\(\int \frac {\Gamma (-3,a+b x)}{(c+d x)^4} \, dx\) [160]

Optimal result

Integrand size = 15, antiderivative size = 248 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^4} \, dx=\frac {b^3 \Gamma (-3,a+b x)}{3 d (b c-a d)^3}-\frac {\Gamma (-3,a+b x)}{3 d (c+d x)^3}-\frac {b^3 \Gamma (-2,a+b x)}{(b c-a d)^4}+\frac {b^3 e^{-a+\frac {b c}{d}} \Gamma \left (-2,\frac {b (c+d x)}{d}\right )}{3 (b c-a d)^4}+\frac {2 b^3 d \Gamma (-1,a+b x)}{(b c-a d)^5}+\frac {4 b^3 d e^{-a+\frac {b c}{d}} \Gamma \left (-1,\frac {b (c+d x)}{d}\right )}{3 (b c-a d)^5}-\frac {10 b^3 d^2 \Gamma (0,a+b x)}{3 (b c-a d)^6}+\frac {10 b^3 d^2 e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{3 (b c-a d)^6} \] Output:

1/3*b^3/(b*x+a)^3*Ei(4,b*x+a)/d/(-a*d+b*c)^3-1/3/(b*x+a)^3*Ei(4,b*x+a)/d/( 
d*x+c)^3-b^3/(b*x+a)^2*Ei(3,b*x+a)/(-a*d+b*c)^4+1/3*b*exp(-a+b*c/d)/(d*x+c 
)^2*d^2*Ei(3,b*(d*x+c)/d)/(-a*d+b*c)^4+2*b^3*d/(b*x+a)*Ei(2,b*x+a)/(-a*d+b 
*c)^5+4/3*b^2*d^2*exp(-a+b*c/d)/(d*x+c)*Ei(2,b*(d*x+c)/d)/(-a*d+b*c)^5-10/ 
3*b^3*d^2*Ei(1,b*x+a)/(-a*d+b*c)^6+10/3*b^3*d^2*exp(-a+b*c/d)*Ei(1,b*(d*x+ 
c)/d)/(-a*d+b*c)^6
 

Mathematica [F]

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

Integrate[Gamma[-3, a + b*x]/(c + d*x)^4,x]
 

Output:

Integrate[Gamma[-3, a + b*x]/(c + d*x)^4, x]
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(573\) vs. \(2(248)=496\).

Time = 1.37 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.31, 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.

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

\(\Big \downarrow \) 7119

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

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {b \int \left (\frac {10 b^2 e^{-a-b x} d^4}{(b c-a d)^6 (c+d x)}+\frac {4 b e^{-a-b x} d^4}{(b c-a d)^5 (c+d x)^2}+\frac {e^{-a-b x} d^4}{(b c-a d)^4 (c+d x)^3}-\frac {10 b^3 e^{-a-b x} d^3}{(b c-a d)^6 (a+b x)}+\frac {6 b^3 e^{-a-b x} d^2}{(b c-a d)^5 (a+b x)^2}-\frac {3 b^3 e^{-a-b x} d}{(b c-a d)^4 (a+b x)^3}+\frac {b^3 e^{-a-b x}}{(b c-a d)^3 (a+b x)^4}\right )dx}{3 d}-\frac {\Gamma (-3,a+b x)}{3 d (c+d x)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b \left (-\frac {10 b^2 d^3 \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^6}+\frac {10 b^2 d^3 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^6}-\frac {6 b^2 d^2 \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^5}-\frac {4 b^2 d^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^5}-\frac {6 b^2 d^2 e^{-a-b x}}{(a+b x) (b c-a d)^5}-\frac {3 b^2 d \operatorname {ExpIntegralEi}(-a-b x)}{2 (b c-a d)^4}+\frac {b^2 d e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{2 (b c-a d)^4}-\frac {b^2 \operatorname {ExpIntegralEi}(-a-b x)}{6 (b c-a d)^3}-\frac {3 b^2 d e^{-a-b x}}{2 (a+b x) (b c-a d)^4}+\frac {3 b^2 d e^{-a-b x}}{2 (a+b x)^2 (b c-a d)^4}-\frac {b^2 e^{-a-b x}}{6 (a+b x) (b c-a d)^3}+\frac {b^2 e^{-a-b x}}{6 (a+b x)^2 (b c-a d)^3}-\frac {b^2 e^{-a-b x}}{3 (a+b x)^3 (b c-a d)^3}-\frac {4 b d^3 e^{-a-b x}}{(c+d x) (b c-a d)^5}-\frac {d^3 e^{-a-b x}}{2 (c+d x)^2 (b c-a d)^4}+\frac {b d^2 e^{-a-b x}}{2 (c+d x) (b c-a d)^4}\right )}{3 d}-\frac {\Gamma (-3,a+b x)}{3 d (c+d x)^3}\)

Input:

Int[Gamma[-3, a + b*x]/(c + d*x)^4,x]
 

Output:

-1/3*(b*(-1/3*(b^2*E^(-a - b*x))/((b*c - a*d)^3*(a + b*x)^3) + (3*b^2*d*E^ 
(-a - b*x))/(2*(b*c - a*d)^4*(a + b*x)^2) + (b^2*E^(-a - b*x))/(6*(b*c - a 
*d)^3*(a + b*x)^2) - (6*b^2*d^2*E^(-a - b*x))/((b*c - a*d)^5*(a + b*x)) - 
(3*b^2*d*E^(-a - b*x))/(2*(b*c - a*d)^4*(a + b*x)) - (b^2*E^(-a - b*x))/(6 
*(b*c - a*d)^3*(a + b*x)) - (d^3*E^(-a - b*x))/(2*(b*c - a*d)^4*(c + d*x)^ 
2) - (4*b*d^3*E^(-a - b*x))/((b*c - a*d)^5*(c + d*x)) + (b*d^2*E^(-a - b*x 
))/(2*(b*c - a*d)^4*(c + d*x)) - (10*b^2*d^3*ExpIntegralEi[-a - b*x])/(b*c 
 - a*d)^6 - (6*b^2*d^2*ExpIntegralEi[-a - b*x])/(b*c - a*d)^5 - (3*b^2*d*E 
xpIntegralEi[-a - b*x])/(2*(b*c - a*d)^4) - (b^2*ExpIntegralEi[-a - b*x])/ 
(6*(b*c - a*d)^3) + (10*b^2*d^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d 
*x))/d)])/(b*c - a*d)^6 - (4*b^2*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*( 
c + d*x))/d)])/(b*c - a*d)^5 + (b^2*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b* 
(c + d*x))/d)])/(2*(b*c - a*d)^4)))/d - Gamma[-3, 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]

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

Input:

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

Output:

int(1/(b*x+a)^3*Ei(4,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. 3116 vs. \(2 (233) = 466\).

Time = 0.35 (sec) , antiderivative size = 3116, normalized size of antiderivative = 12.56 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^4} \, dx=\text {Too large to display} \] Input:

integrate(gamma(-3,b*x+a)/(d*x+c)^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate(uppergamma(-3,b*x+a)/(d*x+c)**4,x)
 

Output:

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

Maxima [F]

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

integrate(gamma(-3,b*x+a)/(d*x+c)^4,x, algorithm="maxima")
 

Output:

integrate(gamma(-3, b*x + a)/(d*x + c)^4, x)
 

Giac [F]

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

integrate(gamma(-3,b*x+a)/(d*x+c)^4,x, algorithm="giac")
 

Output:

integrate(gamma(-3, b*x + a)/(d*x + c)^4, x)
 

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.10 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^4} \, dx=\int \frac {\mathrm {expint}\left (4,a+b\,x\right )}{{\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^4} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^4} \, dx=\int \frac {\mathit {ei} \left (4, b x +a \right )}{b^{3} d^{4} x^{7}+3 a \,b^{2} d^{4} x^{6}+4 b^{3} c \,d^{3} x^{6}+3 a^{2} b \,d^{4} x^{5}+12 a \,b^{2} c \,d^{3} x^{5}+6 b^{3} c^{2} d^{2} x^{5}+a^{3} d^{4} x^{4}+12 a^{2} b c \,d^{3} x^{4}+18 a \,b^{2} c^{2} d^{2} x^{4}+4 b^{3} c^{3} d \,x^{4}+4 a^{3} c \,d^{3} x^{3}+18 a^{2} b \,c^{2} d^{2} x^{3}+12 a \,b^{2} c^{3} d \,x^{3}+b^{3} c^{4} x^{3}+6 a^{3} c^{2} d^{2} x^{2}+12 a^{2} b \,c^{3} d \,x^{2}+3 a \,b^{2} c^{4} x^{2}+4 a^{3} c^{3} d x +3 a^{2} b \,c^{4} x +a^{3} c^{4}}d x \] Input:

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

Output:

int(ei(4,a + b*x)/(a**3*c**4 + 4*a**3*c**3*d*x + 6*a**3*c**2*d**2*x**2 + 4 
*a**3*c*d**3*x**3 + a**3*d**4*x**4 + 3*a**2*b*c**4*x + 12*a**2*b*c**3*d*x* 
*2 + 18*a**2*b*c**2*d**2*x**3 + 12*a**2*b*c*d**3*x**4 + 3*a**2*b*d**4*x**5 
 + 3*a*b**2*c**4*x**2 + 12*a*b**2*c**3*d*x**3 + 18*a*b**2*c**2*d**2*x**4 + 
 12*a*b**2*c*d**3*x**5 + 3*a*b**2*d**4*x**6 + b**3*c**4*x**3 + 4*b**3*c**3 
*d*x**4 + 6*b**3*c**2*d**2*x**5 + 4*b**3*c*d**3*x**6 + b**3*d**4*x**7),x)