\(\int \frac {\Gamma (-1,a+b x)}{(c+d x)^3} \, dx\) [143]

Optimal result

Integrand size = 15, antiderivative size = 153 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^3} \, dx=\frac {b^2 \Gamma (-1,a+b x)}{2 d (b c-a d)^2}-\frac {\Gamma (-1,a+b x)}{2 d (c+d x)^2}+\frac {b^2 e^{-a+\frac {b c}{d}} \Gamma \left (-1,\frac {b (c+d x)}{d}\right )}{2 d (b c-a d)^2}-\frac {b^2 \Gamma (0,a+b x)}{(b c-a d)^3}+\frac {b^2 e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{(b c-a d)^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.14 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^3} \, dx=\frac {1}{2} \left (\frac {b^2 e^{-a-b x}}{d (b c-a d)^2 (a+b x)}+\frac {b e^{-a-b x}}{(b c-a d)^2 (c+d x)}+\frac {b^2 (-b c+(-2+a) d) \operatorname {ExpIntegralEi}(-a-b x)}{d (-b c+a d)^3}+\frac {b^2 (b c-(2+a) d) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d (b c-a d)^3}-\frac {\Gamma (-1,a+b x)}{d (c+d x)^2}\right ) \] Input:

Integrate[Gamma[-1, a + b*x]/(c + d*x)^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.38, 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.

\(\displaystyle \int \frac {\Gamma (-1,a+b x)}{(c+d x)^3} \, dx\)

\(\Big \downarrow \) 7119

\(\displaystyle -\frac {b \int \frac {e^{-a-b x}}{(a+b x)^2 (c+d x)^2}dx}{2 d}-\frac {\Gamma (-1,a+b x)}{2 d (c+d x)^2}\)

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \left (-\frac {b \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^2}-\frac {2 b d \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^3}-\frac {b e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^2}+\frac {2 b d e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^3}-\frac {b e^{-a-b x}}{(a+b x) (b c-a d)^2}-\frac {d e^{-a-b x}}{(c+d x) (b c-a d)^2}\right )}{2 d}-\frac {\Gamma (-1,a+b x)}{2 d (c+d x)^2}\)

Input:

Int[Gamma[-1, a + b*x]/(c + d*x)^3,x]
 

Output:

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

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

Input:

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

Output:

int(1/(b*x+a)*Ei(2,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. 680 vs. \(2 (145) = 290\).

Time = 0.11 (sec) , antiderivative size = 680, normalized size of antiderivative = 4.44 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^3} \, dx=\frac {{\left (a b^{3} c^{3} - {\left (a^{2} + 2 \, a\right )} b^{2} c^{2} d + {\left (b^{4} c d^{2} - {\left (a + 2\right )} b^{3} d^{3}\right )} x^{3} + {\left (2 \, b^{4} c^{2} d - {\left (a + 4\right )} b^{3} c d^{2} - {\left (a^{2} + 2 \, a\right )} b^{2} d^{3}\right )} x^{2} + {\left (b^{4} c^{3} + {\left (a - 2\right )} b^{3} c^{2} d - 2 \, {\left (a^{2} + 2 \, a\right )} b^{2} c d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (a b^{3} c^{3} - {\left (a^{2} - 2 \, a\right )} b^{2} c^{2} d + {\left (b^{4} c d^{2} - {\left (a - 2\right )} b^{3} d^{3}\right )} x^{3} + {\left (2 \, b^{4} c^{2} d - {\left (a - 4\right )} b^{3} c d^{2} - {\left (a^{2} - 2 \, a\right )} b^{2} d^{3}\right )} x^{2} + {\left (b^{4} c^{3} + {\left (a + 2\right )} b^{3} c^{2} d - 2 \, {\left (a^{2} - 2 \, a\right )} b^{2} c d^{2}\right )} x\right )} {\rm Ei}\left (-b x - a\right ) + {\left (b^{3} c^{3} - a^{2} b c d^{2} + 2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + {\left (3 \, b^{3} c^{2} d - 2 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x\right )} e^{\left (-b x - a\right )} - {\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3} + {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x\right )} \Gamma \left (-1, b x + a\right )}{2 \, {\left (a b^{3} c^{5} d - 3 \, a^{2} b^{2} c^{4} d^{2} + 3 \, a^{3} b c^{3} d^{3} - a^{4} c^{2} d^{4} + {\left (b^{4} c^{3} d^{3} - 3 \, a b^{3} c^{2} d^{4} + 3 \, a^{2} b^{2} c d^{5} - a^{3} b d^{6}\right )} x^{3} + {\left (2 \, b^{4} c^{4} d^{2} - 5 \, a b^{3} c^{3} d^{3} + 3 \, a^{2} b^{2} c^{2} d^{4} + a^{3} b c d^{5} - a^{4} d^{6}\right )} x^{2} + {\left (b^{4} c^{5} d - a b^{3} c^{4} d^{2} - 3 \, a^{2} b^{2} c^{3} d^{3} + 5 \, a^{3} b c^{2} d^{4} - 2 \, a^{4} c d^{5}\right )} x\right )}} \] Input:

integrate(gamma(-1,b*x+a)/(d*x+c)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate(uppergamma(-1,b*x+a)/(d*x+c)**3,x)
 

Output:

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

Maxima [F]

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

integrate(gamma(-1,b*x+a)/(d*x+c)^3,x, algorithm="maxima")
 

Output:

integrate(gamma(-1, b*x + a)/(d*x + c)^3, x)
 

Giac [F]

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

integrate(gamma(-1,b*x+a)/(d*x+c)^3,x, algorithm="giac")
 

Output:

integrate(gamma(-1, b*x + a)/(d*x + c)^3, x)
 

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.16 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^3} \, dx=\int \frac {\mathrm {expint}\left (2,a+b\,x\right )}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^3} \, dx=\int \frac {\mathit {ei} \left (2, b x +a \right )}{b \,d^{3} x^{4}+a \,d^{3} x^{3}+3 b c \,d^{2} x^{3}+3 a c \,d^{2} x^{2}+3 b \,c^{2} d \,x^{2}+3 a \,c^{2} d x +b \,c^{3} x +a \,c^{3}}d x \] Input:

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

Output:

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