\(\int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx\) [126]

Optimal result

Integrand size = 15, antiderivative size = 98 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx=-\frac {b^3 (b c-a d) e^{-a+\frac {b c}{d}} \Gamma \left (-2,\frac {b (c+d x)}{d}\right )}{3 d^5}+\frac {b^3 e^{-a+\frac {b c}{d}} \Gamma \left (-1,\frac {b (c+d x)}{d}\right )}{3 d^4}-\frac {\Gamma (2,a+b x)}{3 d (c+d x)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.19 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx=\frac {b^3 (b c-(-2+a) d) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+\frac {d \left (b e^{-a-b x} (c+d x) \left (a d^2+b^2 c (c+d x)+b d (c-a c-(-2+a) d x)\right )-2 d^3 \Gamma (2,a+b x)\right )}{(c+d x)^3}}{6 d^5} \] Input:

Integrate[Gamma[2, a + b*x]/(c + d*x)^4,x]
 

Output:

(b^3*(b*c - (-2 + a)*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] 
 + (d*(b*E^(-a - b*x)*(c + d*x)*(a*d^2 + b^2*c*(c + d*x) + b*d*(c - a*c - 
(-2 + a)*d*x)) - 2*d^3*Gamma[2, a + b*x]))/(c + d*x)^3)/(6*d^5)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7119, 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.

\(\displaystyle \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx\)

\(\Big \downarrow \) 7119

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \left (-\frac {b^2 (b c-a d) e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{2 d^4}-\frac {b^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3}-\frac {b e^{-a-b x} (b c-a d)}{2 d^3 (c+d x)}-\frac {b e^{-a-b x}}{d^2 (c+d x)}+\frac {e^{-a-b x} (b c-a d)}{2 d^2 (c+d x)^2}\right )}{3 d}-\frac {\Gamma (2,a+b x)}{3 d (c+d x)^3}\)

Input:

Int[Gamma[2, a + b*x]/(c + d*x)^4,x]
 

Output:

-1/3*(b*(((b*c - a*d)*E^(-a - b*x))/(2*d^2*(c + d*x)^2) - (b*E^(-a - b*x)) 
/(d^2*(c + d*x)) - (b*(b*c - a*d)*E^(-a - b*x))/(2*d^3*(c + d*x)) - (b^2*E 
^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^3 - (b^2*(b*c - a*d)* 
E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(2*d^4)))/d - Gamma[2, 
 a + b*x]/(3*d*(c + d*x)^3)
 

Defintions of rubi rules used

rule 2009

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

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 7119

Int[Gamma[n_, (a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> 
Block[{$UseGamma = True}, Simp[(c + d*x)^(m + 1)*(Gamma[n, a + b*x]/(d*(m + 
 1))), x] + Simp[b/(d*(m + 1))   Int[(c + d*x)^(m + 1)*((a + b*x)^(n - 1)/E 
^(a + b*x)), x], x]] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || IGtQ 
[n, 0] || IntegersQ[m, n]) && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(411\) vs. \(2(110)=220\).

Time = 0.76 (sec) , antiderivative size = 412, normalized size of antiderivative = 4.20

method	result	size
derivativedivides	\(-\frac {\frac {b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{4}}+\frac {b^{4} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{5}}-\frac {b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{4}}}{b}\)	\(412\)
default	\(-\frac {\frac {b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{4}}+\frac {b^{4} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{5}}-\frac {b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{4}}}{b}\)	\(412\)
risch	\(\frac {b^{3} {\mathrm e}^{-b x -a}}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{3}}-\frac {b^{3} {\mathrm e}^{-b x -a}}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{2}}-\frac {b^{3} {\mathrm e}^{-b x -a}}{3 d^{4} \left (-b x -\frac {c b}{d}\right )}-\frac {b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{3 d^{4}}+\frac {b^{3} {\mathrm e}^{-b x -a} a}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{3}}-\frac {b^{4} {\mathrm e}^{-b x -a} c}{3 d^{5} \left (-b x -\frac {c b}{d}\right )^{3}}+\frac {b^{3} {\mathrm e}^{-b x -a} a}{6 d^{4} \left (-b x -\frac {c b}{d}\right )^{2}}-\frac {b^{4} {\mathrm e}^{-b x -a} c}{6 d^{5} \left (-b x -\frac {c b}{d}\right )^{2}}+\frac {b^{3} {\mathrm e}^{-b x -a} a}{6 d^{4} \left (-b x -\frac {c b}{d}\right )}-\frac {b^{4} {\mathrm e}^{-b x -a} c}{6 d^{5} \left (-b x -\frac {c b}{d}\right )}+\frac {b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a}{6 d^{4}}-\frac {b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c}{6 d^{5}}\)	\(415\)

Input:

int(exp(-b*x-a)*(b*x+a+1)/(d*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/b*(b^4/d^4*(-1/3*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^3-1/6*exp(-b*x-a)/(-b 
*x-a+(a*d-b*c)/d)^2-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)-1/6*exp(-(a*d-b*c 
)/d)*Ei(1,b*x+a-(a*d-b*c)/d))+b^4*(a*d-b*c)/d^5*(-1/3*exp(-b*x-a)/(-b*x-a+ 
(a*d-b*c)/d)^3-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/6*exp(-b*x-a)/(-b* 
x-a+(a*d-b*c)/d)-1/6*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))-b^4/d^4*(- 
1/2*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/2*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d 
)-1/2*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (90) = 180\).

Time = 0.12 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.82 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx=-\frac {2 \, d^{4} \Gamma \left (2, b x + a\right ) - {\left (b^{4} c^{4} - {\left (a - 2\right )} b^{3} c^{3} d + {\left (b^{4} c d^{3} - {\left (a - 2\right )} b^{3} d^{4}\right )} x^{3} + 3 \, {\left (b^{4} c^{2} d^{2} - {\left (a - 2\right )} b^{3} c d^{3}\right )} x^{2} + 3 \, {\left (b^{4} c^{3} d - {\left (a - 2\right )} b^{3} c^{2} d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - {\left (b^{3} c^{3} d - {\left (a - 1\right )} b^{2} c^{2} d^{2} + a b c d^{3} + {\left (b^{3} c d^{3} - {\left (a - 2\right )} b^{2} d^{4}\right )} x^{2} + {\left (2 \, b^{3} c^{2} d^{2} - {\left (2 \, a - 3\right )} b^{2} c d^{3} + a b d^{4}\right )} x\right )} e^{\left (-b x - a\right )}}{6 \, {\left (d^{8} x^{3} + 3 \, c d^{7} x^{2} + 3 \, c^{2} d^{6} x + c^{3} d^{5}\right )}} \] Input:

integrate(gamma(2,b*x+a)/(d*x+c)^4,x, algorithm="fricas")
 

Output:

-1/6*(2*d^4*gamma(2, b*x + a) - (b^4*c^4 - (a - 2)*b^3*c^3*d + (b^4*c*d^3 
- (a - 2)*b^3*d^4)*x^3 + 3*(b^4*c^2*d^2 - (a - 2)*b^3*c*d^3)*x^2 + 3*(b^4* 
c^3*d - (a - 2)*b^3*c^2*d^2)*x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) - ( 
b^3*c^3*d - (a - 1)*b^2*c^2*d^2 + a*b*c*d^3 + (b^3*c*d^3 - (a - 2)*b^2*d^4 
)*x^2 + (2*b^3*c^2*d^2 - (2*a - 3)*b^2*c*d^3 + a*b*d^4)*x)*e^(-b*x - a))/( 
d^8*x^3 + 3*c*d^7*x^2 + 3*c^2*d^6*x + c^3*d^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx=\text {Timed out} \] Input:

integrate(uppergamma(2,b*x+a)/(d*x+c)**4,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{{\left (d x + c\right )}^{4}} \,d x } \] Input:

integrate(gamma(2,b*x+a)/(d*x+c)^4,x, algorithm="maxima")
 

Output:

integrate(gamma(2, b*x + a)/(d*x + c)^4, x)
 

Giac [F]

\[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{{\left (d x + c\right )}^{4}} \,d x } \] Input:

integrate(gamma(2,b*x+a)/(d*x+c)^4,x, algorithm="giac")
 

Output:

integrate(gamma(2, b*x + a)/(d*x + c)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,\left (a+b\,x+1\right )}{{\left (c+d\,x\right )}^4} \,d x \] Input:

int((exp(- a - b*x)*(a + b*x + 1))/(c + d*x)^4,x)
 

Output:

int((exp(- a - b*x)*(a + b*x + 1))/(c + d*x)^4, x)
 

Reduce [F]

\[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^4} \, dx=\text {too large to display} \] Input:

int(exp(-b*x-a)*(b*x+a+1)/(d*x+c)^4,x)
 

Output:

( - e**(b*x)*int(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b 
*c**3*d**2*x**2 + 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3 
*e**(b*x)*c**4*d + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 
12*e**(b*x)*c*d**4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a*b**2*c**4*d - 3*e**(b 
*x)*int(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b*c**3*d** 
2*x**2 + 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3*e**(b*x) 
*c**4*d + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12*e**(b* 
x)*c*d**4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a*b**2*c**3*d**2*x - 3*e**(b*x)* 
int(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b*c**3*d**2*x* 
*2 + 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3*e**(b*x)*c** 
4*d + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12*e**(b*x)*c 
*d**4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a*b**2*c**2*d**3*x**2 - e**(b*x)*int 
(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b*c**3*d**2*x**2 
+ 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3*e**(b*x)*c**4*d 
 + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12*e**(b*x)*c*d* 
*4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a*b**2*c*d**4*x**3 - 3*e**(b*x)*int(x/( 
e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b*c**3*d**2*x**2 + 4* 
e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3*e**(b*x)*c**4*d + 1 
2*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12*e**(b*x)*c*d**4*x 
**3 + 3*e**(b*x)*d**5*x**4),x)*a*b*c**3*d**2 - 9*e**(b*x)*int(x/(e**(b*...