\(\int \frac {\Gamma (-1,a+b x)}{(c+d x)^4} \, dx\) [144]

Optimal result

Integrand size = 15, antiderivative size = 197 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^4} \, dx=\frac {b^3 e^{-a+\frac {b c}{d}} \Gamma \left (-2,\frac {b (c+d x)}{d}\right )}{3 d^2 (b c-a d)^2}+\frac {b^3 \Gamma (-1,a+b x)}{3 d (b c-a d)^3}-\frac {\Gamma (-1,a+b x)}{3 d (c+d x)^3}+\frac {2 b^3 e^{-a+\frac {b c}{d}} \Gamma \left (-1,\frac {b (c+d x)}{d}\right )}{3 d (b c-a d)^3}-\frac {b^3 \Gamma (0,a+b x)}{(b c-a d)^4}+\frac {b^3 e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{(b c-a d)^4} \] Output:

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

Mathematica [F]

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

Integrate[Gamma[-1, a + b*x]/(c + d*x)^4,x]
 

Output:

Integrate[Gamma[-1, a + b*x]/(c + d*x)^4, x]
 

Rubi [A] (verified)

Time = 1.53 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.68, 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.

\(\displaystyle \int \frac {\Gamma (-1,a+b x)}{(c+d x)^4} \, dx\)

\(\Big \downarrow \) 7119

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

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {b \int \left (-\frac {3 d e^{-a-b x} b^3}{(b c-a d)^4 (a+b x)}+\frac {e^{-a-b x} b^3}{(b c-a d)^3 (a+b x)^2}+\frac {3 d^2 e^{-a-b x} b^2}{(b c-a d)^4 (c+d x)}+\frac {2 d^2 e^{-a-b x} b}{(b c-a d)^3 (c+d x)^2}+\frac {d^2 e^{-a-b x}}{(b c-a d)^2 (c+d x)^3}\right )dx}{3 d}-\frac {\Gamma (-1,a+b x)}{3 d (c+d x)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \left (-\frac {b^2 \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^3}-\frac {3 b^2 d \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^4}+\frac {b^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{2 d (b c-a d)^2}-\frac {2 b^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^3}+\frac {3 b^2 d e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^4}-\frac {b^2 e^{-a-b x}}{(a+b x) (b c-a d)^3}+\frac {b e^{-a-b x}}{2 (c+d x) (b c-a d)^2}-\frac {2 b d e^{-a-b x}}{(c+d x) (b c-a d)^3}-\frac {d e^{-a-b x}}{2 (c+d x)^2 (b c-a d)^2}\right )}{3 d}-\frac {\Gamma (-1,a+b x)}{3 d (c+d x)^3}\)

Input:

Int[Gamma[-1, a + b*x]/(c + d*x)^4,x]
 

Output:

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

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1218 vs. \(2 (186) = 372\).

Time = 0.14 (sec) , antiderivative size = 1218, normalized size of antiderivative = 6.18 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \] Input:

integrate(gamma(-1,b*x+a)/(d*x+c)^4,x, algorithm="fricas")
 

Output:

-1/6*((a*b^5*c^5 - 2*(a^2 + 2*a)*b^4*c^4*d + (a^3 + 4*a^2 + 6*a)*b^3*c^3*d 
^2 + (b^6*c^2*d^3 - 2*(a + 2)*b^5*c*d^4 + (a^2 + 4*a + 6)*b^4*d^5)*x^4 + ( 
3*b^6*c^3*d^2 - (5*a + 12)*b^5*c^2*d^3 + (a^2 + 8*a + 18)*b^4*c*d^4 + (a^3 
 + 4*a^2 + 6*a)*b^3*d^5)*x^3 + 3*(b^6*c^4*d - (a + 4)*b^5*c^3*d^2 - (a^2 - 
 6)*b^4*c^2*d^3 + (a^3 + 4*a^2 + 6*a)*b^3*c*d^4)*x^2 + (b^6*c^5 + (a - 4)* 
b^5*c^4*d - (5*a^2 + 8*a - 6)*b^4*c^3*d^2 + 3*(a^3 + 4*a^2 + 6*a)*b^3*c^2* 
d^3)*x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) - 2*(a*b^4*c^4*d - (a^2 - 3 
*a)*b^3*c^3*d^2 + (b^5*c*d^4 - (a - 3)*b^4*d^5)*x^4 + (3*b^5*c^2*d^3 - (2* 
a - 9)*b^4*c*d^4 - (a^2 - 3*a)*b^3*d^5)*x^3 + 3*(b^5*c^3*d^2 + 3*b^4*c^2*d 
^3 - (a^2 - 3*a)*b^3*c*d^4)*x^2 + (b^5*c^4*d + (2*a + 3)*b^4*c^3*d^2 - 3*( 
a^2 - 3*a)*b^3*c^2*d^3)*x)*Ei(-b*x - a) + ((a - 2)*b^4*c^4*d - (2*a^2 + 3* 
a)*b^3*c^3*d^2 - a^3*b*c*d^4 + (a^3 + 6*a^2)*b^2*c^2*d^3 + (b^5*c^2*d^3 - 
2*(a + 3)*b^4*c*d^4 + (a^2 + 6*a)*b^3*d^5)*x^3 + (2*b^5*c^3*d^2 - 3*(a + 5 
)*b^4*c^2*d^3 + 12*a*b^3*c*d^4 + (a^3 + 3*a^2)*b^2*d^5)*x^2 + (b^5*c^4*d - 
 11*b^4*c^3*d^2 - 3*(a^2 - a)*b^3*c^2*d^3 - a^3*b*d^5 + (2*a^3 + 9*a^2)*b^ 
2*c*d^4)*x)*e^(-b*x - a) + 2*(a*b^4*c^4*d - 4*a^2*b^3*c^3*d^2 + 6*a^3*b^2* 
c^2*d^3 - 4*a^4*b*c*d^4 + a^5*d^5 + (b^5*c^4*d - 4*a*b^4*c^3*d^2 + 6*a^2*b 
^3*c^2*d^3 - 4*a^3*b^2*c*d^4 + a^4*b*d^5)*x)*gamma(-1, b*x + a))/(a*b^4*c^ 
7*d^2 - 4*a^2*b^3*c^6*d^3 + 6*a^3*b^2*c^5*d^4 - 4*a^4*b*c^4*d^5 + a^5*c^3* 
d^6 + (b^5*c^4*d^5 - 4*a*b^4*c^3*d^6 + 6*a^2*b^3*c^2*d^7 - 4*a^3*b^2*c*...
 

Sympy [F]

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

integrate(uppergamma(-1,b*x+a)/(d*x+c)**4,x)
 

Output:

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

Maxima [F]

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

integrate(gamma(-1,b*x+a)/(d*x+c)^4,x, algorithm="maxima")
 

Output:

integrate(gamma(-1, b*x + a)/(d*x + c)^4, x)
 

Giac [F]

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

integrate(gamma(-1,b*x+a)/(d*x+c)^4,x, algorithm="giac")
 

Output:

integrate(gamma(-1, b*x + a)/(d*x + c)^4, x)
 

Mupad [B] (verification not implemented)

Time = 0.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.12 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^4} \, dx=\int \frac {\mathrm {expint}\left (2,a+b\,x\right )}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^4} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^4} \, dx=\int \frac {\mathit {ei} \left (2, b x +a \right )}{b \,d^{4} x^{5}+a \,d^{4} x^{4}+4 b c \,d^{3} x^{4}+4 a c \,d^{3} x^{3}+6 b \,c^{2} d^{2} x^{3}+6 a \,c^{2} d^{2} x^{2}+4 b \,c^{3} d \,x^{2}+4 a \,c^{3} d x +b \,c^{4} x +a \,c^{4}}d x \] Input:

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

Output:

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