^3 d^2}-\frac {B (b c-a d)^4 g^2 i^3 n (c+d x)^2}{120 b^2 d^3}-\frac {B (b c-a d)^3 g^2 i^3 n (c+d x)^3}{180 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 n (c+d x)^4}{120 d^3}-\frac {b B (b c-a d) g^2 i^3 n (c+d x)^5}{30 d^3}+\frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log (c+d x)}{60 b^4 d^3} \]
[Out]
-1/60*B*(-a*d+b*c)^5*g^2*i^3*n*x/b^3/d^2-1/120*B*(-a*d+b*c)^4*g^2*i^3*n*(d*x+c)^2/b^2/d^3-1/180*B*(-a*d+b*c)^3
*g^2*i^3*n*(d*x+c)^3/b/d^3+7/120*B*(-a*d+b*c)^2*g^2*i^3*n*(d*x+c)^4/d^3-1/30*b*B*(-a*d+b*c)*g^2*i^3*n*(d*x+c)^
5/d^3+1/4*(-a*d+b*c)^2*g^2*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-2/5*b*(-a*d+b*c)*g^2*i^3*(d*x+c)^
5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3+1/6*b^2*g^2*i^3*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-1/60*B*(-a
*d+b*c)^6*g^2*i^3*n*ln((b*x+a)/(d*x+c))/b^4/d^3-1/60*B*(-a*d+b*c)^6*g^2*i^3*n*ln(d*x+c)/b^4/d^3
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Rubi [A]
time = 0.25, antiderivative size = 387, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2561, 45, 2382,
12, 907} \begin {gather*} \frac {b^2 g^2 i^3 (c+d x)^6 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3}+\frac {g^2 i^3 (c+d x)^4 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^3}-\frac {2 b g^2 i^3 (c+d x)^5 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3}-\frac {B g^2 i^3 n (b c-a d)^6 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac {B g^2 i^3 n (b c-a d)^6 \log (c+d x)}{60 b^4 d^3}-\frac {B g^2 i^3 n x (b c-a d)^5}{60 b^3 d^2}-\frac {B g^2 i^3 n (c+d x)^2 (b c-a d)^4}{120 b^2 d^3}-\frac {B g^2 i^3 n (c+d x)^3 (b c-a d)^3}{180 b d^3}+\frac {7 B g^2 i^3 n (c+d x)^4 (b c-a d)^2}{120 d^3}-\frac {b B g^2 i^3 n (c+d x)^5 (b c-a d)}{30 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]
[Out]
-1/60*(B*(b*c - a*d)^5*g^2*i^3*n*x)/(b^3*d^2) - (B*(b*c - a*d)^4*g^2*i^3*n*(c + d*x)^2)/(120*b^2*d^3) - (B*(b*
c - a*d)^3*g^2*i^3*n*(c + d*x)^3)/(180*b*d^3) + (7*B*(b*c - a*d)^2*g^2*i^3*n*(c + d*x)^4)/(120*d^3) - (b*B*(b*
c - a*d)*g^2*i^3*n*(c + d*x)^5)/(30*d^3) + ((b*c - a*d)^2*g^2*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]))/(4*d^3) - (2*b*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^3) + (b^2*
g^2*i^3*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^3) - (B*(b*c - a*d)^6*g^2*i^3*n*Log[(a + b*x)
/(c + d*x)])/(60*b^4*d^3) - (B*(b*c - a*d)^6*g^2*i^3*n*Log[c + d*x])/(60*b^4*d^3)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 45
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 907
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))
Rule 2382
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[
x^m*(d + e*x)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ
[{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
Rule 2561
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
Rubi steps
\begin {align*} \int (128 c+128 d x)^3 (a g+b g x)^2 \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)^2 g^2 (128 c+128 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^2}-\frac {b (b c-a d) g^2 (128 c+128 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{64 d^2}+\frac {b^2 g^2 (128 c+128 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{16384 d^2}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int (128 c+128 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{16384 d^2}-\frac {\left (b (b c-a d) g^2\right ) \int (128 c+128 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{64 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int (128 c+128 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d^2}\\ &=\frac {524288 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {4194304 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {\left (b^2 B g^2 n\right ) \int \frac {4398046511104 (b c-a d) (c+d x)^5}{a+b x} \, dx}{12582912 d^3}+\frac {\left (b B (b c-a d) g^2 n\right ) \int \frac {34359738368 (b c-a d) (c+d x)^4}{a+b x} \, dx}{40960 d^3}-\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {268435456 (b c-a d) (c+d x)^3}{a+b x} \, dx}{512 d^3}\\ &=\frac {524288 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {4194304 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {\left (1048576 b^2 B (b c-a d) g^2 n\right ) \int \frac {(c+d x)^5}{a+b x} \, dx}{3 d^3}+\frac {\left (4194304 b B (b c-a d)^2 g^2 n\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^3}-\frac {\left (524288 B (b c-a d)^3 g^2 n\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{d^3}\\ &=\frac {524288 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {4194304 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}-\frac {\left (1048576 b^2 B (b c-a d) g^2 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}{3 d^3}+\frac {\left (4194304 b B (b c-a d)^2 g^2 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^3}-\frac {\left (524288 B (b c-a d)^3 g^2 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}{d^3}\\ &=-\frac {524288 B (b c-a d)^5 g^2 n x}{15 b^3 d^2}-\frac {262144 B (b c-a d)^4 g^2 n (c+d x)^2}{15 b^2 d^3}-\frac {524288 B (b c-a d)^3 g^2 n (c+d x)^3}{45 b d^3}+\frac {1835008 B (b c-a d)^2 g^2 n (c+d x)^4}{15 d^3}-\frac {1048576 b B (b c-a d) g^2 n (c+d x)^5}{15 d^3}-\frac {524288 B (b c-a d)^6 g^2 n \log (a+b x)}{15 b^4 d^3}+\frac {524288 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac {4194304 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {1048576 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 441, normalized size = 1.14 \begin {gather*} \frac {g^2 i^3 \left (-15 B (b c-a d)^3 n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+12 B (b c-a d)^2 n \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )-B (b c-a d) n \left (60 b d (b c-a d)^4 x+30 b^2 (b c-a d)^3 (c+d x)^2+20 b^3 (b c-a d)^2 (c+d x)^3+15 b^4 (b c-a d) (c+d x)^4+12 b^5 (c+d x)^5+60 (b c-a d)^5 \log (a+b x)\right )+90 b^4 (b c-a d)^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-144 b^5 (b c-a d) (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+60 b^6 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{360 b^4 d^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]),x]
[Out]
(g^2*i^3*(-15*B*(b*c - a*d)^3*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6
*(b*c - a*d)^3*Log[a + b*x]) + 12*B*(b*c - a*d)^2*n*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2
+ 4*b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]) - B*(b*c - a*d)*n*(60*b*d
*(b*c - a*d)^4*x + 30*b^2*(b*c - a*d)^3*(c + d*x)^2 + 20*b^3*(b*c - a*d)^2*(c + d*x)^3 + 15*b^4*(b*c - a*d)*(c
+ d*x)^4 + 12*b^5*(c + d*x)^5 + 60*(b*c - a*d)^5*Log[a + b*x]) + 90*b^4*(b*c - a*d)^2*(c + d*x)^4*(A + B*Log[
e*((a + b*x)/(c + d*x))^n]) - 144*b^5*(b*c - a*d)*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 60*b^6*
(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(360*b^4*d^3)
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Maple [F]
time = 0.19, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*g*x+a*g)^2*(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)^2*(d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x)
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 1890 vs. \(2 (340) = 680\).
time = 0.32, size = 1890, normalized size = 4.88 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="maxima")
[Out]
-1/6*I*B*b^2*d^3*g^2*x^6*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/6*I*A*b^2*d^3*g^2*x^6 - 3/5*I*B*b^2*c*d^2*
g^2*x^5*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 2/5*I*B*a*b*d^3*g^2*x^5*log((b*x/(d*x + c) + a/(d*x + c))^n*e
) - 3/5*I*A*b^2*c*d^2*g^2*x^5 - 2/5*I*A*a*b*d^3*g^2*x^5 - 3/4*I*B*b^2*c^2*d*g^2*x^4*log((b*x/(d*x + c) + a/(d*
x + c))^n*e) - 3/2*I*B*a*b*c*d^2*g^2*x^4*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/4*I*B*a^2*d^3*g^2*x^4*log(
(b*x/(d*x + c) + a/(d*x + c))^n*e) - 3/4*I*A*b^2*c^2*d*g^2*x^4 - 3/2*I*A*a*b*c*d^2*g^2*x^4 - 1/4*I*A*a^2*d^3*g
^2*x^4 - 1/3*I*B*b^2*c^3*g^2*x^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 2*I*B*a*b*c^2*d*g^2*x^3*log((b*x/(d*
x + c) + a/(d*x + c))^n*e) - I*B*a^2*c*d^2*g^2*x^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/3*I*A*b^2*c^3*g^
2*x^3 - 2*I*A*a*b*c^2*d*g^2*x^3 - I*A*a^2*c*d^2*g^2*x^3 - I*B*a*b*c^3*g^2*x^2*log((b*x/(d*x + c) + a/(d*x + c)
)^n*e) - 3/2*I*B*a^2*c^2*d*g^2*x^2*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - I*A*a*b*c^3*g^2*x^2 - 3/2*I*A*a^2*
c^2*d*g^2*x^2 + 1/360*I*B*b^2*d^3*g^2*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)) - 1/20*I*B*b^2*c*d^2*g^2*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)) - 1/30*I*B*a*b*d^3*g^2*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*lo
g(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)) + 1/8*I*B*b^2*c^2*d*g^2*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
)) + 1/4*I*B*a*b*c*d^2*g^2*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*I*B*a^2*d^3*g^2*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)) - 1/6*I*B*b^2*c^3*g^2*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)) - I*B*a*b*c^2*d*g^2*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)) - 1/2*I*B*a^2*c*d^2*g^2*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)
) + I*B*a*b*c^3*g^2*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*I*B*a^2*c^2*d*
g^2*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) - I*B*a^2*c^3*g^2*n*(a*log(b*x + a)/
b - c*log(d*x + c)/d) - I*B*a^2*c^3*g^2*x*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - I*A*a^2*c^3*g^2*x
________________________________________________________________________________________
Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 894 vs. \(2 (340) = 680\).
time = 0.58, size = 894, normalized size = 2.31 \begin {gather*} -\frac {60 \, {\left (i \, A + i \, B\right )} b^{6} d^{6} g^{2} x^{6} + 12 \, {\left ({\left (-i \, B b^{6} c d^{5} + i \, B a b^{5} d^{6}\right )} g^{2} n + 6 \, {\left (3 \, {\left (i \, A + i \, B\right )} b^{6} c d^{5} + 2 \, {\left (i \, A + i \, B\right )} a b^{5} d^{6}\right )} g^{2}\right )} x^{5} + 3 \, {\left ({\left (-13 i \, B b^{6} c^{2} d^{4} + 6 i \, B a b^{5} c d^{5} + 7 i \, B a^{2} b^{4} d^{6}\right )} g^{2} n + 30 \, {\left (3 \, {\left (i \, A + i \, B\right )} b^{6} c^{2} d^{4} + 6 \, {\left (i \, A + i \, B\right )} a b^{5} c d^{5} + {\left (i \, A + i \, B\right )} a^{2} b^{4} d^{6}\right )} g^{2}\right )} x^{4} + 6 \, {\left (20 i \, B a^{3} b^{3} c^{3} d^{3} - 15 i \, B a^{4} b^{2} c^{2} d^{4} + 6 i \, B a^{5} b c d^{5} - i \, B a^{6} d^{6}\right )} g^{2} n \log \left (\frac {b x + a}{b}\right ) + 6 \, {\left (-i \, B b^{6} c^{6} + 6 i \, B a b^{5} c^{5} d - 15 i \, B a^{2} b^{4} c^{4} d^{2}\right )} g^{2} n \log \left (\frac {d x + c}{d}\right ) + 2 \, {\left ({\left (-19 i \, B b^{6} c^{3} d^{3} - 21 i \, B a b^{5} c^{2} d^{4} + 39 i \, B a^{2} b^{4} c d^{5} + i \, B a^{3} b^{3} d^{6}\right )} g^{2} n + 60 \, {\left ({\left (i \, A + i \, B\right )} b^{6} c^{3} d^{3} + 6 \, {\left (i \, A + i \, B\right )} a b^{5} c^{2} d^{4} + 3 \, {\left (i \, A + i \, B\right )} a^{2} b^{4} c d^{5}\right )} g^{2}\right )} x^{3} + 3 \, {\left ({\left (-i \, B b^{6} c^{4} d^{2} - 34 i \, B a b^{5} c^{3} d^{3} + 30 i \, B a^{2} b^{4} c^{2} d^{4} + 6 i \, B a^{3} b^{3} c d^{5} - i \, B a^{4} b^{2} d^{6}\right )} g^{2} n + 60 \, {\left (2 \, {\left (i \, A + i \, B\right )} a b^{5} c^{3} d^{3} + 3 \, {\left (i \, A + i \, B\right )} a^{2} b^{4} c^{2} d^{4}\right )} g^{2}\right )} x^{2} + 6 \, {\left (60 \, {\left (i \, A + i \, B\right )} a^{2} b^{4} c^{3} d^{3} g^{2} + {\left (i \, B b^{6} c^{5} d - 6 i \, B a b^{5} c^{4} d^{2} - 5 i \, B a^{2} b^{4} c^{3} d^{3} + 15 i \, B a^{3} b^{3} c^{2} d^{4} - 6 i \, B a^{4} b^{2} c d^{5} + i \, B a^{5} b d^{6}\right )} g^{2} n\right )} x + 6 \, {\left (10 i \, B b^{6} d^{6} g^{2} n x^{6} + 60 i \, B a^{2} b^{4} c^{3} d^{3} g^{2} n x + 12 \, {\left (3 i \, B b^{6} c d^{5} + 2 i \, B a b^{5} d^{6}\right )} g^{2} n x^{5} + 15 \, {\left (3 i \, B b^{6} c^{2} d^{4} + 6 i \, B a b^{5} c d^{5} + i \, B a^{2} b^{4} d^{6}\right )} g^{2} n x^{4} + 20 \, {\left (i \, B b^{6} c^{3} d^{3} + 6 i \, B a b^{5} c^{2} d^{4} + 3 i \, B a^{2} b^{4} c d^{5}\right )} g^{2} n x^{3} + 30 \, {\left (2 i \, B a b^{5} c^{3} d^{3} + 3 i \, B a^{2} b^{4} c^{2} d^{4}\right )} g^{2} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{360 \, b^{4} d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="fricas")
[Out]
-1/360*(60*(I*A + I*B)*b^6*d^6*g^2*x^6 + 12*((-I*B*b^6*c*d^5 + I*B*a*b^5*d^6)*g^2*n + 6*(3*(I*A + I*B)*b^6*c*d
^5 + 2*(I*A + I*B)*a*b^5*d^6)*g^2)*x^5 + 3*((-13*I*B*b^6*c^2*d^4 + 6*I*B*a*b^5*c*d^5 + 7*I*B*a^2*b^4*d^6)*g^2*
n + 30*(3*(I*A + I*B)*b^6*c^2*d^4 + 6*(I*A + I*B)*a*b^5*c*d^5 + (I*A + I*B)*a^2*b^4*d^6)*g^2)*x^4 + 6*(20*I*B*
a^3*b^3*c^3*d^3 - 15*I*B*a^4*b^2*c^2*d^4 + 6*I*B*a^5*b*c*d^5 - I*B*a^6*d^6)*g^2*n*log((b*x + a)/b) + 6*(-I*B*b
^6*c^6 + 6*I*B*a*b^5*c^5*d - 15*I*B*a^2*b^4*c^4*d^2)*g^2*n*log((d*x + c)/d) + 2*((-19*I*B*b^6*c^3*d^3 - 21*I*B
*a*b^5*c^2*d^4 + 39*I*B*a^2*b^4*c*d^5 + I*B*a^3*b^3*d^6)*g^2*n + 60*((I*A + I*B)*b^6*c^3*d^3 + 6*(I*A + I*B)*a
*b^5*c^2*d^4 + 3*(I*A + I*B)*a^2*b^4*c*d^5)*g^2)*x^3 + 3*((-I*B*b^6*c^4*d^2 - 34*I*B*a*b^5*c^3*d^3 + 30*I*B*a^
2*b^4*c^2*d^4 + 6*I*B*a^3*b^3*c*d^5 - I*B*a^4*b^2*d^6)*g^2*n + 60*(2*(I*A + I*B)*a*b^5*c^3*d^3 + 3*(I*A + I*B)
*a^2*b^4*c^2*d^4)*g^2)*x^2 + 6*(60*(I*A + I*B)*a^2*b^4*c^3*d^3*g^2 + (I*B*b^6*c^5*d - 6*I*B*a*b^5*c^4*d^2 - 5*
I*B*a^2*b^4*c^3*d^3 + 15*I*B*a^3*b^3*c^2*d^4 - 6*I*B*a^4*b^2*c*d^5 + I*B*a^5*b*d^6)*g^2*n)*x + 6*(10*I*B*b^6*d
^6*g^2*n*x^6 + 60*I*B*a^2*b^4*c^3*d^3*g^2*n*x + 12*(3*I*B*b^6*c*d^5 + 2*I*B*a*b^5*d^6)*g^2*n*x^5 + 15*(3*I*B*b
^6*c^2*d^4 + 6*I*B*a*b^5*c*d^5 + I*B*a^2*b^4*d^6)*g^2*n*x^4 + 20*(I*B*b^6*c^3*d^3 + 6*I*B*a*b^5*c^2*d^4 + 3*I*
B*a^2*b^4*c*d^5)*g^2*n*x^3 + 30*(2*I*B*a*b^5*c^3*d^3 + 3*I*B*a^2*b^4*c^2*d^4)*g^2*n*x^2)*log((b*x + a)/(d*x +
c)))/(b^4*d^3)
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)**2*(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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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 3942 vs. \(2 (340) = 680\).
time = 6.48, size = 3942, normalized size = 10.19 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")
[Out]
-1/360*(6*(I*B*b^9*c^7*g^2*n - 7*I*B*a*b^8*c^6*d*g^2*n - 6*(I*b*x + I*a)*B*b^8*c^7*d*g^2*n/(d*x + c) + 21*I*B*
a^2*b^7*c^5*d^2*g^2*n - 42*(-I*b*x - I*a)*B*a*b^7*c^6*d^2*g^2*n/(d*x + c) + 15*I*(b*x + a)^2*B*b^7*c^7*d^2*g^2
*n/(d*x + c)^2 - 35*I*B*a^3*b^6*c^4*d^3*g^2*n - 126*(I*b*x + I*a)*B*a^2*b^6*c^5*d^3*g^2*n/(d*x + c) - 105*I*(b
*x + a)^2*B*a*b^6*c^6*d^3*g^2*n/(d*x + c)^2 + 35*I*B*a^4*b^5*c^3*d^4*g^2*n - 210*(-I*b*x - I*a)*B*a^3*b^5*c^4*
d^4*g^2*n/(d*x + c) + 315*I*(b*x + a)^2*B*a^2*b^5*c^5*d^4*g^2*n/(d*x + c)^2 - 21*I*B*a^5*b^4*c^2*d^5*g^2*n - 2
10*(I*b*x + I*a)*B*a^4*b^4*c^3*d^5*g^2*n/(d*x + c) - 525*I*(b*x + a)^2*B*a^3*b^4*c^4*d^5*g^2*n/(d*x + c)^2 + 7
*I*B*a^6*b^3*c*d^6*g^2*n - 126*(-I*b*x - I*a)*B*a^5*b^3*c^2*d^6*g^2*n/(d*x + c) + 525*I*(b*x + a)^2*B*a^4*b^3*
c^3*d^6*g^2*n/(d*x + c)^2 - I*B*a^7*b^2*d^7*g^2*n - 42*(I*b*x + I*a)*B*a^6*b^2*c*d^7*g^2*n/(d*x + c) - 315*I*(
b*x + a)^2*B*a^5*b^2*c^2*d^7*g^2*n/(d*x + c)^2 - 6*(-I*b*x - I*a)*B*a^7*b*d^8*g^2*n/(d*x + c) + 105*I*(b*x + a
)^2*B*a^6*b*c*d^8*g^2*n/(d*x + c)^2 - 15*I*(b*x + a)^2*B*a^7*d^9*g^2*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/(
b^6*d^3 - 6*(b*x + a)*b^5*d^4/(d*x + c) + 15*(b*x + a)^2*b^4*d^5/(d*x + c)^2 - 20*(b*x + a)^3*b^3*d^6/(d*x + c
)^3 + 15*(b*x + a)^4*b^2*d^7/(d*x + c)^4 - 6*(b*x + a)^5*b*d^8/(d*x + c)^5 + (b*x + a)^6*d^9/(d*x + c)^6) + (-
2*I*B*b^12*c^7*g^2*n + 14*I*B*a*b^11*c^6*d*g^2*n - 18*(-I*b*x - I*a)*B*b^11*c^7*d*g^2*n/(d*x + c) - 42*I*B*a^2
*b^10*c^5*d^2*g^2*n - 126*(I*b*x + I*a)*B*a*b^10*c^6*d^2*g^2*n/(d*x + c) - 63*I*(b*x + a)^2*B*b^10*c^7*d^2*g^2
*n/(d*x + c)^2 + 70*I*B*a^3*b^9*c^4*d^3*g^2*n - 378*(-I*b*x - I*a)*B*a^2*b^9*c^5*d^3*g^2*n/(d*x + c) + 441*I*(
b*x + a)^2*B*a*b^9*c^6*d^3*g^2*n/(d*x + c)^2 + 74*I*(b*x + a)^3*B*b^9*c^7*d^3*g^2*n/(d*x + c)^3 - 70*I*B*a^4*b
^8*c^3*d^4*g^2*n - 630*(I*b*x + I*a)*B*a^3*b^8*c^4*d^4*g^2*n/(d*x + c) - 1323*I*(b*x + a)^2*B*a^2*b^8*c^5*d^4*
g^2*n/(d*x + c)^2 - 518*I*(b*x + a)^3*B*a*b^8*c^6*d^4*g^2*n/(d*x + c)^3 - 33*I*(b*x + a)^4*B*b^8*c^7*d^4*g^2*n
/(d*x + c)^4 + 42*I*B*a^5*b^7*c^2*d^5*g^2*n - 630*(-I*b*x - I*a)*B*a^4*b^7*c^3*d^5*g^2*n/(d*x + c) + 2205*I*(b
*x + a)^2*B*a^3*b^7*c^4*d^5*g^2*n/(d*x + c)^2 + 1554*I*(b*x + a)^3*B*a^2*b^7*c^5*d^5*g^2*n/(d*x + c)^3 + 231*I
*(b*x + a)^4*B*a*b^7*c^6*d^5*g^2*n/(d*x + c)^4 + 6*I*(b*x + a)^5*B*b^7*c^7*d^5*g^2*n/(d*x + c)^5 - 14*I*B*a^6*
b^6*c*d^6*g^2*n - 378*(I*b*x + I*a)*B*a^5*b^6*c^2*d^6*g^2*n/(d*x + c) - 2205*I*(b*x + a)^2*B*a^4*b^6*c^3*d^6*g
^2*n/(d*x + c)^2 - 2590*I*(b*x + a)^3*B*a^3*b^6*c^4*d^6*g^2*n/(d*x + c)^3 - 693*I*(b*x + a)^4*B*a^2*b^6*c^5*d^
6*g^2*n/(d*x + c)^4 - 42*I*(b*x + a)^5*B*a*b^6*c^6*d^6*g^2*n/(d*x + c)^5 + 2*I*B*a^7*b^5*d^7*g^2*n - 126*(-I*b
*x - I*a)*B*a^6*b^5*c*d^7*g^2*n/(d*x + c) + 1323*I*(b*x + a)^2*B*a^5*b^5*c^2*d^7*g^2*n/(d*x + c)^2 + 2590*I*(b
*x + a)^3*B*a^4*b^5*c^3*d^7*g^2*n/(d*x + c)^3 + 1155*I*(b*x + a)^4*B*a^3*b^5*c^4*d^7*g^2*n/(d*x + c)^4 + 126*I
*(b*x + a)^5*B*a^2*b^5*c^5*d^7*g^2*n/(d*x + c)^5 - 18*(I*b*x + I*a)*B*a^7*b^4*d^8*g^2*n/(d*x + c) - 441*I*(b*x
+ a)^2*B*a^6*b^4*c*d^8*g^2*n/(d*x + c)^2 - 1554*I*(b*x + a)^3*B*a^5*b^4*c^2*d^8*g^2*n/(d*x + c)^3 - 1155*I*(b
*x + a)^4*B*a^4*b^4*c^3*d^8*g^2*n/(d*x + c)^4 - 210*I*(b*x + a)^5*B*a^3*b^4*c^4*d^8*g^2*n/(d*x + c)^5 + 63*I*(
b*x + a)^2*B*a^7*b^3*d^9*g^2*n/(d*x + c)^2 + 518*I*(b*x + a)^3*B*a^6*b^3*c*d^9*g^2*n/(d*x + c)^3 + 693*I*(b*x
+ a)^4*B*a^5*b^3*c^2*d^9*g^2*n/(d*x + c)^4 + 210*I*(b*x + a)^5*B*a^4*b^3*c^3*d^9*g^2*n/(d*x + c)^5 - 74*I*(b*x
+ a)^3*B*a^7*b^2*d^10*g^2*n/(d*x + c)^3 - 231*I*(b*x + a)^4*B*a^6*b^2*c*d^10*g^2*n/(d*x + c)^4 - 126*I*(b*x +
a)^5*B*a^5*b^2*c^2*d^10*g^2*n/(d*x + c)^5 + 33*I*(b*x + a)^4*B*a^7*b*d^11*g^2*n/(d*x + c)^4 + 42*I*(b*x + a)^
5*B*a^6*b*c*d^11*g^2*n/(d*x + c)^5 - 6*I*(b*x + a)^5*B*a^7*d^12*g^2*n/(d*x + c)^5 + 6*I*A*b^12*c^7*g^2 + 6*I*B
*b^12*c^7*g^2 - 42*I*A*a*b^11*c^6*d*g^2 - 42*I*B*a*b^11*c^6*d*g^2 - 36*(I*b*x + I*a)*A*b^11*c^7*d*g^2/(d*x + c
) - 36*(I*b*x + I*a)*B*b^11*c^7*d*g^2/(d*x + c) + 126*I*A*a^2*b^10*c^5*d^2*g^2 + 126*I*B*a^2*b^10*c^5*d^2*g^2
- 252*(-I*b*x - I*a)*A*a*b^10*c^6*d^2*g^2/(d*x + c) - 252*(-I*b*x - I*a)*B*a*b^10*c^6*d^2*g^2/(d*x + c) + 90*I
*(b*x + a)^2*A*b^10*c^7*d^2*g^2/(d*x + c)^2 + 90*I*(b*x + a)^2*B*b^10*c^7*d^2*g^2/(d*x + c)^2 - 210*I*A*a^3*b^
9*c^4*d^3*g^2 - 210*I*B*a^3*b^9*c^4*d^3*g^2 - 756*(I*b*x + I*a)*A*a^2*b^9*c^5*d^3*g^2/(d*x + c) - 756*(I*b*x +
I*a)*B*a^2*b^9*c^5*d^3*g^2/(d*x + c) - 630*I*(b*x + a)^2*A*a*b^9*c^6*d^3*g^2/(d*x + c)^2 - 630*I*(b*x + a)^2*
B*a*b^9*c^6*d^3*g^2/(d*x + c)^2 + 210*I*A*a^4*b^8*c^3*d^4*g^2 + 210*I*B*a^4*b^8*c^3*d^4*g^2 - 1260*(-I*b*x - I
*a)*A*a^3*b^8*c^4*d^4*g^2/(d*x + c) - 1260*(-I*b*x - I*a)*B*a^3*b^8*c^4*d^4*g^2/(d*x + c) + 1890*I*(b*x + a)^2
*A*a^2*b^8*c^5*d^4*g^2/(d*x + c)^2 + 1890*I*(b*x + a)^2*B*a^2*b^8*c^5*d^4*g^2/(d*x + c)^2 - 126*I*A*a^5*b^7*c^
2*d^5*g^2 - 126*I*B*a^5*b^7*c^2*d^5*g^2 - 1260*(I*b*x + I*a)*A*a^4*b^7*c^3*d^5*g^2/(d*x + c) - 1260*(I*b*x + I
*a)*B*a^4*b^7*c^3*d^5*g^2/(d*x + c) - 3150*I*(b*x + a)^2*A*a^3*b^7*c^4*d^5*g^2/(d*x + c)^2 - 3150*I*(b*x + a)^
2*B*a^3*b^7*c^4*d^5*g^2/(d*x + c)^2 + 42*I*A*a^...
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Mupad [B]
time = 6.26, size = 2547, normalized size = 6.58 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n)),x)
[Out]
x^2*((a*c*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))
/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 6
0*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(2*b*d) - ((60*a*d + 60*b*c)*((g^2*i^3*(4*A*a^3*d^3 + 16
*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b
*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^
2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a
^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(60*b*d) - (a*c*((b*d^2*g^2*
i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60))/(b*d)))/(120*b*d)
+ (c*g^2*i^3*(12*A*a^3*d^3 + 3*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^3*n + 36*A*a*b^2*c^2*d + 54*A*a^2*b*c*d^2 -
5*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d^2*n))/(6*b*d)) + x^3*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n -
3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(12*b) + ((60*a*
d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c
))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n +
60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(180*b*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c
+ B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60))/(3*b*d)) - x^4*((((b*d^2*g^2*i^3*(18*A*a*d
+ 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(240*b*d) - (d
*g^2*i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/20 + (A*a
*b*c*d^2*g^2*i^3)/4) + x^5*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/30 - (A*b*d^2*g^2*i^3*(6
0*a*d + 60*b*c))/300) - x*(((60*a*d + 60*b*c)*((a*c*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n
))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2 + 30*A*b
^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(b*d) - ((60*a
*d + 60*b*c)*((g^2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2
*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24
*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*
i^3*(15*A*a^2*d^2 + 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^
2*g^2*i^3))/(60*b*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(6
0*a*d + 60*b*c))/60))/(b*d)))/(60*b*d) + (c*g^2*i^3*(12*A*a^3*d^3 + 3*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^3*n
+ 36*A*a*b^2*c^2*d + 54*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d^2*n))/(3*b*d)))/(60*b*d) + (a*c*((g^
2*i^3*(4*A*a^3*d^3 + 16*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 72*A*a*b^2*c^2*d + 48*A*a^2*b*c*d^2 - 3*B*a*
b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(4*b) + ((60*a*d + 60*b*c)*((((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n
- B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (d*g^2*i^3*(15*A*a^2*d^2
+ 30*A*b^2*c^2 + 2*B*a^2*d^2*n - 3*B*b^2*c^2*n + 60*A*a*b*c*d + B*a*b*c*d*n))/5 + A*a*b*c*d^2*g^2*i^3))/(60*b
*d) - (a*c*((b*d^2*g^2*i^3*(18*A*a*d + 24*A*b*c + B*a*d*n - B*b*c*n))/6 - (A*b*d^2*g^2*i^3*(60*a*d + 60*b*c))/
60))/(b*d)))/(b*d) - (a*c^2*g^2*i^3*(12*A*a^2*d^2 + 6*A*b^2*c^2 + 3*B*a^2*d^2*n - 2*B*b^2*c^2*n + 24*A*a*b*c*d
- B*a*b*c*d*n))/(2*b*d)) + log(e*((a + b*x)/(c + d*x))^n)*(B*a^2*c^3*g^2*i^3*x + (B*c*g^2*i^3*x^3*(3*a^2*d^2
+ b^2*c^2 + 6*a*b*c*d))/3 + (B*d*g^2*i^3*x^4*(a^2*d^2 + 3*b^2*c^2 + 6*a*b*c*d))/4 + (B*b^2*d^3*g^2*i^3*x^6)/6
+ (B*a*c^2*g^2*i^3*x^2*(3*a*d + 2*b*c))/2 + (B*b*d^2*g^2*i^3*x^5*(2*a*d + 3*b*c))/5) - (log(a + b*x)*(B*a^6*d^
3*g^2*i^3*n - 20*B*a^3*b^3*c^3*g^2*i^3*n + 15*B*a^4*b^2*c^2*d*g^2*i^3*n - 6*B*a^5*b*c*d^2*g^2*i^3*n))/(60*b^4)
- (log(c + d*x)*(B*b^2*c^6*g^2*i^3*n + 15*B*a^2*c^4*d^2*g^2*i^3*n - 6*B*a*b*c^5*d*g^2*i^3*n))/(60*d^3) + (A*b
^2*d^3*g^2*i^3*x^6)/6
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