3.6 \(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^2} \, dx\)

Optimal. Leaf size=142 \[ -\frac {d i \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2}-\frac {i (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b g^2 (a+b x)}+\frac {B d i \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}-\frac {B i (c+d x)}{b g^2 (a+b x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.38, antiderivative size = 221, normalized size of antiderivative = 1.56, number of steps used = 15, number of rules used = 11, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B d i \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}+\frac {d i \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2}-\frac {i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2 (a+b x)}-\frac {B i (b c-a d)}{b^2 g^2 (a+b x)}+\frac {B d i \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {B d i \log ^2(a+b x)}{2 b^2 g^2}-\frac {B d i \log (a+b x)}{b^2 g^2}+\frac {B d i \log (c+d x)}{b^2 g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*(b*c - a*d)*i)/(b^2*g^2*(a + b*x))) - (B*d*i*Log[a + b*x])/(b^2*g^2) - (B*d*i*Log[a + b*x]^2)/(2*b^2*g^2)
- ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^2*g^2*(a + b*x)) + (d*i*Log[a + b*x]*(A + B*Log[(e*
(a + b*x))/(c + d*x)]))/(b^2*g^2) + (B*d*i*Log[c + d*x])/(b^2*g^2) + (B*d*i*Log[a + b*x]*Log[(b*(c + d*x))/(b*
c - a*d)])/(b^2*g^2) + (B*d*i*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^2*g^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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

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

Rule 2391

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2524

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

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(6 c+6 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx &=\int \left (\frac {6 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^2 (a+b x)^2}+\frac {6 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^2 (a+b x)}\right ) \, dx\\ &=\frac {(6 d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b g^2}+\frac {(6 (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b g^2}\\ &=-\frac {6 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {6 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}-\frac {(6 B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^2 g^2}+\frac {(6 B (b c-a d)) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac {6 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {6 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}+\frac {\left (6 B (b c-a d)^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}-\frac {(6 B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 e g^2}\\ &=-\frac {6 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {6 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}+\frac {\left (6 B (b c-a d)^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^2}-\frac {(6 B d) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^2 e g^2}\\ &=-\frac {6 B (b c-a d)}{b^2 g^2 (a+b x)}-\frac {6 B d \log (a+b x)}{b^2 g^2}-\frac {6 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {6 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}+\frac {6 B d \log (c+d x)}{b^2 g^2}-\frac {(6 B d) \int \frac {\log (a+b x)}{a+b x} \, dx}{b g^2}+\frac {\left (6 B d^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 g^2}\\ &=-\frac {6 B (b c-a d)}{b^2 g^2 (a+b x)}-\frac {6 B d \log (a+b x)}{b^2 g^2}-\frac {6 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {6 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}+\frac {6 B d \log (c+d x)}{b^2 g^2}+\frac {6 B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {(6 B d) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g^2}-\frac {(6 B d) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g^2}\\ &=-\frac {6 B (b c-a d)}{b^2 g^2 (a+b x)}-\frac {6 B d \log (a+b x)}{b^2 g^2}-\frac {3 B d \log ^2(a+b x)}{b^2 g^2}-\frac {6 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {6 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}+\frac {6 B d \log (c+d x)}{b^2 g^2}+\frac {6 B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {(6 B d) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g^2}\\ &=-\frac {6 B (b c-a d)}{b^2 g^2 (a+b x)}-\frac {6 B d \log (a+b x)}{b^2 g^2}-\frac {3 B d \log ^2(a+b x)}{b^2 g^2}-\frac {6 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}+\frac {6 d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}+\frac {6 B d \log (c+d x)}{b^2 g^2}+\frac {6 B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {6 B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 175, normalized size = 1.23 \[ \frac {i \left (2 d (a+b x) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log \left (\frac {b (c+d x)}{b c-a d}\right )+A-B\right )-2 (A+B) (b c-a d)+2 (a B d-b B c) \log \left (\frac {e (a+b x)}{c+d x}\right )+2 B d (a+b x) \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+2 B d (a+b x) \log (c+d x)-B d (a+b x) \log ^2(a+b x)\right )}{2 b^2 g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d i x + A c i + {\left (B d i x + B c i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="fricas")

[Out]

integral((A*d*i*x + A*c*i + (B*d*i*x + B*c*i)*log((b*e*x + a*e)/(d*x + c)))/(b^2*g^2*x^2 + 2*a*b*g^2*x + a^2*g
^2), x)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [B]  time = 0.06, size = 1025, normalized size = 7.22 \[ -\frac {B a d e i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) b \,g^{2}}-\frac {B a \,d^{2} i \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) b^{2} g^{2}}+\frac {B a \,d^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right ) b^{2} g^{2}}+\frac {B c d i \ln \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) b \,g^{2}}-\frac {B c d i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right ) b \,g^{2}}+\frac {B c e i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}-\frac {A a d e i}{\left (a d -b c \right ) \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) b \,g^{2}}-\frac {A a \,d^{2} i \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{\left (a d -b c \right ) b^{2} g^{2}}+\frac {A a \,d^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) b^{2} g^{2}}+\frac {A c d i \ln \left (-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d \right )}{\left (a d -b c \right ) b \,g^{2}}-\frac {A c d i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right ) b \,g^{2}}+\frac {A c e i}{\left (a d -b c \right ) \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}}-\frac {B a d e i}{\left (a d -b c \right ) \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) b \,g^{2}}-\frac {B a \,d^{2} i \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{\left (a d -b c \right ) b^{2} g^{2}}+\frac {B c d i \dilog \left (-\frac {-b e +\left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right ) d}{b e}\right )}{\left (a d -b c \right ) b \,g^{2}}+\frac {B c e i}{\left (a d -b c \right ) \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right ) g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -B d i {\left (\frac {{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )} \log \left (d x + c\right )}{b^{3} g^{2} x + a b^{2} g^{2}} - \int \frac {b^{2} d x^{2} \log \relax (e) + a^{2} d + {\left (b^{2} c \log \relax (e) + a b d\right )} x + {\left (2 \, b^{2} d x^{2} + a^{2} d + {\left (b^{2} c + 2 \, a b d\right )} x\right )} \log \left (b x + a\right )}{b^{4} d g^{2} x^{3} + a^{2} b^{2} c g^{2} + {\left (b^{4} c g^{2} + 2 \, a b^{3} d g^{2}\right )} x^{2} + {\left (2 \, a b^{3} c g^{2} + a^{2} b^{2} d g^{2}\right )} x}\,{d x}\right )} + A d i {\left (\frac {a}{b^{3} g^{2} x + a b^{2} g^{2}} + \frac {\log \left (b x + a\right )}{b^{2} g^{2}}\right )} - B c i {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {A c i}{b^{2} g^{2} x + a b g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algorithm="maxima")

[Out]

-B*d*i*(((b*x + a)*log(b*x + a) + a)*log(d*x + c)/(b^3*g^2*x + a*b^2*g^2) - integrate((b^2*d*x^2*log(e) + a^2*
d + (b^2*c*log(e) + a*b*d)*x + (2*b^2*d*x^2 + a^2*d + (b^2*c + 2*a*b*d)*x)*log(b*x + a))/(b^4*d*g^2*x^3 + a^2*
b^2*c*g^2 + (b^4*c*g^2 + 2*a*b^3*d*g^2)*x^2 + (2*a*b^3*c*g^2 + a^2*b^2*d*g^2)*x), x)) + A*d*i*(a/(b^3*g^2*x +
a*b^2*g^2) + log(b*x + a)/(b^2*g^2)) - B*c*i*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^2*g^2*x + a*b*g^2) + 1/(
b^2*g^2*x + a*b*g^2) + d*log(b*x + a)/((b^2*c - a*b*d)*g^2) - d*log(d*x + c)/((b^2*c - a*b*d)*g^2)) - A*c*i/(b
^2*g^2*x + a*b*g^2)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (c\,i+d\,i\,x\right )\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^2,x)

[Out]

int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^2, x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________