3.2.37 \(\int \frac {(a g+b g x) (A+B \log (e (\frac {a+b x}{c+d x})^n))}{c i+d i x} \, dx\) [137]

Optimal. Leaf size=134 \[ \frac {g (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d i}+\frac {(b c-a d) g \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^2 i}+\frac {B (b c-a d) g n \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]

[Out]

g*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d/i+(-a*d+b*c)*g*(A+B*n+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b
/(d*x+c))/d^2/i+B*(-a*d+b*c)*g*n*polylog(2,d*(b*x+a)/b/(d*x+c))/d^2/i

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Rubi [A]
time = 0.10, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2561, 2384, 2354, 2438} \begin {gather*} \frac {B g n (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i}+\frac {g (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A+B n\right )}{d^2 i}+\frac {g (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*i) + ((b*c - a*d)*g*(A + B*n + B*Log[e*((a + b*x)/(c +
d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(d^2*i) + (B*(b*c - a*d)*g*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))
])/(d^2*i)

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 \frac {(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{137 c+137 d x} \, dx &=\int \left (\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{137 d}+\frac {(-b c+a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{137 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{137 d}-\frac {((b c-a d) g) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{137 d}\\ &=\frac {A b g x}{137 d}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac {(b B g) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{137 d}+\frac {(B (b c-a d) g n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{137 d^2}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac {(B (b c-a d) g n) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{137 d^2}-\frac {(B (b c-a d) g n) \int \frac {1}{c+d x} \, dx}{137 d}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {B (b c-a d) g n \log (c+d x)}{137 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac {(b B (b c-a d) g n) \int \frac {\log (c+d x)}{a+b x} \, dx}{137 d^2}-\frac {(B (b c-a d) g n) \int \frac {\log (c+d x)}{c+d x} \, dx}{137 d}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac {B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac {(B (b c-a d) g n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{137 d^2}-\frac {(B (b c-a d) g n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{137 d}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac {B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac {B (b c-a d) g n \log ^2(c+d x)}{274 d^2}-\frac {(B (b c-a d) g n) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{137 d^2}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac {B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac {B (b c-a d) g n \log ^2(c+d x)}{274 d^2}+\frac {B (b c-a d) g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{137 d^2}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 170, normalized size = 1.27 \begin {gather*} \frac {g \left (2 A b d x+2 B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B (b c-a d) n \log (c+d x)-2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{2 d^2 i} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g*(2*A*b*d*x + 2*B*d*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] - 2*B*(b*c - a*d)*n*Log[c + d*x] - 2*(b*c - a*d
)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - a*d)*n*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)]
- Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(2*d^2*i)

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (b g x +a g \right ) \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}{d i x +c i}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*g*x+a*g)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(d*i*x+c*i),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. 289 vs. \(2 (129) = 258\).
time = 0.56, size = 289, normalized size = 2.16 \begin {gather*} A b g {\left (-\frac {i \, x}{d} + \frac {i \, c \log \left (d x + c\right )}{d^{2}}\right )} - \frac {i \, A a g \log \left (i \, d x + i \, c\right )}{d} - \frac {{\left (-i \, b c g n + i \, a d g n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{2}} - \frac {{\left (b c g {\left (-i \, n - i\right )} + i \, a d g\right )} B \log \left (d x + c\right )}{d^{2}} + \frac {-2 i \, B a d g n \log \left (b x + a\right ) - 2 i \, B b d g x - 2 \, {\left (i \, b c g n - i \, a d g n\right )} B \log \left (b x + a\right ) \log \left (d x + c\right ) + {\left (i \, b c g n - i \, a d g n\right )} B \log \left (d x + c\right )^{2} - 2 \, {\left (i \, B b d g x + {\left (-i \, b c g + i \, a d g\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (-i \, B b d g x + {\left (i \, b c g - i \, a d g\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*b*g*(-I*x/d + I*c*log(d*x + c)/d^2) - I*A*a*g*log(I*d*x + I*c)/d - (-I*b*c*g*n + I*a*d*g*n)*(log(b*x + a)*lo
g((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B/d^2 - (b*c*g*(-I*n - I) + I*a*d*g)*B*l
og(d*x + c)/d^2 + 1/2*(-2*I*B*a*d*g*n*log(b*x + a) - 2*I*B*b*d*g*x - 2*(I*b*c*g*n - I*a*d*g*n)*B*log(b*x + a)*
log(d*x + c) + (I*b*c*g*n - I*a*d*g*n)*B*log(d*x + c)^2 - 2*(I*B*b*d*g*x + (-I*b*c*g + I*a*d*g)*B*log(d*x + c)
)*log((b*x + a)^n) - 2*(-I*B*b*d*g*x + (I*b*c*g - I*a*d*g)*B*log(d*x + c))*log((d*x + c)^n))/d^2

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((-I*A - I*B)*b*g*x + (-I*A - I*B)*a*g + (-I*B*b*g*n*x - I*B*a*g*n)*log((b*x + a)/(d*x + c)))/(d*x +
c), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {g \left (\int \frac {A a}{c + d x}\, dx + \int \frac {A b x}{c + d x}\, dx + \int \frac {B a \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {B b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx\right )}{i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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. 1279 vs. \(2 (129) = 258\).
time = 75.17, size = 1279, normalized size = 9.54 \begin {gather*} -\frac {1}{6} \, {\left (\frac {{\left (-i \, B b^{5} c^{4} g n + 4 i \, B a b^{4} c^{3} d g n - \frac {3 \, {\left (-i \, b x - i \, a\right )} B b^{4} c^{4} d g n}{d x + c} - 6 i \, B a^{2} b^{3} c^{2} d^{2} g n - \frac {12 \, {\left (i \, b x + i \, a\right )} B a b^{3} c^{3} d^{2} g n}{d x + c} + 4 i \, B a^{3} b^{2} c d^{3} g n - \frac {18 \, {\left (-i \, b x - i \, a\right )} B a^{2} b^{2} c^{2} d^{3} g n}{d x + c} - i \, B a^{4} b d^{4} g n - \frac {12 \, {\left (i \, b x + i \, a\right )} B a^{3} b c d^{4} g n}{d x + c} - \frac {3 \, {\left (-i \, b x - i \, a\right )} B a^{4} d^{5} g n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{3} d^{2} - \frac {3 \, {\left (b x + a\right )} b^{2} d^{3}}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} b d^{4}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{3} d^{5}}{{\left (d x + c\right )}^{3}}} + \frac {\frac {{\left (-i \, b x - i \, a\right )} B b^{5} c^{4} d g n}{d x + c} - \frac {4 \, {\left (-i \, b x - i \, a\right )} B a b^{4} c^{3} d^{2} g n}{d x + c} + \frac {i \, {\left (b x + a\right )}^{2} B b^{4} c^{4} d^{2} g n}{{\left (d x + c\right )}^{2}} - \frac {6 \, {\left (i \, b x + i \, a\right )} B a^{2} b^{3} c^{2} d^{3} g n}{d x + c} - \frac {4 i \, {\left (b x + a\right )}^{2} B a b^{3} c^{3} d^{3} g n}{{\left (d x + c\right )}^{2}} - \frac {4 \, {\left (-i \, b x - i \, a\right )} B a^{3} b^{2} c d^{4} g n}{d x + c} + \frac {6 i \, {\left (b x + a\right )}^{2} B a^{2} b^{2} c^{2} d^{4} g n}{{\left (d x + c\right )}^{2}} + \frac {{\left (-i \, b x - i \, a\right )} B a^{4} b d^{5} g n}{d x + c} - \frac {4 i \, {\left (b x + a\right )}^{2} B a^{3} b c d^{5} g n}{{\left (d x + c\right )}^{2}} + \frac {i \, {\left (b x + a\right )}^{2} B a^{4} d^{6} g n}{{\left (d x + c\right )}^{2}} - i \, A b^{6} c^{4} g - i \, B b^{6} c^{4} g + 4 i \, A a b^{5} c^{3} d g + 4 i \, B a b^{5} c^{3} d g - \frac {3 \, {\left (-i \, b x - i \, a\right )} A b^{5} c^{4} d g}{d x + c} - \frac {3 \, {\left (-i \, b x - i \, a\right )} B b^{5} c^{4} d g}{d x + c} - 6 i \, A a^{2} b^{4} c^{2} d^{2} g - 6 i \, B a^{2} b^{4} c^{2} d^{2} g - \frac {12 \, {\left (i \, b x + i \, a\right )} A a b^{4} c^{3} d^{2} g}{d x + c} - \frac {12 \, {\left (i \, b x + i \, a\right )} B a b^{4} c^{3} d^{2} g}{d x + c} + 4 i \, A a^{3} b^{3} c d^{3} g + 4 i \, B a^{3} b^{3} c d^{3} g - \frac {18 \, {\left (-i \, b x - i \, a\right )} A a^{2} b^{3} c^{2} d^{3} g}{d x + c} - \frac {18 \, {\left (-i \, b x - i \, a\right )} B a^{2} b^{3} c^{2} d^{3} g}{d x + c} - i \, A a^{4} b^{2} d^{4} g - i \, B a^{4} b^{2} d^{4} g - \frac {12 \, {\left (i \, b x + i \, a\right )} A a^{3} b^{2} c d^{4} g}{d x + c} - \frac {12 \, {\left (i \, b x + i \, a\right )} B a^{3} b^{2} c d^{4} g}{d x + c} - \frac {3 \, {\left (-i \, b x - i \, a\right )} A a^{4} b d^{5} g}{d x + c} - \frac {3 \, {\left (-i \, b x - i \, a\right )} B a^{4} b d^{5} g}{d x + c}}{b^{4} d^{2} - \frac {3 \, {\left (b x + a\right )} b^{3} d^{3}}{d x + c} + \frac {3 \, {\left (b x + a\right )}^{2} b^{2} d^{4}}{{\left (d x + c\right )}^{2}} - \frac {{\left (b x + a\right )}^{3} b d^{5}}{{\left (d x + c\right )}^{3}}} - \frac {{\left (i \, B b^{4} c^{4} g n - 4 i \, B a b^{3} c^{3} d g n + 6 i \, B a^{2} b^{2} c^{2} d^{2} g n - 4 i \, B a^{3} b c d^{3} g n + i \, B a^{4} d^{4} g n\right )} \log \left (b - \frac {{\left (b x + a\right )} d}{d x + c}\right )}{b^{2} d^{2}} - \frac {{\left (-i \, B b^{4} c^{4} g n + 4 i \, B a b^{3} c^{3} d g n - 6 i \, B a^{2} b^{2} c^{2} d^{2} g n + 4 i \, B a^{3} b c d^{3} g n - i \, B a^{4} d^{4} g n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{b^{2} d^{2}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*((-I*B*b^5*c^4*g*n + 4*I*B*a*b^4*c^3*d*g*n - 3*(-I*b*x - I*a)*B*b^4*c^4*d*g*n/(d*x + c) - 6*I*B*a^2*b^3*c
^2*d^2*g*n - 12*(I*b*x + I*a)*B*a*b^3*c^3*d^2*g*n/(d*x + c) + 4*I*B*a^3*b^2*c*d^3*g*n - 18*(-I*b*x - I*a)*B*a^
2*b^2*c^2*d^3*g*n/(d*x + c) - I*B*a^4*b*d^4*g*n - 12*(I*b*x + I*a)*B*a^3*b*c*d^4*g*n/(d*x + c) - 3*(-I*b*x - I
*a)*B*a^4*d^5*g*n/(d*x + c))*log((b*x + a)/(d*x + c))/(b^3*d^2 - 3*(b*x + a)*b^2*d^3/(d*x + c) + 3*(b*x + a)^2
*b*d^4/(d*x + c)^2 - (b*x + a)^3*d^5/(d*x + c)^3) + ((-I*b*x - I*a)*B*b^5*c^4*d*g*n/(d*x + c) - 4*(-I*b*x - I*
a)*B*a*b^4*c^3*d^2*g*n/(d*x + c) + I*(b*x + a)^2*B*b^4*c^4*d^2*g*n/(d*x + c)^2 - 6*(I*b*x + I*a)*B*a^2*b^3*c^2
*d^3*g*n/(d*x + c) - 4*I*(b*x + a)^2*B*a*b^3*c^3*d^3*g*n/(d*x + c)^2 - 4*(-I*b*x - I*a)*B*a^3*b^2*c*d^4*g*n/(d
*x + c) + 6*I*(b*x + a)^2*B*a^2*b^2*c^2*d^4*g*n/(d*x + c)^2 + (-I*b*x - I*a)*B*a^4*b*d^5*g*n/(d*x + c) - 4*I*(
b*x + a)^2*B*a^3*b*c*d^5*g*n/(d*x + c)^2 + I*(b*x + a)^2*B*a^4*d^6*g*n/(d*x + c)^2 - I*A*b^6*c^4*g - I*B*b^6*c
^4*g + 4*I*A*a*b^5*c^3*d*g + 4*I*B*a*b^5*c^3*d*g - 3*(-I*b*x - I*a)*A*b^5*c^4*d*g/(d*x + c) - 3*(-I*b*x - I*a)
*B*b^5*c^4*d*g/(d*x + c) - 6*I*A*a^2*b^4*c^2*d^2*g - 6*I*B*a^2*b^4*c^2*d^2*g - 12*(I*b*x + I*a)*A*a*b^4*c^3*d^
2*g/(d*x + c) - 12*(I*b*x + I*a)*B*a*b^4*c^3*d^2*g/(d*x + c) + 4*I*A*a^3*b^3*c*d^3*g + 4*I*B*a^3*b^3*c*d^3*g -
18*(-I*b*x - I*a)*A*a^2*b^3*c^2*d^3*g/(d*x + c) - 18*(-I*b*x - I*a)*B*a^2*b^3*c^2*d^3*g/(d*x + c) - I*A*a^4*b
^2*d^4*g - I*B*a^4*b^2*d^4*g - 12*(I*b*x + I*a)*A*a^3*b^2*c*d^4*g/(d*x + c) - 12*(I*b*x + I*a)*B*a^3*b^2*c*d^4
*g/(d*x + c) - 3*(-I*b*x - I*a)*A*a^4*b*d^5*g/(d*x + c) - 3*(-I*b*x - I*a)*B*a^4*b*d^5*g/(d*x + c))/(b^4*d^2 -
3*(b*x + a)*b^3*d^3/(d*x + c) + 3*(b*x + a)^2*b^2*d^4/(d*x + c)^2 - (b*x + a)^3*b*d^5/(d*x + c)^3) - (I*B*b^4
*c^4*g*n - 4*I*B*a*b^3*c^3*d*g*n + 6*I*B*a^2*b^2*c^2*d^2*g*n - 4*I*B*a^3*b*c*d^3*g*n + I*B*a^4*d^4*g*n)*log(b
- (b*x + a)*d/(d*x + c))/(b^2*d^2) - (-I*B*b^4*c^4*g*n + 4*I*B*a*b^3*c^3*d*g*n - 6*I*B*a^2*b^2*c^2*d^2*g*n + 4
*I*B*a^3*b*c*d^3*g*n - I*B*a^4*d^4*g*n)*log((b*x + a)/(d*x + c))/(b^2*d^2))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*
d)^2)^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a\,g+b\,g\,x\right )\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{c\,i+d\,i\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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