3.1.33 \(\int \frac {(a g+b g x) (A+B \log (\frac {e (a+b x)}{c+d x}))}{c i+d i x} \, dx\) [33]

Optimal. Leaf size=125 \[ \frac {g (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d i}+\frac {(b c-a d) g \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}+\frac {B (b c-a d) g \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2562, 2384, 2354, 2438} \begin {gather*} \frac {B g (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 (\frac {e (a+b x)}{c+d x}\right )+A+B\right )}{d^2 i}+\frac {g (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\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)]))/(c*i + d*i*x),x]

[Out]

(g*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*i) + ((b*c - a*d)*g*Log[(b*c - a*d)/(b*(c + d*x))]*(A +
B + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^2*i) + (B*(b*c - a*d)*g*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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(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] && EqQ[n + mn, 0] && IGtQ[n, 0] && 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 (\frac {e (a+b x)}{c+d x}\right )\right )}{33 c+33 d x} \, dx &=\int \left (\frac {b g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{33 d}+\frac {(-b c+a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{33 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{33 d}-\frac {((b c-a d) g) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{33 d}\\ &=\frac {A b g x}{33 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac {(b B g) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{33 d}+\frac {(B (b c-a d) g) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{33 d^2}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac {(B (b c-a d) g) \int \frac {1}{c+d x} \, dx}{33 d}+\frac {(B (b c-a d) g) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{33 d^2 e}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac {(B (b c-a d) g) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{33 d^2 e}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}+\frac {(b B (b c-a d) g) \int \frac {\log (c+d x)}{a+b x} \, dx}{33 d^2}-\frac {(B (b c-a d) g) \int \frac {\log (c+d x)}{c+d x} \, dx}{33 d}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}+\frac {B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac {(B (b c-a d) g) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{33 d^2}-\frac {(B (b c-a d) g) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{33 d}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}+\frac {B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac {B (b c-a d) g \log ^2(c+d x)}{66 d^2}-\frac {(B (b c-a d) g) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{33 d^2}\\ &=\frac {A b g x}{33 d}+\frac {B g (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{33 d}-\frac {B (b c-a d) g \log (c+d x)}{33 d^2}+\frac {B (b c-a d) g \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{33 d^2}-\frac {(b c-a d) g \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{33 d^2}-\frac {B (b c-a d) g \log ^2(c+d x)}{66 d^2}+\frac {B (b c-a d) g \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{33 d^2}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 162, normalized size = 1.30 \begin {gather*} \frac {g \left (2 A b d x+2 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 B (b c-a d) \log (c+d x)-2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \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)]))/(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)] - 2*B*(b*c - a*d)*Log[c + d*x] - 2*(b*c - a*d)*(A
+ B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + B*(b*c - a*d)*((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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(379\) vs. \(2(125)=250\).
time = 1.33, size = 380, normalized size = 3.04

method	result	size
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {g A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g A b}{i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e i}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e i}+\frac {g B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g d B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d^{2}}\)	\(380\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {g A \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g A b}{i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}+\frac {g B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e i}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right )}{e i}+\frac {g B \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e i}+\frac {g d B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{d^{2}}\)	\(380\)
risch	\(\frac {g A b x}{i d}+\frac {g A \ln \left (d x +c \right ) a}{i d}-\frac {g A \ln \left (d x +c \right ) c b}{i \,d^{2}}-\frac {g B \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right ) a}{i d}+\frac {g B b \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right ) c}{i \,d^{2}}+\frac {g B b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a}{i d \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right )}-\frac {g B \,b^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c}{i \,d^{2} \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right )}+\frac {g B e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a^{2}}{i \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}-\frac {2 g B b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a c}{i d \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}+\frac {g B \,b^{2} e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c^{2}}{i \,d^{2} \left (\frac {e d a}{d x +c}-\frac {e c b}{d x +c}\right ) \left (d x +c \right )}-\frac {g B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right ) a}{i d}+\frac {g B \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right ) c b}{i \,d^{2}}-\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right ) a}{i d}+\frac {g B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d}{b e}\right ) c b}{i \,d^{2}}\)	\(769\)

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)))/(d*i*x+c*i),x,method=_RETURNVERBOSE)

[Out]

-1/d^2*e*(a*d-b*c)*(g/e/i*A*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)+g/i*A*b/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))
*d)+g/e/i*B*dilog(-(-b*e+(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)/b/e)+g/e/i*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(-(-b
*e+(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)/b/e)+g/e/i*B*ln(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d)+g*d/e/i*B*ln(b*e/d+(a
*d-b*c)*e/d/(d*x+c))*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(b*e-(b*e/d+(a*d-b*c)*e/d/(d*x+c))*d))

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Maxima [A]
time = 0.33, size = 201, normalized size = 1.61 \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 + i \, a d g\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 (-2 i \, b c g + i \, a d g\right )} B \log \left (d x + c\right )}{d^{2}} + \frac {2 i \, B b d g x \log \left (d x + c\right ) - 2 i \, B b d g x + {\left (-i \, b c g + i \, a d g\right )} B \log \left (d x + c\right )^{2} - 2 \, {\left (i \, B b d g x + i \, B a d g\right )} \log \left (b x + a\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)))/(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 + I*a*d*g)*(log(b*x + a)*log((b
*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B/d^2 - (-2*I*b*c*g + I*a*d*g)*B*log(d*x + c
)/d^2 + 1/2*(2*I*B*b*d*g*x*log(d*x + c) - 2*I*B*b*d*g*x + (-I*b*c*g + I*a*d*g)*B*log(d*x + c)^2 - 2*(I*B*b*d*g
*x + I*B*a*d*g)*log(b*x + a))/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)))/(d*i*x+c*i),x, algorithm="fricas")

[Out]

integral((-I*A*b*g*x - I*A*a*g + (-I*B*b*g*x - I*B*a*g)*log((b*x + a)*e/(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 (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}{c + d x}\, dx + \int \frac {B b x \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \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)))/(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(a*e/(c + d*x) + b*e*x/(c + d*x
))/(c + d*x), x) + Integral(B*b*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(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. 2364 vs. \(2 (120) = 240\).
time = 49.16, size = 2364, normalized size = 18.91 \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)*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x, algorithm="giac")

[Out]

1/6*(I*B*b^7*c^4*g*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 4*I*B*a*b^6*c^3*d*g*e^4*log(-b*e + (b*x*e + a*e
)*d/(d*x + c)) + 6*I*B*a^2*b^5*c^2*d^2*g*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 4*I*B*a^3*b^4*c*d^3*g*e^4
*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) + I*B*a^4*b^3*d^4*g*e^4*log(-b*e + (b*x*e + a*e)*d/(d*x + c)) - 3*I*(b*
x*e + a*e)*B*b^6*c^4*d*g*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 12*I*(b*x*e + a*e)*B*a*b^5*c^3*
d^2*g*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) - 18*I*(b*x*e + a*e)*B*a^2*b^4*c^2*d^3*g*e^3*log(-b*
e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c) + 12*I*(b*x*e + a*e)*B*a^3*b^3*c*d^4*g*e^3*log(-b*e + (b*x*e + a*e)*d
/(d*x + c))/(d*x + c) - 3*I*(b*x*e + a*e)*B*a^4*b^2*d^5*g*e^3*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)
+ 3*I*(b*x*e + a*e)^2*B*b^5*c^4*d^2*g*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 12*I*(b*x*e + a*
e)^2*B*a*b^4*c^3*d^3*g*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 18*I*(b*x*e + a*e)^2*B*a^2*b^3*
c^2*d^4*g*e^2*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 - 12*I*(b*x*e + a*e)^2*B*a^3*b^2*c*d^5*g*e^2*l
og(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^2 + 3*I*(b*x*e + a*e)^2*B*a^4*b*d^6*g*e^2*log(-b*e + (b*x*e + a
*e)*d/(d*x + c))/(d*x + c)^2 - I*(b*x*e + a*e)^3*B*b^4*c^4*d^3*g*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x
+ c)^3 + 4*I*(b*x*e + a*e)^3*B*a*b^3*c^3*d^4*g*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 6*I*(b*x*
e + a*e)^3*B*a^2*b^2*c^2*d^5*g*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 + 4*I*(b*x*e + a*e)^3*B*a^3
*b*c*d^6*g*e*log(-b*e + (b*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - I*(b*x*e + a*e)^3*B*a^4*d^7*g*e*log(-b*e + (b
*x*e + a*e)*d/(d*x + c))/(d*x + c)^3 - 3*I*(b*x*e + a*e)^2*B*b^5*c^4*d^2*g*e^2*log((b*x*e + a*e)/(d*x + c))/(d
*x + c)^2 + 12*I*(b*x*e + a*e)^2*B*a*b^4*c^3*d^3*g*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 18*I*(b*x*e
+ a*e)^2*B*a^2*b^3*c^2*d^4*g*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 12*I*(b*x*e + a*e)^2*B*a^3*b^2*c*d
^5*g*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 - 3*I*(b*x*e + a*e)^2*B*a^4*b*d^6*g*e^2*log((b*x*e + a*e)/(d
*x + c))/(d*x + c)^2 + I*(b*x*e + a*e)^3*B*b^4*c^4*d^3*g*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 4*I*(b*x
*e + a*e)^3*B*a*b^3*c^3*d^4*g*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 6*I*(b*x*e + a*e)^3*B*a^2*b^2*c^2*d
^5*g*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 4*I*(b*x*e + a*e)^3*B*a^3*b*c*d^6*g*e*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c)^3 + I*(b*x*e + a*e)^3*B*a^4*d^7*g*e*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + I*A*b^7*c^4*g*
e^4 - 4*I*A*a*b^6*c^3*d*g*e^4 + 6*I*A*a^2*b^5*c^2*d^2*g*e^4 - 4*I*A*a^3*b^4*c*d^3*g*e^4 + I*A*a^4*b^3*d^4*g*e^
4 - 3*I*(b*x*e + a*e)*A*b^6*c^4*d*g*e^3/(d*x + c) + I*(b*x*e + a*e)*B*b^6*c^4*d*g*e^3/(d*x + c) + 12*I*(b*x*e
+ a*e)*A*a*b^5*c^3*d^2*g*e^3/(d*x + c) - 4*I*(b*x*e + a*e)*B*a*b^5*c^3*d^2*g*e^3/(d*x + c) - 18*I*(b*x*e + a*e
)*A*a^2*b^4*c^2*d^3*g*e^3/(d*x + c) + 6*I*(b*x*e + a*e)*B*a^2*b^4*c^2*d^3*g*e^3/(d*x + c) + 12*I*(b*x*e + a*e)
*A*a^3*b^3*c*d^4*g*e^3/(d*x + c) - 4*I*(b*x*e + a*e)*B*a^3*b^3*c*d^4*g*e^3/(d*x + c) - 3*I*(b*x*e + a*e)*A*a^4
*b^2*d^5*g*e^3/(d*x + c) + I*(b*x*e + a*e)*B*a^4*b^2*d^5*g*e^3/(d*x + c) - I*(b*x*e + a*e)^2*B*b^5*c^4*d^2*g*e
^2/(d*x + c)^2 + 4*I*(b*x*e + a*e)^2*B*a*b^4*c^3*d^3*g*e^2/(d*x + c)^2 - 6*I*(b*x*e + a*e)^2*B*a^2*b^3*c^2*d^4
*g*e^2/(d*x + c)^2 + 4*I*(b*x*e + a*e)^2*B*a^3*b^2*c*d^5*g*e^2/(d*x + c)^2 - I*(b*x*e + a*e)^2*B*a^4*b*d^6*g*e
^2/(d*x + c)^2)*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2/(b^5*d^2*e^3 - 3*(b*
x*e + a*e)*b^4*d^3*e^2/(d*x + c) + 3*(b*x*e + a*e)^2*b^3*d^4*e/(d*x + c)^2 - (b*x*e + a*e)^3*b^2*d^5/(d*x + c)
^3)

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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 (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\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))))/(c*i + d*i*x),x)

[Out]

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

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