\(\int \frac {(b e-a e e^{c+d x}) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx\) [500]

Optimal result

Integrand size = 47, antiderivative size = 150 \[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\frac {x^2}{2}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 d}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 d}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{2 d^2}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 d^2} \] Output:

1/2*x^2-1/2*x*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/d-1/2*x*ln(1+b*exp(d* 
x+c)/(a+(a^2+b^2)^(1/2)))/d-1/2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2) 
))/d^2-1/2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/d^2
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(398\) vs. \(2(150)=300\).

Time = 0.76 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.65 \[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\frac {-a d x \log \left (1+\frac {\left (a-\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )-\sqrt {a^2+b^2} d x \log \left (1+\frac {\left (a-\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )+a d x \log \left (1+\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )-\sqrt {a^2+b^2} d x \log \left (1+\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )+a d x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-a d x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+\left (a+\sqrt {a^2+b^2}\right ) \operatorname {PolyLog}\left (2,\frac {\left (-a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )+\left (-a+\sqrt {a^2+b^2}\right ) \operatorname {PolyLog}\left (2,-\frac {\left (a+\sqrt {a^2+b^2}\right ) e^{-c-d x}}{b}\right )+a \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-a \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{2 \sqrt {a^2+b^2} d^2} \] Input:

Integrate[((b*e - a*e*E^(c + d*x))*x)/(b*e - 2*a*e*E^(c + d*x) - b*e*E^(2* 
(c + d*x))),x]
 

Output:

(-(a*d*x*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b]) - Sqrt[a^2 + b^2 
]*d*x*Log[1 + ((a - Sqrt[a^2 + b^2])*E^(-c - d*x))/b] + a*d*x*Log[1 + ((a 
+ Sqrt[a^2 + b^2])*E^(-c - d*x))/b] - Sqrt[a^2 + b^2]*d*x*Log[1 + ((a + Sq 
rt[a^2 + b^2])*E^(-c - d*x))/b] + a*d*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[ 
a^2 + b^2])] - a*d*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + (a + 
 Sqrt[a^2 + b^2])*PolyLog[2, ((-a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b] + (- 
a + Sqrt[a^2 + b^2])*PolyLog[2, -(((a + Sqrt[a^2 + b^2])*E^(-c - d*x))/b)] 
 + a*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - a*PolyLog[2, -(( 
b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(2*Sqrt[a^2 + b^2]*d^2)
 

Rubi [A] (verified)

Time = 1.75 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {2695, 27, 2615, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x \left (b e-a e e^{c+d x}\right )}{-2 a e e^{c+d x}-b e e^{2 (c+d x)}+b e} \, dx\)

\(\Big \downarrow \) 2695

\(\displaystyle -\left (e \left (a-\sqrt {a^2+b^2}\right ) \int -\frac {x}{2 \left (b e^{c+d x} e+\left (a-\sqrt {a^2+b^2}\right ) e\right )}dx\right )-e \left (\sqrt {a^2+b^2}+a\right ) \int -\frac {x}{2 \left (b e^{c+d x} e+\left (a+\sqrt {a^2+b^2}\right ) e\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \int \frac {x}{b e^{c+d x} e+\left (a-\sqrt {a^2+b^2}\right ) e}dx+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \int \frac {x}{b e^{c+d x} e+\left (a+\sqrt {a^2+b^2}\right ) e}dx\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \left (\frac {x^2}{2 e \left (a-\sqrt {a^2+b^2}\right )}-\frac {b \int \frac {e^{c+d x} x}{b e^{c+d x} e+\left (a-\sqrt {a^2+b^2}\right ) e}dx}{a-\sqrt {a^2+b^2}}\right )+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \left (\frac {x^2}{2 e \left (\sqrt {a^2+b^2}+a\right )}-\frac {b \int \frac {e^{c+d x} x}{b e^{c+d x} e+\left (a+\sqrt {a^2+b^2}\right ) e}dx}{\sqrt {a^2+b^2}+a}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \left (\frac {x^2}{2 e \left (a-\sqrt {a^2+b^2}\right )}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d e}-\frac {\int \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )dx}{b d e}\right )}{a-\sqrt {a^2+b^2}}\right )+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \left (\frac {x^2}{2 e \left (\sqrt {a^2+b^2}+a\right )}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d e}-\frac {\int \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )dx}{b d e}\right )}{\sqrt {a^2+b^2}+a}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \left (\frac {x^2}{2 e \left (a-\sqrt {a^2+b^2}\right )}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d e}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2 e}\right )}{a-\sqrt {a^2+b^2}}\right )+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \left (\frac {x^2}{2 e \left (\sqrt {a^2+b^2}+a\right )}-\frac {b \left (\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d e}-\frac {\int e^{-c-d x} \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right )de^{c+d x}}{b d^2 e}\right )}{\sqrt {a^2+b^2}+a}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {1}{2} e \left (a-\sqrt {a^2+b^2}\right ) \left (\frac {x^2}{2 e \left (a-\sqrt {a^2+b^2}\right )}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b d^2 e}+\frac {x \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b d e}\right )}{a-\sqrt {a^2+b^2}}\right )+\frac {1}{2} e \left (\sqrt {a^2+b^2}+a\right ) \left (\frac {x^2}{2 e \left (\sqrt {a^2+b^2}+a\right )}-\frac {b \left (\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b d^2 e}+\frac {x \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b d e}\right )}{\sqrt {a^2+b^2}+a}\right )\)

Input:

Int[((b*e - a*e*E^(c + d*x))*x)/(b*e - 2*a*e*E^(c + d*x) - b*e*E^(2*(c + d 
*x))),x]
 

Output:

((a - Sqrt[a^2 + b^2])*e*(x^2/(2*(a - Sqrt[a^2 + b^2])*e) - (b*((x*Log[1 + 
 (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b*d*e) + PolyLog[2, -((b*E^(c + 
d*x))/(a - Sqrt[a^2 + b^2]))]/(b*d^2*e)))/(a - Sqrt[a^2 + b^2])))/2 + ((a 
+ Sqrt[a^2 + b^2])*e*(x^2/(2*(a + Sqrt[a^2 + b^2])*e) - (b*((x*Log[1 + (b* 
E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b*d*e) + PolyLog[2, -((b*E^(c + d*x) 
)/(a + Sqrt[a^2 + b^2]))]/(b*d^2*e)))/(a + Sqrt[a^2 + b^2])))/2
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2615

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x 
_))))^(n_.)), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(a*d*(m + 1)), x] - Simp[ 
b/a   Int[(c + d*x)^m*((F^(g*(e + f*x)))^n/(a + b*(F^(g*(e + f*x)))^n)), x] 
, x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2695

Int[(((i_.)*(F_)^(u_) + (h_))*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F 
_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Sim 
p[(Simplify[(2*c*h - b*i)/q] + i)   Int[(f + g*x)^m/(b - q + 2*c*F^u), x], 
x] - Simp[(Simplify[(2*c*h - b*i)/q] - i)   Int[(f + g*x)^m/(b + q + 2*c*F^ 
u), x], x]] /; FreeQ[{F, a, b, c, f, g, h, i}, x] && EqQ[v, 2*u] && LinearQ 
[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

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

rule 2838

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(317\) vs. \(2(128)=256\).

Time = 0.15 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.12

method	result	size
risch	\(\frac {c \ln \left (2 \,{\mathrm e}^{d x +c} a +{\mathrm e}^{2 d x +2 c} b -b \right )}{2 d^{2}}-\frac {c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2}}-\frac {\ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) x}{2 d}-\frac {\ln \left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right ) c}{2 d^{2}}-\frac {\ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) x}{2 d}-\frac {\ln \left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right ) c}{2 d^{2}}-\frac {\operatorname {dilog}\left (\frac {-b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}-a}{-a +\sqrt {a^{2}+b^{2}}}\right )}{2 d^{2}}-\frac {\operatorname {dilog}\left (\frac {b \,{\mathrm e}^{d x +c}+\sqrt {a^{2}+b^{2}}+a}{a +\sqrt {a^{2}+b^{2}}}\right )}{2 d^{2}}+\frac {x^{2}}{2}+\frac {c x}{d}+\frac {c^{2}}{2 d^{2}}\)	\(318\)
default	\(\text {Expression too large to display}\)	\(965\)

Input:

int((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)),x,met 
hod=_RETURNVERBOSE)
 

Output:

1/2/d^2*c*ln(2*exp(d*x+c)*a+exp(2*d*x+2*c)*b-b)-1/d^2*c*ln(exp(d*x+c))-1/2 
/d*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/2/d^2*ln 
((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/2/d*ln((b*exp 
(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/2/d^2*ln((b*exp(d*x+c) 
+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/2/d^2*dilog((-b*exp(d*x+c)+(a 
^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/2/d^2*dilog((b*exp(d*x+c)+(a^2+b^ 
2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))+1/2*x^2+c*x/d+1/2/d^2*c^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.67 \[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\frac {d^{2} x^{2} + c \log \left (2 \, b e^{\left (d x + c\right )} + 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) + c \log \left (2 \, b e^{\left (d x + c\right )} - 2 \, b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} + 2 \, a\right ) - {\left (d x + c\right )} \log \left (-\frac {b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} e^{\left (d x + c\right )} + a e^{\left (d x + c\right )} - b}{b}\right ) - {\left (d x + c\right )} \log \left (\frac {b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} e^{\left (d x + c\right )} - a e^{\left (d x + c\right )} + b}{b}\right ) - {\rm Li}_2\left (\frac {b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} e^{\left (d x + c\right )} + a e^{\left (d x + c\right )} - b}{b} + 1\right ) - {\rm Li}_2\left (-\frac {b \sqrt {\frac {a^{2} + b^{2}}{b^{2}}} e^{\left (d x + c\right )} - a e^{\left (d x + c\right )} + b}{b} + 1\right )}{2 \, d^{2}} \] Input:

integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)) 
,x, algorithm="fricas")
 

Output:

1/2*(d^2*x^2 + c*log(2*b*e^(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 
c*log(2*b*e^(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - (d*x + c)*log(- 
(b*sqrt((a^2 + b^2)/b^2)*e^(d*x + c) + a*e^(d*x + c) - b)/b) - (d*x + c)*l 
og((b*sqrt((a^2 + b^2)/b^2)*e^(d*x + c) - a*e^(d*x + c) + b)/b) - dilog((b 
*sqrt((a^2 + b^2)/b^2)*e^(d*x + c) + a*e^(d*x + c) - b)/b + 1) - dilog(-(b 
*sqrt((a^2 + b^2)/b^2)*e^(d*x + c) - a*e^(d*x + c) + b)/b + 1))/d^2
 

Sympy [F]

\[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\int \frac {x \left (a e^{c} e^{d x} - b\right )}{2 a e^{c} e^{d x} + b e^{2 c} e^{2 d x} - b}\, dx \] Input:

integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)) 
,x)
 

Output:

Integral(x*(a*exp(c)*exp(d*x) - b)/(2*a*exp(c)*exp(d*x) + b*exp(2*c)*exp(2 
*d*x) - b), x)
 

Maxima [F]

\[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\int { \frac {{\left (a e e^{\left (d x + c\right )} - b e\right )} x}{b e e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e e^{\left (d x + c\right )} - b e} \,d x } \] Input:

integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)) 
,x, algorithm="maxima")
 

Output:

integrate((a*e*e^(d*x + c) - b*e)*x/(b*e*e^(2*d*x + 2*c) + 2*a*e*e^(d*x + 
c) - b*e), x)
 

Giac [F]

\[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\int { \frac {{\left (a e e^{\left (d x + c\right )} - b e\right )} x}{b e e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e e^{\left (d x + c\right )} - b e} \,d x } \] Input:

integrate((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)) 
,x, algorithm="giac")
 

Output:

integrate((a*e*e^(d*x + c) - b*e)*x/(b*e*e^(2*d*x + 2*c) + 2*a*e*e^(d*x + 
c) - b*e), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=\int -\frac {x\,\left (b\,e-a\,e\,{\mathrm {e}}^{c+d\,x}\right )}{2\,a\,e\,{\mathrm {e}}^{c+d\,x}-b\,e+b\,e\,{\mathrm {e}}^{2\,c+2\,d\,x}} \,d x \] Input:

int(-(x*(b*e - a*e*exp(c + d*x)))/(2*a*e*exp(c + d*x) - b*e + b*e*exp(2*c 
+ 2*d*x)),x)
 

Output:

int(-(x*(b*e - a*e*exp(c + d*x)))/(2*a*e*exp(c + d*x) - b*e + b*e*exp(2*c 
+ 2*d*x)), x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\left (b e-a e e^{c+d x}\right ) x}{b e-2 a e e^{c+d x}-b e e^{2 (c+d x)}} \, dx=e^{c} \left (\int \frac {e^{d x} x}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) a -\left (\int \frac {x}{e^{2 d x +2 c} b +2 e^{d x +c} a -b}d x \right ) b \] Input:

int((b*e-a*e*exp(d*x+c))*x/(b*e-2*a*e*exp(d*x+c)-b*e*exp(2*d*x+2*c)),x)
 

Output:

e**c*int((e**(d*x)*x)/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*a - i 
nt(x/(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b),x)*b