\(\int \frac {\Gamma (2,a+b x)}{c+d x} \, dx\) [123]

Optimal result

Integrand size = 15, antiderivative size = 81 \[ \int \frac {\Gamma (2,a+b x)}{c+d x} \, dx=-\frac {e^{-a-b x}}{d}+\frac {e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d}-\frac {(b c-a d) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^2} \] Output:

-exp(-b*x-a)/d+exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d-(-a*d+b*c)*exp(-a+b*c/d)*E 
i(-b*(d*x+c)/d)/d^2
 

Mathematica [F]

\[ \int \frac {\Gamma (2,a+b x)}{c+d x} \, dx=\int \frac {\Gamma (2,a+b x)}{c+d x} \, dx \] Input:

Integrate[Gamma[2, a + b*x]/(c + d*x),x]
 

Output:

Integrate[Gamma[2, a + b*x]/(c + d*x), x]
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {7118, 2609, 2629, 2009}

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.

\(\displaystyle \int \frac {\Gamma (2,a+b x)}{c+d x} \, dx\)

\(\Big \downarrow \) 7118

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

\(\Big \downarrow \) 2609

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {(b c-a d) e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^2}+\frac {e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d}-\frac {e^{-a-b x}}{d}\)

Input:

Int[Gamma[2, a + b*x]/(c + d*x),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2609

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si 
mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F 
reeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]
 

rule 2629

Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte 
grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ 
Px, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]
 

rule 7118

Int[Gamma[n_, (a_.) + (b_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Int[(a 
 + b*x)^(n - 1)/((c + d*x)*E^(a + b*x)), x] + Simp[(n - 1)   Int[Gamma[n - 
1, a + b*x]/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 1]
 

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.32

method	result	size
derivativedivides	\(-\frac {\frac {b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d}+\frac {b \,{\mathrm e}^{-b x -a}}{d}+\frac {\left (a d -c b \right ) b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{2}}}{b}\)	\(107\)
default	\(-\frac {\frac {b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d}+\frac {b \,{\mathrm e}^{-b x -a}}{d}+\frac {\left (a d -c b \right ) b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{2}}}{b}\)	\(107\)
risch	\(-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d}-\frac {{\mathrm e}^{-b x -a}}{d}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a}{d}+\frac {b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c}{d^{2}}\)	\(135\)

Input:

int(exp(-b*x-a)*(b*x+a+1)/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

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

Output:

integral(gamma(2, b*x + a)/(d*x + c), x)
 

Sympy [F]

\[ \int \frac {\Gamma (2,a+b x)}{c+d x} \, dx=\left (\int \frac {a}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {b x}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {1}{c e^{b x} + d x e^{b x}}\, dx\right ) e^{- a} \] Input:

integrate(uppergamma(2,b*x+a)/(d*x+c),x)
 

Output:

(Integral(a/(c*exp(b*x) + d*x*exp(b*x)), x) + Integral(b*x/(c*exp(b*x) + d 
*x*exp(b*x)), x) + Integral(1/(c*exp(b*x) + d*x*exp(b*x)), x))*exp(-a)
 

Maxima [F]

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

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

Output:

integrate(gamma(2, b*x + a)/(d*x + c), x)
 

Giac [F]

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

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

Output:

integrate(gamma(2, b*x + a)/(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\Gamma (2,a+b x)}{c+d x} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,\left (a+b\,x+1\right )}{c+d\,x} \,d x \] Input:

int((exp(- a - b*x)*(a + b*x + 1))/(c + d*x),x)
 

Output:

int((exp(- a - b*x)*(a + b*x + 1))/(c + d*x), x)
 

Reduce [F]

\[ \int \frac {\Gamma (2,a+b x)}{c+d x} \, dx=\frac {e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) a d -e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) b c +e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) d -1}{e^{b x +a} d} \] Input:

int(exp(-b*x-a)*(b*x+a+1)/(d*x+c),x)
 

Output:

(e**(b*x)*int(1/(e**(b*x)*c + e**(b*x)*d*x),x)*a*d - e**(b*x)*int(1/(e**(b 
*x)*c + e**(b*x)*d*x),x)*b*c + e**(b*x)*int(1/(e**(b*x)*c + e**(b*x)*d*x), 
x)*d - 1)/(e**(a + b*x)*d)