3.2.8 \(\int (a g+b g x)^3 (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [108]

Optimal. Leaf size=223 \[ -\frac {B (b c-a d)^4 g^3 i n x}{20 b d^3}+\frac {B (b c-a d)^3 g^3 i n (a+b x)^2}{40 b^2 d^2}-\frac {B (b c-a d)^2 g^3 i n (a+b x)^3}{60 b^2 d}+\frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}+\frac {(b c-a d) g^3 i (a+b x)^4 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^2}+\frac {B (b c-a d)^5 g^3 i n \log (c+d x)}{20 b^2 d^4} \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2559, 2547, 21, 45} \begin {gather*} \frac {g^3 i (a+b x)^4 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A-B n\right )}{20 b^2}+\frac {g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}+\frac {B g^3 i n (b c-a d)^5 \log (c+d x)}{20 b^2 d^4}+\frac {B g^3 i n (a+b x)^2 (b c-a d)^3}{40 b^2 d^2}-\frac {B g^3 i n (a+b x)^3 (b c-a d)^2}{60 b^2 d}-\frac {B g^3 i n x (b c-a d)^4}{20 b d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + 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 2547

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x
_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 1))), x] - Dist[B*n*((b*c -
a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, m
, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, -2]

Rule 2559

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.)*((
h_.) + (i_.)*(x_)), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(h + i*x)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(
m + 2))), x] + Dist[i*((b*c - a*d)/(b*d*(m + 2))), Int[(f + g*x)^m*(A - B*n + B*Log[e*((a + b*x)/(c + d*x))^n]
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && Eq
Q[d*h - c*i, 0] && IGtQ[m, -2]

Rubi steps

\begin {align*} \int (108 c+108 d x) (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 {108 (b c-a d) (a g+b g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}+\frac {108 d (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g}\right ) \, dx\\ &=\frac {(108 (b c-a d)) \int (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}{b}+\frac {(108 d) \int (a g+b g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b g}\\ &=\frac {27 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac {108 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^2}-\frac {(108 B d n) \int \frac {(b c-a d) g^5 (a+b x)^4}{c+d x} \, dx}{5 b^2 g^2}-\frac {(27 B (b c-a d) n) \int \frac {(b c-a d) g^4 (a+b x)^3}{c+d x} \, dx}{b^2 g}\\ &=\frac {27 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac {108 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^2}-\frac {\left (108 B d (b c-a d) g^3 n\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{5 b^2}-\frac {\left (27 B (b c-a d)^2 g^3 n\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{b^2}\\ &=\frac {27 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac {108 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^2}-\frac {\left (108 B d (b c-a d) g^3 n\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{5 b^2}-\frac {\left (27 B (b c-a d)^2 g^3 n\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{b^2}\\ &=-\frac {27 B (b c-a d)^4 g^3 n x}{5 b d^3}+\frac {27 B (b c-a d)^3 g^3 n (a+b x)^2}{10 b^2 d^2}-\frac {9 B (b c-a d)^2 g^3 n (a+b x)^3}{5 b^2 d}-\frac {27 B (b c-a d) g^3 n (a+b x)^4}{5 b^2}+\frac {27 (b c-a d) g^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac {108 d g^3 (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b^2}+\frac {27 B (b c-a d)^5 g^3 n \log (c+d x)}{5 b^2 d^4}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 269, normalized size = 1.21 \begin {gather*} \frac {g^3 i \left (30 (b c-a d) (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+24 d (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {5 B (b c-a d)^2 n \left (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)\right )}{d^4}+\frac {2 B (b c-a d) n \left (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)\right )}{d^4}\right )}{120 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*i*(30*(b*c - a*d)*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*d*(a + b*x)^5*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]) - (5*B*(b*c - a*d)^2*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]))/d^4 + (2*B*(b*c - a*d)*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]))/d^4))
/(120*b^2)

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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{3} \left (d i x +c i \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*g*x+a*g)^3*(d*i*x+c*i)*(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. 1109 vs. \(2 (209) = 418\).
time = 0.32, size = 1109, normalized size = 4.97 \begin {gather*} \frac {1}{5} i \, B b^{3} d g^{3} 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^{3} d g^{3} x^{5} + \frac {1}{4} i \, B b^{3} c g^{3} x^{4} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {3}{4} i \, B a b^{2} d g^{3} x^{4} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {1}{4} i \, A b^{3} c g^{3} x^{4} + \frac {3}{4} i \, A a b^{2} d g^{3} x^{4} + i \, B a b^{2} c g^{3} x^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + i \, B a^{2} b d g^{3} x^{3} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + i \, A a b^{2} c g^{3} x^{3} + i \, A a^{2} b d g^{3} x^{3} + \frac {3}{2} i \, B a^{2} b c g^{3} x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {1}{2} i \, B a^{3} d g^{3} x^{2} \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + \frac {3}{2} i \, A a^{2} b c g^{3} x^{2} + \frac {1}{2} i \, A a^{3} d g^{3} x^{2} + \frac {1}{60} i \, B b^{3} d g^{3} 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}{24} i \, B b^{3} c g^{3} 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}{8} i \, B a b^{2} d g^{3} 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 a b^{2} c g^{3} 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^{2} b d g^{3} 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 {3}{2} i \, B a^{2} b c g^{3} 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 {1}{2} i \, B a^{3} d g^{3} 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^{3} c g^{3} n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + i \, B a^{3} c g^{3} x \log \left ({\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n} e\right ) + i \, A a^{3} c g^{3} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*I*B*b^3*d*g^3*x^5*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 1/5*I*A*b^3*d*g^3*x^5 + 1/4*I*B*b^3*c*g^3*x^4*l
og((b*x/(d*x + c) + a/(d*x + c))^n*e) + 3/4*I*B*a*b^2*d*g^3*x^4*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 1/4*I
*A*b^3*c*g^3*x^4 + 3/4*I*A*a*b^2*d*g^3*x^4 + I*B*a*b^2*c*g^3*x^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + I*B*
a^2*b*d*g^3*x^3*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + I*A*a*b^2*c*g^3*x^3 + I*A*a^2*b*d*g^3*x^3 + 3/2*I*B*a
^2*b*c*g^3*x^2*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 1/2*I*B*a^3*d*g^3*x^2*log((b*x/(d*x + c) + a/(d*x + c)
)^n*e) + 3/2*I*A*a^2*b*c*g^3*x^2 + 1/2*I*A*a^3*d*g^3*x^2 + 1/60*I*B*b^3*d*g^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*I*B*b^3*c*g^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)) - 1/8*I*B*a*b^2*d*g^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*I*B*a*b^2*c*g^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)) + 1/2*I*B*
a^2*b*d*g^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*I*B*a^2*b*c*g^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))
- 1/2*I*B*a^3*d*g^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)) + I*B*a^3*c*g^3*n*(a
*log(b*x + a)/b - c*log(d*x + c)/d) + I*B*a^3*c*g^3*x*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + I*A*a^3*c*g^3*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. 635 vs. \(2 (209) = 418\).
time = 0.50, size = 635, normalized size = 2.85 \begin {gather*} -\frac {24 \, {\left (-i \, A - i \, B\right )} b^{5} d^{5} g^{3} x^{5} + 6 \, {\left (-5 i \, B a^{4} b c d^{4} + i \, B a^{5} d^{5}\right )} g^{3} 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 - 10 i \, B a^{2} b^{3} c^{3} d^{2} + 10 i \, B a^{3} b^{2} c^{2} d^{3}\right )} g^{3} n \log \left (\frac {d x + c}{d}\right ) + 6 \, {\left ({\left (i \, B b^{5} c d^{4} - i \, B a b^{4} d^{5}\right )} g^{3} n + 5 \, {\left ({\left (-i \, A - i \, B\right )} b^{5} c d^{4} + 3 \, {\left (-i \, A - i \, B\right )} a b^{4} d^{5}\right )} g^{3}\right )} x^{4} + 2 \, {\left ({\left (i \, B b^{5} c^{2} d^{3} + 10 i \, B a b^{4} c d^{4} - 11 i \, B a^{2} b^{3} d^{5}\right )} g^{3} n + 60 \, {\left ({\left (-i \, A - i \, B\right )} a b^{4} c d^{4} + {\left (-i \, A - i \, B\right )} a^{2} b^{3} d^{5}\right )} g^{3}\right )} x^{3} + 3 \, {\left ({\left (-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} - 9 i \, B a^{3} b^{2} d^{5}\right )} g^{3} n + 20 \, {\left (3 \, {\left (-i \, A - i \, B\right )} a^{2} b^{3} c d^{4} + {\left (-i \, A - i \, B\right )} a^{3} b^{2} d^{5}\right )} g^{3}\right )} x^{2} + 6 \, {\left (20 \, {\left (-i \, A - i \, B\right )} a^{3} b^{2} c d^{4} g^{3} + {\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^{3} n\right )} x + 6 \, {\left (-4 i \, B b^{5} d^{5} g^{3} n x^{5} - 20 i \, B a^{3} b^{2} c d^{4} g^{3} n x + 5 \, {\left (-i \, B b^{5} c d^{4} - 3 i \, B a b^{4} d^{5}\right )} g^{3} n x^{4} + 20 \, {\left (-i \, B a b^{4} c d^{4} - i \, B a^{2} b^{3} d^{5}\right )} g^{3} n x^{3} + 10 \, {\left (-3 i \, B a^{2} b^{3} c d^{4} - i \, B a^{3} b^{2} d^{5}\right )} g^{3} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{120 \, b^{2} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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)**3*(d*i*x+c*i)*(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. 3847 vs. \(2 (209) = 418\).
time = 3.50, size = 3847, normalized size = 17.25 \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)^3*(d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, algorithm="giac")

[Out]

-1/120*(6*(I*B*b^9*c^6*g^3*n - 6*I*B*a*b^8*c^5*d*g^3*n + 5*(-I*b*x - I*a)*B*b^8*c^6*d*g^3*n/(d*x + c) + 15*I*B
*a^2*b^7*c^4*d^2*g^3*n + 30*(I*b*x + I*a)*B*a*b^7*c^5*d^2*g^3*n/(d*x + c) + 10*I*(b*x + a)^2*B*b^7*c^6*d^2*g^3
*n/(d*x + c)^2 - 20*I*B*a^3*b^6*c^3*d^3*g^3*n + 75*(-I*b*x - I*a)*B*a^2*b^6*c^4*d^3*g^3*n/(d*x + c) - 60*I*(b*
x + a)^2*B*a*b^6*c^5*d^3*g^3*n/(d*x + c)^2 - 10*I*(b*x + a)^3*B*b^6*c^6*d^3*g^3*n/(d*x + c)^3 + 15*I*B*a^4*b^5
*c^2*d^4*g^3*n + 100*(I*b*x + I*a)*B*a^3*b^5*c^3*d^4*g^3*n/(d*x + c) + 150*I*(b*x + a)^2*B*a^2*b^5*c^4*d^4*g^3
*n/(d*x + c)^2 + 60*I*(b*x + a)^3*B*a*b^5*c^5*d^4*g^3*n/(d*x + c)^3 - 6*I*B*a^5*b^4*c*d^5*g^3*n + 75*(-I*b*x -
I*a)*B*a^4*b^4*c^2*d^5*g^3*n/(d*x + c) - 200*I*(b*x + a)^2*B*a^3*b^4*c^3*d^5*g^3*n/(d*x + c)^2 - 150*I*(b*x +
a)^3*B*a^2*b^4*c^4*d^5*g^3*n/(d*x + c)^3 + I*B*a^6*b^3*d^6*g^3*n + 30*(I*b*x + I*a)*B*a^5*b^3*c*d^6*g^3*n/(d*
x + c) + 150*I*(b*x + a)^2*B*a^4*b^3*c^2*d^6*g^3*n/(d*x + c)^2 + 200*I*(b*x + a)^3*B*a^3*b^3*c^3*d^6*g^3*n/(d*
x + c)^3 + 5*(-I*b*x - I*a)*B*a^6*b^2*d^7*g^3*n/(d*x + c) - 60*I*(b*x + a)^2*B*a^5*b^2*c*d^7*g^3*n/(d*x + c)^2
- 150*I*(b*x + a)^3*B*a^4*b^2*c^2*d^7*g^3*n/(d*x + c)^3 + 10*I*(b*x + a)^2*B*a^6*b*d^8*g^3*n/(d*x + c)^2 + 60
*I*(b*x + a)^3*B*a^5*b*c*d^8*g^3*n/(d*x + c)^3 - 10*I*(b*x + a)^3*B*a^6*d^9*g^3*n/(d*x + c)^3)*log((b*x + a)/(
d*x + c))/(b^5*d^4 - 5*(b*x + a)*b^4*d^5/(d*x + c) + 10*(b*x + a)^2*b^3*d^6/(d*x + c)^2 - 10*(b*x + a)^3*b^2*d
^7/(d*x + c)^3 + 5*(b*x + a)^4*b*d^8/(d*x + c)^4 - (b*x + a)^5*d^9/(d*x + c)^5) - (-5*I*B*b^10*c^6*g^3*n + 30*
I*B*a*b^9*c^5*d*g^3*n - 19*(-I*b*x - I*a)*B*b^9*c^6*d*g^3*n/(d*x + c) - 75*I*B*a^2*b^8*c^4*d^2*g^3*n - 114*(I*
b*x + I*a)*B*a*b^8*c^5*d^2*g^3*n/(d*x + c) - 23*I*(b*x + a)^2*B*b^8*c^6*d^2*g^3*n/(d*x + c)^2 + 100*I*B*a^3*b^
7*c^3*d^3*g^3*n - 285*(-I*b*x - I*a)*B*a^2*b^7*c^4*d^3*g^3*n/(d*x + c) + 138*I*(b*x + a)^2*B*a*b^7*c^5*d^3*g^3
*n/(d*x + c)^2 + 3*I*(b*x + a)^3*B*b^7*c^6*d^3*g^3*n/(d*x + c)^3 - 75*I*B*a^4*b^6*c^2*d^4*g^3*n - 380*(I*b*x +
I*a)*B*a^3*b^6*c^3*d^4*g^3*n/(d*x + c) - 345*I*(b*x + a)^2*B*a^2*b^6*c^4*d^4*g^3*n/(d*x + c)^2 - 18*I*(b*x +
a)^3*B*a*b^6*c^5*d^4*g^3*n/(d*x + c)^3 + 6*I*(b*x + a)^4*B*b^6*c^6*d^4*g^3*n/(d*x + c)^4 + 30*I*B*a^5*b^5*c*d^
5*g^3*n - 285*(-I*b*x - I*a)*B*a^4*b^5*c^2*d^5*g^3*n/(d*x + c) + 460*I*(b*x + a)^2*B*a^3*b^5*c^3*d^5*g^3*n/(d*
x + c)^2 + 45*I*(b*x + a)^3*B*a^2*b^5*c^4*d^5*g^3*n/(d*x + c)^3 - 36*I*(b*x + a)^4*B*a*b^5*c^5*d^5*g^3*n/(d*x
+ c)^4 - 5*I*B*a^6*b^4*d^6*g^3*n - 114*(I*b*x + I*a)*B*a^5*b^4*c*d^6*g^3*n/(d*x + c) - 345*I*(b*x + a)^2*B*a^4
*b^4*c^2*d^6*g^3*n/(d*x + c)^2 - 60*I*(b*x + a)^3*B*a^3*b^4*c^3*d^6*g^3*n/(d*x + c)^3 + 90*I*(b*x + a)^4*B*a^2
*b^4*c^4*d^6*g^3*n/(d*x + c)^4 - 19*(-I*b*x - I*a)*B*a^6*b^3*d^7*g^3*n/(d*x + c) + 138*I*(b*x + a)^2*B*a^5*b^3
*c*d^7*g^3*n/(d*x + c)^2 + 45*I*(b*x + a)^3*B*a^4*b^3*c^2*d^7*g^3*n/(d*x + c)^3 - 120*I*(b*x + a)^4*B*a^3*b^3*
c^3*d^7*g^3*n/(d*x + c)^4 - 23*I*(b*x + a)^2*B*a^6*b^2*d^8*g^3*n/(d*x + c)^2 - 18*I*(b*x + a)^3*B*a^5*b^2*c*d^
8*g^3*n/(d*x + c)^3 + 90*I*(b*x + a)^4*B*a^4*b^2*c^2*d^8*g^3*n/(d*x + c)^4 + 3*I*(b*x + a)^3*B*a^6*b*d^9*g^3*n
/(d*x + c)^3 - 36*I*(b*x + a)^4*B*a^5*b*c*d^9*g^3*n/(d*x + c)^4 + 6*I*(b*x + a)^4*B*a^6*d^10*g^3*n/(d*x + c)^4
- 6*I*A*b^10*c^6*g^3 - 6*I*B*b^10*c^6*g^3 + 36*I*A*a*b^9*c^5*d*g^3 + 36*I*B*a*b^9*c^5*d*g^3 - 30*(-I*b*x - I*
a)*A*b^9*c^6*d*g^3/(d*x + c) - 30*(-I*b*x - I*a)*B*b^9*c^6*d*g^3/(d*x + c) - 90*I*A*a^2*b^8*c^4*d^2*g^3 - 90*I
*B*a^2*b^8*c^4*d^2*g^3 - 180*(I*b*x + I*a)*A*a*b^8*c^5*d^2*g^3/(d*x + c) - 180*(I*b*x + I*a)*B*a*b^8*c^5*d^2*g
^3/(d*x + c) - 60*I*(b*x + a)^2*A*b^8*c^6*d^2*g^3/(d*x + c)^2 - 60*I*(b*x + a)^2*B*b^8*c^6*d^2*g^3/(d*x + c)^2
+ 120*I*A*a^3*b^7*c^3*d^3*g^3 + 120*I*B*a^3*b^7*c^3*d^3*g^3 - 450*(-I*b*x - I*a)*A*a^2*b^7*c^4*d^3*g^3/(d*x +
c) - 450*(-I*b*x - I*a)*B*a^2*b^7*c^4*d^3*g^3/(d*x + c) + 360*I*(b*x + a)^2*A*a*b^7*c^5*d^3*g^3/(d*x + c)^2 +
360*I*(b*x + a)^2*B*a*b^7*c^5*d^3*g^3/(d*x + c)^2 + 60*I*(b*x + a)^3*A*b^7*c^6*d^3*g^3/(d*x + c)^3 + 60*I*(b*
x + a)^3*B*b^7*c^6*d^3*g^3/(d*x + c)^3 - 90*I*A*a^4*b^6*c^2*d^4*g^3 - 90*I*B*a^4*b^6*c^2*d^4*g^3 - 600*(I*b*x
+ I*a)*A*a^3*b^6*c^3*d^4*g^3/(d*x + c) - 600*(I*b*x + I*a)*B*a^3*b^6*c^3*d^4*g^3/(d*x + c) - 900*I*(b*x + a)^2
*A*a^2*b^6*c^4*d^4*g^3/(d*x + c)^2 - 900*I*(b*x + a)^2*B*a^2*b^6*c^4*d^4*g^3/(d*x + c)^2 - 360*I*(b*x + a)^3*A
*a*b^6*c^5*d^4*g^3/(d*x + c)^3 - 360*I*(b*x + a)^3*B*a*b^6*c^5*d^4*g^3/(d*x + c)^3 + 36*I*A*a^5*b^5*c*d^5*g^3
+ 36*I*B*a^5*b^5*c*d^5*g^3 - 450*(-I*b*x - I*a)*A*a^4*b^5*c^2*d^5*g^3/(d*x + c) - 450*(-I*b*x - I*a)*B*a^4*b^5
*c^2*d^5*g^3/(d*x + c) + 1200*I*(b*x + a)^2*A*a^3*b^5*c^3*d^5*g^3/(d*x + c)^2 + 1200*I*(b*x + a)^2*B*a^3*b^5*c
^3*d^5*g^3/(d*x + c)^2 + 900*I*(b*x + a)^3*A*a^2*b^5*c^4*d^5*g^3/(d*x + c)^3 + 900*I*(b*x + a)^3*B*a^2*b^5*c^4
*d^5*g^3/(d*x + c)^3 - 6*I*A*a^6*b^4*d^6*g^3 - 6*I*B*a^6*b^4*d^6*g^3 - 180*(I*b*x + I*a)*A*a^5*b^4*c*d^6*g^3/(
d*x + c) - 180*(I*b*x + I*a)*B*a^5*b^4*c*d^6*g^3/(d*x + c) - 900*I*(b*x + a)^2*A*a^4*b^4*c^2*d^6*g^3/(d*x + c)
^2 - 900*I*(b*x + a)^2*B*a^4*b^4*c^2*d^6*g^3/(d...

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Mupad [B]
time = 5.62, size = 1237, normalized size = 5.55 \begin {gather*} x\,\left (\frac {a\,c\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {b^2\,g^3\,i\,\left (20\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,b^2\,g^3\,i\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{20\,b\,d}-\frac {b\,g^3\,i\,\left (24\,A\,a^2\,d^2+4\,A\,b^2\,c^2+3\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+32\,A\,a\,b\,c\,d-2\,B\,a\,b\,c\,d\,n\right )}{4\,d}+A\,a\,b^2\,c\,g^3\,i\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 {b^2\,g^3\,i\,\left (20\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,b^2\,g^3\,i\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{20\,b\,d}-\frac {b\,g^3\,i\,\left (24\,A\,a^2\,d^2+4\,A\,b^2\,c^2+3\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+32\,A\,a\,b\,c\,d-2\,B\,a\,b\,c\,d\,n\right )}{4\,d}+A\,a\,b^2\,c\,g^3\,i\right )}{20\,b\,d}-\frac {a\,c\,\left (\frac {b^2\,g^3\,i\,\left (20\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,b^2\,g^3\,i\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{b\,d}+\frac {a\,g^3\,i\,\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 )}{d}\right )}{20\,b\,d}+\frac {a^2\,g^3\,i\,\left (2\,A\,a^2\,d^2+12\,A\,b^2\,c^2+B\,a^2\,d^2\,n-3\,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^2\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {b^2\,g^3\,i\,\left (20\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,b^2\,g^3\,i\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{20\,b\,d}-\frac {b\,g^3\,i\,\left (24\,A\,a^2\,d^2+4\,A\,b^2\,c^2+3\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+32\,A\,a\,b\,c\,d-2\,B\,a\,b\,c\,d\,n\right )}{4\,d}+A\,a\,b^2\,c\,g^3\,i\right )}{40\,b\,d}-\frac {a\,c\,\left (\frac {b^2\,g^3\,i\,\left (20\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,b^2\,g^3\,i\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{2\,b\,d}+\frac {a\,g^3\,i\,\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\,d}\right )-x^3\,\left (\frac {\left (20\,a\,d+20\,b\,c\right )\,\left (\frac {b^2\,g^3\,i\,\left (20\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{5}-\frac {A\,b^2\,g^3\,i\,\left (20\,a\,d+20\,b\,c\right )}{20}\right )}{60\,b\,d}-\frac {b\,g^3\,i\,\left (24\,A\,a^2\,d^2+4\,A\,b^2\,c^2+3\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+32\,A\,a\,b\,c\,d-2\,B\,a\,b\,c\,d\,n\right )}{12\,d}+\frac {A\,a\,b^2\,c\,g^3\,i}{3}\right )+\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,a^2\,g^3\,i\,x^2\,\left (a\,d+3\,b\,c\right )}{2}+\frac {B\,b^2\,g^3\,i\,x^4\,\left (3\,a\,d+b\,c\right )}{4}+B\,a^3\,c\,g^3\,i\,x+\frac {B\,b^3\,d\,g^3\,i\,x^5}{5}+B\,a\,b\,g^3\,i\,x^3\,\left (a\,d+b\,c\right )\right )+x^4\,\left (\frac {b^2\,g^3\,i\,\left (20\,A\,a\,d+10\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{20}-\frac {A\,b^2\,g^3\,i\,\left (20\,a\,d+20\,b\,c\right )}{80}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^5\,d\,g^3\,i\,n-5\,B\,a^4\,b\,c\,g^3\,i\,n\right )}{20\,b^2}+\frac {\ln \left (c+d\,x\right )\,\left (-10\,B\,i\,n\,a^3\,c^2\,d^3\,g^3+10\,B\,i\,n\,a^2\,b\,c^3\,d^2\,g^3-5\,B\,i\,n\,a\,b^2\,c^4\,d\,g^3+B\,i\,n\,b^3\,c^5\,g^3\right )}{20\,d^4}+\frac {A\,b^3\,d\,g^3\,i\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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