\(\int \frac {\Gamma (-2,a+b x)}{(c+d x)^3} \, dx\) [151]

Optimal result

Integrand size = 15, antiderivative size = 179 \[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^3} \, dx=\frac {b^2 \Gamma (-2,a+b x)}{2 d (b c-a d)^2}-\frac {\Gamma (-2,a+b x)}{2 d (c+d x)^2}-\frac {b^2 \Gamma (-1,a+b x)}{(b c-a d)^3}-\frac {b^2 e^{-a+\frac {b c}{d}} \Gamma \left (-1,\frac {b (c+d x)}{d}\right )}{2 (b c-a d)^3}+\frac {3 b^2 d \Gamma (0,a+b x)}{2 (b c-a d)^4}-\frac {3 b^2 d e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{2 (b c-a d)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.73 \[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^3} \, dx=-\frac {\frac {b^2 \left (b^2 c^2-2 (-2+a) b c d+\left (6-4 a+a^2\right ) d^2\right ) \operatorname {ExpIntegralEi}(-a-b x)}{d}+2 b^2 (b c-(3+a) d) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+\frac {(b c-a d) \left (2 b d^2 e^{-a-b x} (a+b x)^2 (c+d x)-b^3 c e^{-a-b x} (c+d x)^2+a b^2 d e^{-a-b x} (c+d x)^2+b^3 c e^{-a-b x} (a+b x) (c+d x)^2+4 b^2 d e^{-a-b x} (a+b x) (c+d x)^2-a b^2 d e^{-a-b x} (a+b x) (c+d x)^2+2 (b c-a d)^3 (a+b x)^2 \Gamma (-2,a+b x)\right )}{d (a+b x)^2 (c+d x)^2}}{4 (b c-a d)^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.71, 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.

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

\(\Big \downarrow \) 7119

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1284 vs. \(2 (167) = 334\).

Time = 0.13 (sec) , antiderivative size = 1284, normalized size of antiderivative = 7.17 \[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 19.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.13 \[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^3} \, dx=\int \frac {\mathrm {expint}\left (3,a+b\,x\right )}{{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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