\(\int \frac {\Gamma (3,a+b x)}{c+d x} \, dx\) [132]

Optimal result

Integrand size = 15, antiderivative size = 162 \[ \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx=-\frac {3 e^{-a-b x}}{d}+\frac {(b c-a d) e^{-a-b x}}{d^2}-\frac {e^{-a-b x} (a+b x)}{d}+\frac {2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d}-\frac {2 (b c-a d) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^2}+\frac {(b c-a d)^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3} \] Output:

-3*exp(-b*x-a)/d+(-a*d+b*c)*exp(-b*x-a)/d^2-exp(-b*x-a)*(b*x+a)/d+2*exp(-a 
+b*c/d)*Ei(-b*(d*x+c)/d)/d-2*(-a*d+b*c)*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^2 
+(-a*d+b*c)^2*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^3
 

Mathematica [F]

\[ \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx=\int \frac {\Gamma (3,a+b x)}{c+d x} \, dx \] Input:

Integrate[Gamma[3, a + b*x]/(c + d*x),x]
 

Output:

Integrate[Gamma[3, a + b*x]/(c + d*x), x]
 

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {7118, 2629, 2009, 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.

\(\displaystyle \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx\)

\(\Big \downarrow \) 7118

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

\(\displaystyle 2 \int \frac {\Gamma (2,a+b x)}{c+d x}dx+\frac {e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3}+\frac {e^{-a-b x} (b c-a d)}{d^2}-\frac {e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)}{d}\)

\(\Big \downarrow \) 7118

\(\displaystyle 2 \left (\int \frac {e^{-a-b x}}{c+d x}dx+\int \frac {e^{-a-b x} (a+b x)}{c+d x}dx\right )+\frac {e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3}+\frac {e^{-a-b x} (b c-a d)}{d^2}-\frac {e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)}{d}\)

\(\Big \downarrow \) 2609

\(\displaystyle 2 \left (\int \frac {e^{-a-b x} (a+b x)}{c+d x}dx+\frac {e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d}\right )+\frac {e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3}+\frac {e^{-a-b x} (b c-a d)}{d^2}-\frac {e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)}{d}\)

\(\Big \downarrow \) 2629

\(\displaystyle 2 \left (\int \left (\frac {e^{-a-b x} b}{d}+\frac {(a d-b c) e^{-a-b x}}{d (c+d x)}\right )dx+\frac {e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d}\right )+\frac {e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3}+\frac {e^{-a-b x} (b c-a d)}{d^2}-\frac {e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3}+2 \left (-\frac {(b c-a d) e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^2}+\frac {e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d}-\frac {e^{-a-b x}}{d}\right )+\frac {e^{-a-b x} (b c-a d)}{d^2}-\frac {e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)}{d}\)

Input:

Int[Gamma[3, a + b*x]/(c + d*x),x]
 

Output:

-(E^(-a - b*x)/d) + ((b*c - a*d)*E^(-a - b*x))/d^2 - (E^(-a - b*x)*(a + b* 
x))/d + ((b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)]) 
/d^3 + 2*(-(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 = 0.84 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.48

method	result	size
derivativedivides	\(-\frac {\frac {b a \,{\mathrm e}^{-b x -a}}{d}-\frac {b^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{3}}+\frac {2 b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d}+\frac {2 b \,{\mathrm e}^{-b x -a}}{d}+\frac {2 \left (a d -c b \right ) b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{2}}}{b}\)	\(239\)
default	\(-\frac {\frac {b a \,{\mathrm e}^{-b x -a}}{d}-\frac {b^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{3}}+\frac {2 b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d}+\frac {2 b \,{\mathrm e}^{-b x -a}}{d}+\frac {2 \left (a d -c b \right ) b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{2}}}{b}\)	\(239\)
risch	\(-\frac {2 a \,{\mathrm e}^{-b x -a}}{d}+\frac {b c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b \,{\mathrm e}^{-b x -a} x}{d}-\frac {3 \,{\mathrm e}^{-b x -a}}{d}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{2}}{d}+\frac {2 b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a c}{d^{2}}-\frac {b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c^{2}}{d^{3}}-\frac {2 \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d}-\frac {2 \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a}{d}+\frac {2 b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c}{d^{2}}\)	\(311\)

Input:

int(2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/b*(b/d*a*exp(-b*x-a)-b^2/d^2*c*exp(-b*x-a)-1/d*b*((-b*x-a)*exp(-b*x-a)- 
exp(-b*x-a))+(a^2*d^2-2*a*b*c*d+b^2*c^2)*b/d^3*exp(-(a*d-b*c)/d)*Ei(1,b*x+ 
a-(a*d-b*c)/d)+2*b/d*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)+2*b/d*exp(- 
b*x-a)+2*(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]

\[ \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{d x + c} \,d x } \] Input:

integrate(gamma(3,b*x+a)/(d*x+c),x, algorithm="fricas")
 

Output:

integral(gamma(3, b*x + a)/(d*x + c), x)
 

Sympy [F]

\[ \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx=\left (\int \frac {2 a}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {a^{2}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {2 b x}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {b^{2} x^{2}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {2 a b x}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {2}{c e^{b x} + d x e^{b x}}\, dx\right ) e^{- a} \] Input:

integrate(uppergamma(3,b*x+a)/(d*x+c),x)
 

Output:

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

Maxima [F]

\[ \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{d x + c} \,d x } \] Input:

integrate(gamma(3,b*x+a)/(d*x+c),x, algorithm="maxima")
 

Output:

integrate(gamma(3, b*x + a)/(d*x + c), x)
 

Giac [F]

\[ \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{d x + c} \,d x } \] Input:

integrate(gamma(3,b*x+a)/(d*x+c),x, algorithm="giac")
 

Output:

integrate(gamma(3, b*x + a)/(d*x + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx=\int \frac {2\,{\mathrm {e}}^{-a-b\,x}\,\left (a+b\,x+\frac {{\left (a+b\,x\right )}^2}{2}+1\right )}{c+d\,x} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\Gamma (3,a+b x)}{c+d x} \, dx=\frac {e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) a^{2} d^{2}-2 e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) a b c d +2 e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) a \,d^{2}+e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) b^{2} c^{2}-2 e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) b c d +2 e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) d^{2}-2 a d +b c -b d x -3 d}{e^{b x +a} d^{2}} \] Input:

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

Output:

(e**(b*x)*int(1/(e**(b*x)*c + e**(b*x)*d*x),x)*a**2*d**2 - 2*e**(b*x)*int( 
1/(e**(b*x)*c + e**(b*x)*d*x),x)*a*b*c*d + 2*e**(b*x)*int(1/(e**(b*x)*c + 
e**(b*x)*d*x),x)*a*d**2 + e**(b*x)*int(1/(e**(b*x)*c + e**(b*x)*d*x),x)*b* 
*2*c**2 - 2*e**(b*x)*int(1/(e**(b*x)*c + e**(b*x)*d*x),x)*b*c*d + 2*e**(b* 
x)*int(1/(e**(b*x)*c + e**(b*x)*d*x),x)*d**2 - 2*a*d + b*c - b*d*x - 3*d)/ 
(e**(a + b*x)*d**2)