3.2.29 \(\int (a g+b g x) (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [129]

Optimal. Leaf size=283 \[ \frac {B (b c-a d)^4 g i^3 n x}{20 b^3 d}+\frac {B (b c-a d)^3 g i^3 n (c+d x)^2}{40 b^2 d^2}+\frac {B (b c-a d)^2 g i^3 n (c+d x)^3}{60 b d^2}-\frac {B (b c-a d) g i^3 n (c+d x)^4}{20 d^2}-\frac {(b c-a d) g i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}+\frac {B (b c-a d)^5 g i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{20 b^4 d^2}+\frac {B (b c-a d)^5 g i^3 n \log (c+d x)}{20 b^4 d^2} \]

[Out]

1/20*B*(-a*d+b*c)^4*g*i^3*n*x/b^3/d+1/40*B*(-a*d+b*c)^3*g*i^3*n*(d*x+c)^2/b^2/d^2+1/60*B*(-a*d+b*c)^2*g*i^3*n*
(d*x+c)^3/b/d^2-1/20*B*(-a*d+b*c)*g*i^3*n*(d*x+c)^4/d^2-1/4*(-a*d+b*c)*g*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x
+c))^n))/d^2+1/5*b*g*i^3*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^2+1/20*B*(-a*d+b*c)^5*g*i^3*n*ln((b*x+a)/
(d*x+c))/b^4/d^2+1/20*B*(-a*d+b*c)^5*g*i^3*n*ln(d*x+c)/b^4/d^2

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Rubi [A]
time = 0.17, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2561, 45, 2382, 12, 78} \begin {gather*} -\frac {g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2}+\frac {b g i^3 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^2}+\frac {B g i^3 n (b c-a d)^5 \log \left (\frac {a+b x}{c+d x}\right )}{20 b^4 d^2}+\frac {B g i^3 n (b c-a d)^5 \log (c+d x)}{20 b^4 d^2}+\frac {B g i^3 n x (b c-a d)^4}{20 b^3 d}+\frac {B g i^3 n (c+d x)^2 (b c-a d)^3}{40 b^2 d^2}+\frac {B g i^3 n (c+d x)^3 (b c-a d)^2}{60 b d^2}-\frac {B g i^3 n (c+d x)^4 (b c-a d)}{20 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^4*g*i^3*n*x)/(20*b^3*d) + (B*(b*c - a*d)^3*g*i^3*n*(c + d*x)^2)/(40*b^2*d^2) + (B*(b*c - a*d)^2
*g*i^3*n*(c + d*x)^3)/(60*b*d^2) - (B*(b*c - a*d)*g*i^3*n*(c + d*x)^4)/(20*d^2) - ((b*c - a*d)*g*i^3*(c + d*x)
^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^2) + (b*g*i^3*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/(5*d^2) + (B*(b*c - a*d)^5*g*i^3*n*Log[(a + b*x)/(c + d*x)])/(20*b^4*d^2) + (B*(b*c - a*d)^5*g*i^3*n*Log[c
+ d*x])/(20*b^4*d^2)

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 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

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 (129 c+129 d x)^3 (a g+b g x) \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) g (129 c+129 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac {b g (129 c+129 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{129 d}\right ) \, dx\\ &=\frac {(b g) \int (129 c+129 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{129 d}+\frac {((-b c+a d) g) \int (129 c+129 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}\\ &=-\frac {2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac {(b B g n) \int \frac {35723051649 (b c-a d) (c+d x)^4}{a+b x} \, dx}{83205 d^2}+\frac {(B (b c-a d) g n) \int \frac {276922881 (b c-a d) (c+d x)^3}{a+b x} \, dx}{516 d^2}\\ &=-\frac {2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac {(2146689 b B (b c-a d) g n) \int \frac {(c+d x)^4}{a+b x} \, dx}{5 d^2}+\frac {\left (2146689 B (b c-a d)^2 g n\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{4 d^2}\\ &=-\frac {2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}-\frac {(2146689 b B (b c-a d) g n) \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^2}+\frac {\left (2146689 B (b c-a d)^2 g 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^2}\\ &=\frac {2146689 B (b c-a d)^4 g n x}{20 b^3 d}+\frac {2146689 B (b c-a d)^3 g n (c+d x)^2}{40 b^2 d^2}+\frac {715563 B (b c-a d)^2 g n (c+d x)^3}{20 b d^2}-\frac {2146689 B (b c-a d) g n (c+d x)^4}{20 d^2}+\frac {2146689 B (b c-a d)^5 g n \log (a+b x)}{20 b^4 d^2}-\frac {2146689 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {2146689 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^2}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 269, normalized size = 0.95 \begin {gather*} \frac {g i^3 \left (\frac {5 B (b c-a d)^2 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 )}{b^4}-\frac {2 B (b c-a d) 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^4}-30 (b c-a d) (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+24 b (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{120 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g*i^3*((5*B*(b*c - a*d)^2*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]))/b^4 - (2*B*(b*c - a*d)*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^4 - 30*(b*c - a*d)*(c
+ d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*b*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])))/(1
20*d^2)

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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{3} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)*(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)*(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. 1061 vs. \(2 (245) = 490\).
time = 0.31, size = 1061, normalized size = 3.75 \begin {gather*} -\frac {1}{5} i \, B b d^{3} g x^{5} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - \frac {1}{5} i \, A b d^{3} g x^{5} - \frac {3}{4} i \, B b c d^{2} g x^{4} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - \frac {1}{4} i \, B a d^{3} g x^{4} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - \frac {3}{4} i \, A b c d^{2} g x^{4} - \frac {1}{4} i \, A a d^{3} g x^{4} - i \, B b c^{2} d g x^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - i \, B a c d^{2} g x^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - i \, A b c^{2} d g x^{3} - i \, A a c d^{2} g x^{3} - \frac {1}{2} i \, B b c^{3} g x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - \frac {3}{2} i \, B a c^{2} d g x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - \frac {1}{2} i \, A b c^{3} g x^{2} - \frac {3}{2} i \, A a c^{2} d g x^{2} - \frac {1}{60} i \, B b d^{3} g n {\left (\frac {12 \, a^{5} \log \left (b x + a\right )}{b^{5}} - \frac {12 \, c^{5} \log \left (d x + c\right )}{d^{5}} - \frac {3 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{4} - 4 \, {\left (b^{4} c^{2} d^{2} - a^{2} b^{2} d^{4}\right )} x^{3} + 6 \, {\left (b^{4} c^{3} d - a^{3} b d^{4}\right )} x^{2} - 12 \, {\left (b^{4} c^{4} - a^{4} d^{4}\right )} x}{b^{4} d^{4}}\right )} + \frac {1}{8} i \, B b c d^{2} g n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} + \frac {1}{24} i \, B a d^{3} g n {\left (\frac {6 \, a^{4} \log \left (b x + a\right )}{b^{4}} - \frac {6 \, c^{4} \log \left (d x + c\right )}{d^{4}} + \frac {2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{3} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x}{b^{3} d^{3}}\right )} - \frac {1}{2} i \, B b c^{2} d g n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} - \frac {1}{2} i \, B a c d^{2} g n {\left (\frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} + \frac {1}{2} i \, B b c^{3} g n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} + \frac {3}{2} i \, B a c^{2} d g n {\left (\frac {a^{2} \log \left (b x + a\right )}{b^{2}} - \frac {c^{2} \log \left (d x + c\right )}{d^{2}} + \frac {{\left (b c - a d\right )} x}{b d}\right )} - i \, B a c^{3} g n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} - i \, B a c^{3} g x \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) - i \, A a c^{3} g x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*I*B*b*d^3*g*x^5*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/5*I*A*b*d^3*g*x^5 - 3/4*I*B*b*c*d^2*g*x^4*log(
(b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/4*I*B*a*d^3*g*x^4*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 3/4*I*A*b*c*
d^2*g*x^4 - 1/4*I*A*a*d^3*g*x^4 - I*B*b*c^2*d*g*x^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - I*B*a*c*d^2*g*x^3
*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - I*A*b*c^2*d*g*x^3 - I*A*a*c*d^2*g*x^3 - 1/2*I*B*b*c^3*g*x^2*log((b*x
/(d*x + c) + a/(d*x + c))^n*e) - 3/2*I*B*a*c^2*d*g*x^2*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - 1/2*I*A*b*c^3*
g*x^2 - 3/2*I*A*a*c^2*d*g*x^2 - 1/60*I*B*b*d^3*g*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)) + 1/8*I*B*b*c*d^2*g*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*d^3*g*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*I*B*b*c^2*d*g*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*c*d^2*g*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*b*c^3*g*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*c^2*d*g*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*c^3*g*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) - I*B*a
*c^3*g*x*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - I*A*a*c^3*g*x

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 603 vs. \(2 (245) = 490\).
time = 0.50, size = 603, normalized size = 2.13 \begin {gather*} -\frac {24 \, {\left (i \, A + i \, B\right )} b^{5} d^{5} g x^{5} + 6 \, {\left ({\left (-i \, B b^{5} c d^{4} + i \, B a b^{4} d^{5}\right )} g n + 5 \, {\left (3 \, {\left (i \, A + i \, B\right )} b^{5} c d^{4} + {\left (i \, A + i \, B\right )} a b^{4} d^{5}\right )} g\right )} x^{4} + 2 \, {\left ({\left (-11 i \, B b^{5} c^{2} d^{3} + 10 i \, B a b^{4} c d^{4} + i \, B a^{2} b^{3} d^{5}\right )} g n + 60 \, {\left ({\left (i \, A + i \, B\right )} b^{5} c^{2} d^{3} + {\left (i \, A + i \, B\right )} a b^{4} c d^{4}\right )} g\right )} x^{3} + 6 \, {\left (10 i \, B a^{2} b^{3} c^{3} d^{2} - 10 i \, B a^{3} b^{2} c^{2} d^{3} + 5 i \, B a^{4} b c d^{4} - i \, B a^{5} d^{5}\right )} g n \log \left (\frac {b x + a}{b}\right ) + 6 \, {\left (i \, B b^{5} c^{5} - 5 i \, B a b^{4} c^{4} d\right )} g n \log \left (\frac {d x + c}{d}\right ) + 3 \, {\left ({\left (-9 i \, B b^{5} c^{3} d^{2} + 5 i \, B a b^{4} c^{2} d^{3} + 5 i \, B a^{2} b^{3} c d^{4} - i \, B a^{3} b^{2} d^{5}\right )} g n + 20 \, {\left ({\left (i \, A + i \, B\right )} b^{5} c^{3} d^{2} + 3 \, {\left (i \, A + i \, B\right )} a b^{4} c^{2} d^{3}\right )} g\right )} x^{2} + 6 \, {\left (20 \, {\left (i \, A + i \, B\right )} a b^{4} c^{3} d^{2} g + {\left (-i \, B b^{5} c^{4} d - 5 i \, B a b^{4} c^{3} d^{2} + 10 i \, B a^{2} b^{3} c^{2} d^{3} - 5 i \, B a^{3} b^{2} c d^{4} + i \, B a^{4} b d^{5}\right )} g n\right )} x + 6 \, {\left (4 i \, B b^{5} d^{5} g n x^{5} + 20 i \, B a b^{4} c^{3} d^{2} g n x + 5 \, {\left (3 i \, B b^{5} c d^{4} + i \, B a b^{4} d^{5}\right )} g n x^{4} + 20 \, {\left (i \, B b^{5} c^{2} d^{3} + i \, B a b^{4} c d^{4}\right )} g n x^{3} + 10 \, {\left (i \, B b^{5} c^{3} d^{2} + 3 i \, B a b^{4} c^{2} d^{3}\right )} g n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{120 \, b^{4} d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(24*(I*A + I*B)*b^5*d^5*g*x^5 + 6*((-I*B*b^5*c*d^4 + I*B*a*b^4*d^5)*g*n + 5*(3*(I*A + I*B)*b^5*c*d^4 +
(I*A + I*B)*a*b^4*d^5)*g)*x^4 + 2*((-11*I*B*b^5*c^2*d^3 + 10*I*B*a*b^4*c*d^4 + I*B*a^2*b^3*d^5)*g*n + 60*((I*A
+ I*B)*b^5*c^2*d^3 + (I*A + I*B)*a*b^4*c*d^4)*g)*x^3 + 6*(10*I*B*a^2*b^3*c^3*d^2 - 10*I*B*a^3*b^2*c^2*d^3 + 5
*I*B*a^4*b*c*d^4 - I*B*a^5*d^5)*g*n*log((b*x + a)/b) + 6*(I*B*b^5*c^5 - 5*I*B*a*b^4*c^4*d)*g*n*log((d*x + c)/d
) + 3*((-9*I*B*b^5*c^3*d^2 + 5*I*B*a*b^4*c^2*d^3 + 5*I*B*a^2*b^3*c*d^4 - I*B*a^3*b^2*d^5)*g*n + 20*((I*A + I*B
)*b^5*c^3*d^2 + 3*(I*A + I*B)*a*b^4*c^2*d^3)*g)*x^2 + 6*(20*(I*A + I*B)*a*b^4*c^3*d^2*g + (-I*B*b^5*c^4*d - 5*
I*B*a*b^4*c^3*d^2 + 10*I*B*a^2*b^3*c^2*d^3 - 5*I*B*a^3*b^2*c*d^4 + I*B*a^4*b*d^5)*g*n)*x + 6*(4*I*B*b^5*d^5*g*
n*x^5 + 20*I*B*a*b^4*c^3*d^2*g*n*x + 5*(3*I*B*b^5*c*d^4 + I*B*a*b^4*d^5)*g*n*x^4 + 20*(I*B*b^5*c^2*d^3 + I*B*a
*b^4*c*d^4)*g*n*x^3 + 10*(I*B*b^5*c^3*d^2 + 3*I*B*a*b^4*c^2*d^3)*g*n*x^2)*log((b*x + a)/(d*x + c)))/(b^4*d^2)

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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)*(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. 2372 vs. \(2 (245) = 490\).
time = 4.09, size = 2372, normalized size = 8.38 \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)*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

1/120*(6*(I*B*b^7*c^6*g*n - 6*I*B*a*b^6*c^5*d*g*n + 5*(-I*b*x - I*a)*B*b^6*c^6*d*g*n/(d*x + c) + 15*I*B*a^2*b^
5*c^4*d^2*g*n + 30*(I*b*x + I*a)*B*a*b^5*c^5*d^2*g*n/(d*x + c) - 20*I*B*a^3*b^4*c^3*d^3*g*n + 75*(-I*b*x - I*a
)*B*a^2*b^4*c^4*d^3*g*n/(d*x + c) + 15*I*B*a^4*b^3*c^2*d^4*g*n + 100*(I*b*x + I*a)*B*a^3*b^3*c^3*d^4*g*n/(d*x
+ c) - 6*I*B*a^5*b^2*c*d^5*g*n + 75*(-I*b*x - I*a)*B*a^4*b^2*c^2*d^5*g*n/(d*x + c) + I*B*a^6*b*d^6*g*n + 30*(I
*b*x + I*a)*B*a^5*b*c*d^6*g*n/(d*x + c) + 5*(-I*b*x - I*a)*B*a^6*d^7*g*n/(d*x + c))*log((b*x + a)/(d*x + c))/(
b^5*d^2 - 5*(b*x + a)*b^4*d^3/(d*x + c) + 10*(b*x + a)^2*b^3*d^4/(d*x + c)^2 - 10*(b*x + a)^3*b^2*d^5/(d*x + c
)^3 + 5*(b*x + a)^4*b*d^6/(d*x + c)^4 - (b*x + a)^5*d^7/(d*x + c)^5) + (-5*I*B*b^10*c^6*g*n + 30*I*B*a*b^9*c^5
*d*g*n - 31*(-I*b*x - I*a)*B*b^9*c^6*d*g*n/(d*x + c) - 75*I*B*a^2*b^8*c^4*d^2*g*n - 186*(I*b*x + I*a)*B*a*b^8*
c^5*d^2*g*n/(d*x + c) - 47*I*(b*x + a)^2*B*b^8*c^6*d^2*g*n/(d*x + c)^2 + 100*I*B*a^3*b^7*c^3*d^3*g*n - 465*(-I
*b*x - I*a)*B*a^2*b^7*c^4*d^3*g*n/(d*x + c) + 282*I*(b*x + a)^2*B*a*b^7*c^5*d^3*g*n/(d*x + c)^2 + 27*I*(b*x +
a)^3*B*b^7*c^6*d^3*g*n/(d*x + c)^3 - 75*I*B*a^4*b^6*c^2*d^4*g*n - 620*(I*b*x + I*a)*B*a^3*b^6*c^3*d^4*g*n/(d*x
+ c) - 705*I*(b*x + a)^2*B*a^2*b^6*c^4*d^4*g*n/(d*x + c)^2 - 162*I*(b*x + a)^3*B*a*b^6*c^5*d^4*g*n/(d*x + c)^
3 - 6*I*(b*x + a)^4*B*b^6*c^6*d^4*g*n/(d*x + c)^4 + 30*I*B*a^5*b^5*c*d^5*g*n - 465*(-I*b*x - I*a)*B*a^4*b^5*c^
2*d^5*g*n/(d*x + c) + 940*I*(b*x + a)^2*B*a^3*b^5*c^3*d^5*g*n/(d*x + c)^2 + 405*I*(b*x + a)^3*B*a^2*b^5*c^4*d^
5*g*n/(d*x + c)^3 + 36*I*(b*x + a)^4*B*a*b^5*c^5*d^5*g*n/(d*x + c)^4 - 5*I*B*a^6*b^4*d^6*g*n - 186*(I*b*x + I*
a)*B*a^5*b^4*c*d^6*g*n/(d*x + c) - 705*I*(b*x + a)^2*B*a^4*b^4*c^2*d^6*g*n/(d*x + c)^2 - 540*I*(b*x + a)^3*B*a
^3*b^4*c^3*d^6*g*n/(d*x + c)^3 - 90*I*(b*x + a)^4*B*a^2*b^4*c^4*d^6*g*n/(d*x + c)^4 - 31*(-I*b*x - I*a)*B*a^6*
b^3*d^7*g*n/(d*x + c) + 282*I*(b*x + a)^2*B*a^5*b^3*c*d^7*g*n/(d*x + c)^2 + 405*I*(b*x + a)^3*B*a^4*b^3*c^2*d^
7*g*n/(d*x + c)^3 + 120*I*(b*x + a)^4*B*a^3*b^3*c^3*d^7*g*n/(d*x + c)^4 - 47*I*(b*x + a)^2*B*a^6*b^2*d^8*g*n/(
d*x + c)^2 - 162*I*(b*x + a)^3*B*a^5*b^2*c*d^8*g*n/(d*x + c)^3 - 90*I*(b*x + a)^4*B*a^4*b^2*c^2*d^8*g*n/(d*x +
c)^4 + 27*I*(b*x + a)^3*B*a^6*b*d^9*g*n/(d*x + c)^3 + 36*I*(b*x + a)^4*B*a^5*b*c*d^9*g*n/(d*x + c)^4 - 6*I*(b
*x + a)^4*B*a^6*d^10*g*n/(d*x + c)^4 + 6*I*A*b^10*c^6*g + 6*I*B*b^10*c^6*g - 36*I*A*a*b^9*c^5*d*g - 36*I*B*a*b
^9*c^5*d*g - 30*(I*b*x + I*a)*A*b^9*c^6*d*g/(d*x + c) - 30*(I*b*x + I*a)*B*b^9*c^6*d*g/(d*x + c) + 90*I*A*a^2*
b^8*c^4*d^2*g + 90*I*B*a^2*b^8*c^4*d^2*g - 180*(-I*b*x - I*a)*A*a*b^8*c^5*d^2*g/(d*x + c) - 180*(-I*b*x - I*a)
*B*a*b^8*c^5*d^2*g/(d*x + c) - 120*I*A*a^3*b^7*c^3*d^3*g - 120*I*B*a^3*b^7*c^3*d^3*g - 450*(I*b*x + I*a)*A*a^2
*b^7*c^4*d^3*g/(d*x + c) - 450*(I*b*x + I*a)*B*a^2*b^7*c^4*d^3*g/(d*x + c) + 90*I*A*a^4*b^6*c^2*d^4*g + 90*I*B
*a^4*b^6*c^2*d^4*g - 600*(-I*b*x - I*a)*A*a^3*b^6*c^3*d^4*g/(d*x + c) - 600*(-I*b*x - I*a)*B*a^3*b^6*c^3*d^4*g
/(d*x + c) - 36*I*A*a^5*b^5*c*d^5*g - 36*I*B*a^5*b^5*c*d^5*g - 450*(I*b*x + I*a)*A*a^4*b^5*c^2*d^5*g/(d*x + c)
- 450*(I*b*x + I*a)*B*a^4*b^5*c^2*d^5*g/(d*x + c) + 6*I*A*a^6*b^4*d^6*g + 6*I*B*a^6*b^4*d^6*g - 180*(-I*b*x -
I*a)*A*a^5*b^4*c*d^6*g/(d*x + c) - 180*(-I*b*x - I*a)*B*a^5*b^4*c*d^6*g/(d*x + c) - 30*(I*b*x + I*a)*A*a^6*b^
3*d^7*g/(d*x + c) - 30*(I*b*x + I*a)*B*a^6*b^3*d^7*g/(d*x + c))/(b^8*d^2 - 5*(b*x + a)*b^7*d^3/(d*x + c) + 10*
(b*x + a)^2*b^6*d^4/(d*x + c)^2 - 10*(b*x + a)^3*b^5*d^5/(d*x + c)^3 + 5*(b*x + a)^4*b^4*d^6/(d*x + c)^4 - (b*
x + a)^5*b^3*d^7/(d*x + c)^5) + 6*(I*B*b^6*c^6*g*n - 6*I*B*a*b^5*c^5*d*g*n + 15*I*B*a^2*b^4*c^4*d^2*g*n - 20*I
*B*a^3*b^3*c^3*d^3*g*n + 15*I*B*a^4*b^2*c^2*d^4*g*n - 6*I*B*a^5*b*c*d^5*g*n + I*B*a^6*d^6*g*n)*log(b - (b*x +
a)*d/(d*x + c))/(b^4*d^2) + 6*(-I*B*b^6*c^6*g*n + 6*I*B*a*b^5*c^5*d*g*n - 15*I*B*a^2*b^4*c^4*d^2*g*n + 20*I*B*
a^3*b^3*c^3*d^3*g*n - 15*I*B*a^4*b^2*c^2*d^4*g*n + 6*I*B*a^5*b*c*d^5*g*n - I*B*a^6*d^6*g*n)*log((b*x + a)/(d*x
+ c))/(b^4*d^2))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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Mupad [B]
time = 5.40, size = 1234, normalized size = 4.36 \begin {gather*} x\,\left (\frac {a\,c\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {d^2\,g\,i^3\,\left (10\,A\,a\,d+20\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,d^2\,g\,i^3\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{20\,b\,d}-\frac {d\,g\,i^3\,\left (4\,A\,a^2\,d^2+24\,A\,b^2\,c^2+B\,a^2\,d^2\,n-3\,B\,b^2\,c^2\,n+32\,A\,a\,b\,c\,d+2\,B\,a\,b\,c\,d\,n\right )}{4\,b}+A\,a\,c\,d^2\,g\,i^3\right )}{b\,d}-\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {d^2\,g\,i^3\,\left (10\,A\,a\,d+20\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,d^2\,g\,i^3\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{20\,b\,d}-\frac {d\,g\,i^3\,\left (4\,A\,a^2\,d^2+24\,A\,b^2\,c^2+B\,a^2\,d^2\,n-3\,B\,b^2\,c^2\,n+32\,A\,a\,b\,c\,d+2\,B\,a\,b\,c\,d\,n\right )}{4\,b}+A\,a\,c\,d^2\,g\,i^3\right )}{20\,b\,d}-\frac {a\,c\,\left (\frac {d^2\,g\,i^3\,\left (10\,A\,a\,d+20\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,d^2\,g\,i^3\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{b\,d}+\frac {c\,g\,i^3\,\left (4\,A\,a^2\,d^2+4\,A\,b^2\,c^2+B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+12\,A\,a\,b\,c\,d\right )}{b}\right )}{20\,b\,d}+\frac {c^2\,g\,i^3\,\left (12\,A\,a^2\,d^2+2\,A\,b^2\,c^2+3\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+16\,A\,a\,b\,c\,d-2\,B\,a\,b\,c\,d\,n\right )}{2\,b\,d}\right )-x^3\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {d^2\,g\,i^3\,\left (10\,A\,a\,d+20\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,d^2\,g\,i^3\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{60\,b\,d}-\frac {d\,g\,i^3\,\left (4\,A\,a^2\,d^2+24\,A\,b^2\,c^2+B\,a^2\,d^2\,n-3\,B\,b^2\,c^2\,n+32\,A\,a\,b\,c\,d+2\,B\,a\,b\,c\,d\,n\right )}{12\,b}+\frac {A\,a\,c\,d^2\,g\,i^3}{3}\right )+x^2\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {d^2\,g\,i^3\,\left (10\,A\,a\,d+20\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,d^2\,g\,i^3\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{20\,b\,d}-\frac {d\,g\,i^3\,\left (4\,A\,a^2\,d^2+24\,A\,b^2\,c^2+B\,a^2\,d^2\,n-3\,B\,b^2\,c^2\,n+32\,A\,a\,b\,c\,d+2\,B\,a\,b\,c\,d\,n\right )}{4\,b}+A\,a\,c\,d^2\,g\,i^3\right )}{40\,b\,d}-\frac {a\,c\,\left (\frac {d^2\,g\,i^3\,\left (10\,A\,a\,d+20\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,d^2\,g\,i^3\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{2\,b\,d}+\frac {c\,g\,i^3\,\left (4\,A\,a^2\,d^2+4\,A\,b^2\,c^2+B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+12\,A\,a\,b\,c\,d\right )}{2\,b}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,c^2\,g\,i^3\,x^2\,\left (3\,a\,d+b\,c\right )}{2}+\frac {B\,d^2\,g\,i^3\,x^4\,\left (a\,d+3\,b\,c\right )}{4}+B\,a\,c^3\,g\,i^3\,x+\frac {B\,b\,d^3\,g\,i^3\,x^5}{5}+B\,c\,d\,g\,i^3\,x^3\,\left (a\,d+b\,c\right )\right )+x^4\,\left (\frac {d^2\,g\,i^3\,\left (10\,A\,a\,d+20\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{20}-\frac {A\,d^2\,g\,i^3\,\left (20\,a\,d+20\,b\,c\right )}{80}\right )+\frac {\ln \left (c+d\,x\right )\,\left (B\,b\,c^5\,g\,i^3\,n-5\,B\,a\,c^4\,d\,g\,i^3\,n\right )}{20\,d^2}-\frac {\ln \left (a+b\,x\right )\,\left (B\,g\,n\,a^5\,d^3\,i^3-5\,B\,g\,n\,a^4\,b\,c\,d^2\,i^3+10\,B\,g\,n\,a^3\,b^2\,c^2\,d\,i^3-10\,B\,g\,n\,a^2\,b^3\,c^3\,i^3\right )}{20\,b^4}+\frac {A\,b\,d^3\,g\,i^3\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((a*c*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d +
20*b*c))/20))/(20*b*d) - (d*g*i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2
*B*a*b*c*d*n))/(4*b) + A*a*c*d^2*g*i^3))/(b*d) - ((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((
d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20*b*d) - (d*g*
i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2*B*a*b*c*d*n))/(4*b) + A*a*c*d
^2*g*i^3))/(20*b*d) - (a*c*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 2
0*b*c))/20))/(b*d) + (c*g*i^3*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 12*A*a*b*c*d))/b))/(20*
b*d) + (c^2*g*i^3*(12*A*a^2*d^2 + 2*A*b^2*c^2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 16*A*a*b*c*d - 2*B*a*b*c*d*n))/(
2*b*d)) - x^3*(((20*a*d + 20*b*c)*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*
a*d + 20*b*c))/20))/(60*b*d) - (d*g*i^3*(4*A*a^2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c
*d + 2*B*a*b*c*d*n))/(12*b) + (A*a*c*d^2*g*i^3)/3) + x^2*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((d^2*g*i^3*(
10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20))/(20*b*d) - (d*g*i^3*(4*A*a^
2*d^2 + 24*A*b^2*c^2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 32*A*a*b*c*d + 2*B*a*b*c*d*n))/(4*b) + A*a*c*d^2*g*i^3))/
(40*b*d) - (a*c*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/20)
)/(2*b*d) + (c*g*i^3*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c^2*n + 12*A*a*b*c*d))/(2*b)) + log(e*((
a + b*x)/(c + d*x))^n)*((B*c^2*g*i^3*x^2*(3*a*d + b*c))/2 + (B*d^2*g*i^3*x^4*(a*d + 3*b*c))/4 + B*a*c^3*g*i^3*
x + (B*b*d^3*g*i^3*x^5)/5 + B*c*d*g*i^3*x^3*(a*d + b*c)) + x^4*((d^2*g*i^3*(10*A*a*d + 20*A*b*c + B*a*d*n - B*
b*c*n))/20 - (A*d^2*g*i^3*(20*a*d + 20*b*c))/80) + (log(c + d*x)*(B*b*c^5*g*i^3*n - 5*B*a*c^4*d*g*i^3*n))/(20*
d^2) - (log(a + b*x)*(B*a^5*d^3*g*i^3*n - 10*B*a^2*b^3*c^3*g*i^3*n - 5*B*a^4*b*c*d^2*g*i^3*n + 10*B*a^3*b^2*c^
2*d*g*i^3*n))/(20*b^4) + (A*b*d^3*g*i^3*x^5)/5

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