\(\int \frac {\Gamma (-3,a+b x)}{(c+d x)^3} \, dx\) [159]

Optimal result

Integrand size = 15, antiderivative size = 205 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^3} \, dx=\frac {b^2 \Gamma (-3,a+b x)}{2 d (b c-a d)^2}-\frac {\Gamma (-3,a+b x)}{2 d (c+d x)^2}-\frac {b^2 \Gamma (-2,a+b x)}{(b c-a d)^3}+\frac {3 b^2 d \Gamma (-1,a+b x)}{2 (b c-a d)^4}+\frac {b^2 d e^{-a+\frac {b c}{d}} \Gamma \left (-1,\frac {b (c+d x)}{d}\right )}{2 (b c-a d)^4}-\frac {2 b^2 d^2 \Gamma (0,a+b x)}{(b c-a d)^5}+\frac {2 b^2 d^2 e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{(b c-a d)^5} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(877\) vs. \(2(205)=410\).

Time = 6.91 (sec) , antiderivative size = 877, normalized size of antiderivative = 4.28 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((2*((E^(-a - b*x)*(2*a - (3 + a)*(a + b*x) + (3 + a)*(a + b*x)^2))/(b*(a 
+ b*x)^3) + (3*ExpIntegralEi[-a - b*x])/b + (a*ExpIntegralEi[-a - b*x])/b 
+ 6*x*Gamma[-3, a + b*x]))/(c + d*x)^3 + d*(E^(-a - b*x)*((-2*b^2*(b*c - 3 
*a*d))/(d^2*(-(b*c) + a*d)^3*(a + b*x)^3) - (b^2*(b^2*c^2 - 4*a*b*c*d + 3* 
(-4 + a)*a*d^2))/(d^2*(b*c - a*d)^4*(a + b*x)^2) - (b^2*(b^3*c^3 - 5*a*b^2 
*c^2*d + a*(-6 + 7*a)*b*c*d^2 - 3*a*(8 - 2*a + a^2)*d^3))/(d^2*(-(b*c) + a 
*d)^5*(a + b*x)) + (2*((3 + a)*b^2*c^2 + (3 - 5*a - 2*a^2)*b*c*d + a*(-1 + 
 2*a + a^2)*d^2))/(b*(b*c - a*d)^3*(c + d*x)^3) + (2*((3 + a)*b^2*c^2 - 2* 
(-3 + 2*a + a^2)*b*c*d + a^2*(1 + a)*d^2))/((b*c - a*d)^4*(c + d*x)^2) + ( 
2*b*((3 + a)*b^2*c^2 + (12 - 3*a - 2*a^2)*b*c*d + a^3*d^2))/((b*c - a*d)^5 
*(c + d*x))) + (18*b^3*c*ExpIntegralEi[-a - b*x])/(b*c - a*d)^5 - (12*a*b^ 
3*c*ExpIntegralEi[-a - b*x])/(b*c - a*d)^5 + (3*a^2*b^3*c*ExpIntegralEi[-a 
 - b*x])/(b*c - a*d)^5 + (b^5*c^3*ExpIntegralEi[-a - b*x])/(d^2*(b*c - a*d 
)^5) + (6*b^4*c^2*ExpIntegralEi[-a - b*x])/(d*(b*c - a*d)^5) + (24*b^2*d*E 
xpIntegralEi[-a - b*x])/(b*c - a*d)^5 + (3*a*b^4*c^2*ExpIntegralEi[-a - b* 
x])/(d*(-(b*c) + a*d)^5) + (18*a*b^2*d*ExpIntegralEi[-a - b*x])/(-(b*c) + 
a*d)^5 - (6*a^2*b^2*d*ExpIntegralEi[-a - b*x])/(-(b*c) + a*d)^5 + (a^3*b^2 
*d*ExpIntegralEi[-a - b*x])/(-(b*c) + a*d)^5 - (2*(3 + a)*ExpIntegralEi[-a 
 - b*x])/(b*d*(c + d*x)^3) + (6*b^3*c*E^(-a + (b*c)/d)*ExpIntegralEi[-((b* 
(c + d*x))/d)])/(b*c - a*d)^5 - (24*b^2*d*E^(-a + (b*c)/d)*ExpIntegralE...
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(430\) vs. \(2(205)=410\).

Time = 1.17 (sec) , antiderivative size = 430, normalized size of antiderivative = 2.10, 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.

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

\(\Big \downarrow \) 7119

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

int(1/(b*x+a)^3*Ei(4,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. 2285 vs. \(2 (195) = 390\).

Time = 0.21 (sec) , antiderivative size = 2285, normalized size of antiderivative = 11.15 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/12*(6*(a^3*b^3*c^3*d^2 - (a^4 + 4*a^3)*b^2*c^2*d^3 + (b^6*c*d^4 - (a + 4 
)*b^5*d^5)*x^5 + (2*b^6*c^2*d^3 + (a - 8)*b^5*c*d^4 - 3*(a^2 + 4*a)*b^4*d^ 
5)*x^4 + (b^6*c^3*d^2 + (5*a - 4)*b^5*c^2*d^3 - 3*(a^2 + 8*a)*b^4*c*d^4 - 
3*(a^3 + 4*a^2)*b^3*d^5)*x^3 + (3*a*b^5*c^3*d^2 + 3*(a^2 - 4*a)*b^4*c^2*d^ 
3 - (5*a^3 + 24*a^2)*b^3*c*d^4 - (a^4 + 4*a^3)*b^2*d^5)*x^2 + (3*a^2*b^4*c 
^3*d^2 - (a^3 + 12*a^2)*b^3*c^2*d^3 - 2*(a^4 + 4*a^3)*b^2*c*d^4)*x)*Ei(-(b 
*d*x + b*c)/d)*e^((b*c - a*d)/d) + (a^3*b^5*c^5 - 3*(a^4 - 2*a^3)*b^4*c^4* 
d + 3*(a^5 - 4*a^4 + 6*a^3)*b^3*c^3*d^2 - (a^6 - 6*a^5 + 18*a^4 - 24*a^3)* 
b^2*c^2*d^3 + (b^8*c^3*d^2 - 3*(a - 2)*b^7*c^2*d^3 + 3*(a^2 - 4*a + 6)*b^6 
*c*d^4 - (a^3 - 6*a^2 + 18*a - 24)*b^5*d^5)*x^5 + (2*b^8*c^4*d - 3*(a - 4) 
*b^7*c^3*d^2 - 3*(a^2 + 2*a - 12)*b^6*c^2*d^3 + (7*a^3 - 24*a^2 + 18*a + 4 
8)*b^5*c*d^4 - 3*(a^4 - 6*a^3 + 18*a^2 - 24*a)*b^4*d^5)*x^4 + (b^8*c^5 + 3 
*(a + 2)*b^7*c^4*d - 6*(2*a^2 - 4*a - 3)*b^6*c^3*d^2 + 2*(4*a^3 - 24*a^2 + 
 45*a + 12)*b^5*c^2*d^3 + 3*(a^4 - 18*a^2 + 48*a)*b^4*c*d^4 - 3*(a^5 - 6*a 
^4 + 18*a^3 - 24*a^2)*b^3*d^5)*x^3 + (3*a*b^7*c^5 - 3*(a^2 - 6*a)*b^6*c^4* 
d - 2*(4*a^3 - 27*a)*b^5*c^3*d^2 + 6*(2*a^4 - 8*a^3 + 9*a^2 + 12*a)*b^4*c^ 
2*d^3 - 3*(a^5 - 8*a^4 + 30*a^3 - 48*a^2)*b^3*c*d^4 - (a^6 - 6*a^5 + 18*a^ 
4 - 24*a^3)*b^2*d^5)*x^2 + (3*a^2*b^6*c^5 - (7*a^3 - 18*a^2)*b^5*c^4*d + 3 
*(a^4 - 8*a^3 + 18*a^2)*b^4*c^3*d^2 + 3*(a^5 - 2*a^4 - 6*a^3 + 24*a^2)*b^3 
*c^2*d^3 - 2*(a^6 - 6*a^5 + 18*a^4 - 24*a^3)*b^2*c*d^4)*x)*Ei(-b*x - a)...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.12 \[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^3} \, dx=\int \frac {\mathrm {expint}\left (4,a+b\,x\right )}{{\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\Gamma (-3,a+b x)}{(c+d x)^3} \, dx=\int \frac {\mathit {ei} \left (4, b x +a \right )}{b^{3} d^{3} x^{6}+3 a \,b^{2} d^{3} x^{5}+3 b^{3} c \,d^{2} x^{5}+3 a^{2} b \,d^{3} x^{4}+9 a \,b^{2} c \,d^{2} x^{4}+3 b^{3} c^{2} d \,x^{4}+a^{3} d^{3} x^{3}+9 a^{2} b c \,d^{2} x^{3}+9 a \,b^{2} c^{2} d \,x^{3}+b^{3} c^{3} x^{3}+3 a^{3} c \,d^{2} x^{2}+9 a^{2} b \,c^{2} d \,x^{2}+3 a \,b^{2} c^{3} x^{2}+3 a^{3} c^{2} d x +3 a^{2} b \,c^{3} x +a^{3} c^{3}}d x \] Input:

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

Output:

int(ei(4,a + b*x)/(a**3*c**3 + 3*a**3*c**2*d*x + 3*a**3*c*d**2*x**2 + a**3 
*d**3*x**3 + 3*a**2*b*c**3*x + 9*a**2*b*c**2*d*x**2 + 9*a**2*b*c*d**2*x**3 
 + 3*a**2*b*d**3*x**4 + 3*a*b**2*c**3*x**2 + 9*a*b**2*c**2*d*x**3 + 9*a*b* 
*2*c*d**2*x**4 + 3*a*b**2*d**3*x**5 + b**3*c**3*x**3 + 3*b**3*c**2*d*x**4 
+ 3*b**3*c*d**2*x**5 + b**3*d**3*x**6),x)