\(\int \frac {\Gamma (-3,a+b x)}{(c+d x)^2} \, dx\) [158]

Optimal result

Integrand size = 15, antiderivative size = 144 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^2} \, dx=\frac {b \Gamma (-3,a+b x)}{d (b c-a d)}-\frac {\Gamma (-3,a+b x)}{d (c+d x)}-\frac {b \Gamma (-2,a+b x)}{(b c-a d)^2}+\frac {b d \Gamma (-1,a+b x)}{(b c-a d)^3}-\frac {b d^2 \Gamma (0,a+b x)}{(b c-a d)^4}+\frac {b d^2 e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{(b c-a d)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.90 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^2} \, dx=\frac {1}{6} \left (\frac {3 b e^{-a-b x} (-1+a+b x)}{(b c-a d)^2 (a+b x)^2}+\frac {6 b d e^{-a-b x}}{(b c-a d)^3 (a+b x)}+\frac {b e^{-a-b x} \left (2-a-b x+(a+b x)^2\right )}{d (b c-a d) (a+b x)^3}+\frac {6 b d^2 \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^4}+\frac {6 b d \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^3}+\frac {3 b \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^2}+\frac {b \operatorname {ExpIntegralEi}(-a-b x)}{b c d-a d^2}-\frac {6 b d^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^4}-\frac {6 \Gamma (-3,a+b x)}{d (c+d x)}\right ) \] Input:

Integrate[Gamma[-3, a + b*x]/(c + d*x)^2,x]
 

Output:

((3*b*E^(-a - b*x)*(-1 + a + b*x))/((b*c - a*d)^2*(a + b*x)^2) + (6*b*d*E^ 
(-a - b*x))/((b*c - a*d)^3*(a + b*x)) + (b*E^(-a - b*x)*(2 - a - b*x + (a 
+ b*x)^2))/(d*(b*c - a*d)*(a + b*x)^3) + (6*b*d^2*ExpIntegralEi[-a - b*x]) 
/(b*c - a*d)^4 + (6*b*d*ExpIntegralEi[-a - b*x])/(b*c - a*d)^3 + (3*b*ExpI 
ntegralEi[-a - b*x])/(b*c - a*d)^2 + (b*ExpIntegralEi[-a - b*x])/(b*c*d - 
a*d^2) - (6*b*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c 
 - a*d)^4 - (6*Gamma[-3, a + b*x])/(d*(c + d*x)))/6
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(349\) vs. \(2(144)=288\).

Time = 1.02 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.42, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7119, 7293, 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.

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

\(\Big \downarrow \) 7119

\(\displaystyle -\frac {b \int \frac {e^{-a-b x}}{(a+b x)^4 (c+d x)}dx}{d}-\frac {\Gamma (-3,a+b x)}{d (c+d x)}\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \left (-\frac {d^3 \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^4}+\frac {d^3 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^4}-\frac {d^2 \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^3}-\frac {d^2 e^{-a-b x}}{(a+b x) (b c-a d)^3}-\frac {d \operatorname {ExpIntegralEi}(-a-b x)}{2 (b c-a d)^2}-\frac {\operatorname {ExpIntegralEi}(-a-b x)}{6 (b c-a d)}-\frac {d e^{-a-b x}}{2 (a+b x) (b c-a d)^2}+\frac {d e^{-a-b x}}{2 (a+b x)^2 (b c-a d)^2}-\frac {e^{-a-b x}}{6 (a+b x) (b c-a d)}+\frac {e^{-a-b x}}{6 (a+b x)^2 (b c-a d)}-\frac {e^{-a-b x}}{3 (a+b x)^3 (b c-a d)}\right )}{d}-\frac {\Gamma (-3,a+b x)}{d (c+d x)}\)

Input:

Int[Gamma[-3, a + b*x]/(c + d*x)^2,x]
 

Output:

-((b*(-1/3*E^(-a - b*x)/((b*c - a*d)*(a + b*x)^3) + (d*E^(-a - b*x))/(2*(b 
*c - a*d)^2*(a + b*x)^2) + E^(-a - b*x)/(6*(b*c - a*d)*(a + b*x)^2) - (d^2 
*E^(-a - b*x))/((b*c - a*d)^3*(a + b*x)) - (d*E^(-a - b*x))/(2*(b*c - a*d) 
^2*(a + b*x)) - E^(-a - b*x)/(6*(b*c - a*d)*(a + b*x)) - (d^3*ExpIntegralE 
i[-a - b*x])/(b*c - a*d)^4 - (d^2*ExpIntegralEi[-a - b*x])/(b*c - a*d)^3 - 
 (d*ExpIntegralEi[-a - b*x])/(2*(b*c - a*d)^2) - ExpIntegralEi[-a - b*x]/( 
6*(b*c - a*d)) + (d^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/ 
(b*c - a*d)^4))/d) - Gamma[-3, a + b*x]/(d*(c + d*x))
 

Defintions of rubi rules used

rule 2009

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

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]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [F]

\[\int \frac {\operatorname {expIntegral}_{4}\left (b x +a \right )}{\left (b x +a \right )^{3} \left (d x +c \right )^{2}}d x\]

Input:

int(1/(b*x+a)^3*Ei(4,b*x+a)/(d*x+c)^2,x)
 

Output:

int(1/(b*x+a)^3*Ei(4,b*x+a)/(d*x+c)^2,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1517 vs. \(2 (143) = 286\).

Time = 0.14 (sec) , antiderivative size = 1517, normalized size of antiderivative = 10.53 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:

integrate(gamma(-3,b*x+a)/(d*x+c)^2,x, algorithm="fricas")
 

Output:

-1/6*(6*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a* 
b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*Ei(-(b*d*x 
 + b*c)/d)*e^((b*c - a*d)/d) - (a^3*b^4*c^4 - 3*(a^4 - a^3)*b^3*c^3*d + 3* 
(a^5 - 2*a^4 + 2*a^3)*b^2*c^2*d^2 - (a^6 - 3*a^5 + 6*a^4 - 6*a^3)*b*c*d^3 
+ (b^7*c^3*d - 3*(a - 1)*b^6*c^2*d^2 + 3*(a^2 - 2*a + 2)*b^5*c*d^3 - (a^3 
- 3*a^2 + 6*a - 6)*b^4*d^4)*x^4 + (b^7*c^4 + 3*b^6*c^3*d - 3*(2*a^2 - a - 
2)*b^5*c^2*d^2 + (8*a^3 - 15*a^2 + 12*a + 6)*b^4*c*d^3 - 3*(a^4 - 3*a^3 + 
6*a^2 - 6*a)*b^3*d^4)*x^3 + 3*(a*b^6*c^4 - (2*a^2 - 3*a)*b^5*c^3*d - 3*(a^ 
2 - 2*a)*b^4*c^2*d^2 + (2*a^4 - 3*a^3 + 6*a)*b^3*c*d^3 - (a^5 - 3*a^4 + 6* 
a^3 - 6*a^2)*b^2*d^4)*x^2 + (3*a^2*b^5*c^4 - (8*a^3 - 9*a^2)*b^4*c^3*d + 3 
*(2*a^4 - 5*a^3 + 6*a^2)*b^3*c^2*d^2 + 3*(a^4 - 4*a^3 + 6*a^2)*b^2*c*d^3 - 
 (a^6 - 3*a^5 + 6*a^4 - 6*a^3)*b*d^4)*x)*Ei(-b*x - a) - ((a^2 - a + 2)*b^4 
*c^4 - 3*(a^3 - 2*a^2 + 3*a)*b^3*c^3*d + 3*(a^4 - 3*a^3 + 6*a^2)*b^2*c^2*d 
^2 - (a^5 - 4*a^4 + 11*a^3)*b*c*d^3 + (b^6*c^3*d - 3*(a - 1)*b^5*c^2*d^2 + 
 3*(a^2 - 2*a + 2)*b^4*c*d^3 - (a^3 - 3*a^2 + 6*a)*b^3*d^4)*x^3 + (b^6*c^4 
 - (a - 2)*b^5*c^3*d - 3*(a^2 - a - 1)*b^4*c^2*d^2 + (5*a^3 - 12*a^2 + 12* 
a)*b^3*c*d^3 - (2*a^4 - 7*a^3 + 15*a^2)*b^2*d^4)*x^2 + ((2*a - 1)*b^5*c^4 
- (5*a^2 - 8*a + 1)*b^4*c^3*d + 3*(a^3 - 3*a^2 + 3*a)*b^3*c^2*d^2 + (a^4 - 
 2*a^3 + 3*a^2)*b^2*c*d^3 - (a^5 - 4*a^4 + 11*a^3)*b*d^4)*x)*e^(-b*x - a) 
+ 6*(a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 ...
 

Sympy [F]

\[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^2} \, dx=\int \frac {\operatorname {E}_{4}\left (a + b x\right )}{\left (a + b x\right )^{3} \left (c + d x\right )^{2}}\, dx \] Input:

integrate(uppergamma(-3,b*x+a)/(d*x+c)**2,x)
 

Output:

Integral(expint(4, a + b*x)/((a + b*x)**3*(c + d*x)**2), x)
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate(gamma(-3,b*x+a)/(d*x+c)^2,x, algorithm="maxima")
 

Output:

integrate(gamma(-3, b*x + a)/(d*x + c)^2, x)
 

Giac [F]

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

integrate(gamma(-3,b*x+a)/(d*x+c)^2,x, algorithm="giac")
 

Output:

integrate(gamma(-3, b*x + a)/(d*x + c)^2, x)
 

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.17 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {expint}\left (4,a+b\,x\right )}{{\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \] Input:

int(expint(4, a + b*x)/((a + b*x)^3*(c + d*x)^2),x)
 

Output:

int(expint(4, a + b*x)/((a + b*x)^3*(c + d*x)^2), x)
 

Reduce [F]

\[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathit {ei} \left (4, b x +a \right )}{b^{3} d^{2} x^{5}+3 a \,b^{2} d^{2} x^{4}+2 b^{3} c d \,x^{4}+3 a^{2} b \,d^{2} x^{3}+6 a \,b^{2} c d \,x^{3}+b^{3} c^{2} x^{3}+a^{3} d^{2} x^{2}+6 a^{2} b c d \,x^{2}+3 a \,b^{2} c^{2} x^{2}+2 a^{3} c d x +3 a^{2} b \,c^{2} x +a^{3} c^{2}}d x \] Input:

int(1/(b*x+a)^3*Ei(4,b*x+a)/(d*x+c)^2,x)
 

Output:

int(ei(4,a + b*x)/(a**3*c**2 + 2*a**3*c*d*x + a**3*d**2*x**2 + 3*a**2*b*c* 
*2*x + 6*a**2*b*c*d*x**2 + 3*a**2*b*d**2*x**3 + 3*a*b**2*c**2*x**2 + 6*a*b 
**2*c*d*x**3 + 3*a*b**2*d**2*x**4 + b**3*c**2*x**3 + 2*b**3*c*d*x**4 + b** 
3*d**2*x**5),x)