\(\int \frac {\Gamma (2,a+b x)}{(c+d x)^5} \, dx\) [127]

Optimal result

Integrand size = 15, antiderivative size = 98 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^5} \, dx=-\frac {b^4 (b c-a d) e^{-a+\frac {b c}{d}} \Gamma \left (-3,\frac {b (c+d x)}{d}\right )}{4 d^6}+\frac {b^4 e^{-a+\frac {b c}{d}} \Gamma \left (-2,\frac {b (c+d x)}{d}\right )}{4 d^5}-\frac {\Gamma (2,a+b x)}{4 d (c+d x)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.95 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^5} \, dx=-\frac {\frac {b d e^{-a-b x} \left (2 d^2 (b c-a d)-b d (b c-(-3+a) d) (c+d x)+b^2 (b c-(-3+a) d) (c+d x)^2\right )}{(c+d x)^3}+b^5 c e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+3 b^4 d e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-a b^4 d e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+\frac {6 d^5 \Gamma (2,a+b x)}{(c+d x)^4}}{24 d^6} \] Input:

Integrate[Gamma[2, a + b*x]/(c + d*x)^5,x]
 

Output:

-1/24*((b*d*E^(-a - b*x)*(2*d^2*(b*c - a*d) - b*d*(b*c - (-3 + a)*d)*(c + 
d*x) + b^2*(b*c - (-3 + a)*d)*(c + d*x)^2))/(c + d*x)^3 + b^5*c*E^(-a + (b 
*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] + 3*b^4*d*E^(-a + (b*c)/d)*ExpInt 
egralEi[-((b*(c + d*x))/d)] - a*b^4*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b* 
(c + d*x))/d)] + (6*d^5*Gamma[2, a + b*x])/(c + d*x)^4)/d^6
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(259\) vs. \(2(98)=196\).

Time = 0.65 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.64, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7119, 2629, 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.

\(\displaystyle \int \frac {\Gamma (2,a+b x)}{(c+d x)^5} \, dx\)

\(\Big \downarrow \) 7119

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

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

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

Input:

Int[Gamma[2, a + b*x]/(c + d*x)^5,x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2629

Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte 
grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ 
Px, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(110)=220\).

Time = 0.80 (sec) , antiderivative size = 510, normalized size of antiderivative = 5.20

method	result	size
risch	\(-\frac {b^{4} {\mathrm e}^{-b x -a}}{4 d^{5} \left (-b x -\frac {c b}{d}\right )^{4}}+\frac {b^{4} {\mathrm e}^{-b x -a}}{4 d^{5} \left (-b x -\frac {c b}{d}\right )^{3}}+\frac {b^{4} {\mathrm e}^{-b x -a}}{8 d^{5} \left (-b x -\frac {c b}{d}\right )^{2}}+\frac {b^{4} {\mathrm e}^{-b x -a}}{8 d^{5} \left (-b x -\frac {c b}{d}\right )}+\frac {b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{8 d^{5}}-\frac {b^{4} {\mathrm e}^{-b x -a} a}{4 d^{5} \left (-b x -\frac {c b}{d}\right )^{4}}+\frac {b^{5} {\mathrm e}^{-b x -a} c}{4 d^{6} \left (-b x -\frac {c b}{d}\right )^{4}}-\frac {b^{4} {\mathrm e}^{-b x -a} a}{12 d^{5} \left (-b x -\frac {c b}{d}\right )^{3}}+\frac {b^{5} {\mathrm e}^{-b x -a} c}{12 d^{6} \left (-b x -\frac {c b}{d}\right )^{3}}-\frac {b^{4} {\mathrm e}^{-b x -a} a}{24 d^{5} \left (-b x -\frac {c b}{d}\right )^{2}}+\frac {b^{5} {\mathrm e}^{-b x -a} c}{24 d^{6} \left (-b x -\frac {c b}{d}\right )^{2}}-\frac {b^{4} {\mathrm e}^{-b x -a} a}{24 d^{5} \left (-b x -\frac {c b}{d}\right )}+\frac {b^{5} {\mathrm e}^{-b x -a} c}{24 d^{6} \left (-b x -\frac {c b}{d}\right )}-\frac {b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a}{24 d^{5}}+\frac {b^{5} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c}{24 d^{6}}\)	\(510\)
derivativedivides	\(-\frac {-\frac {b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{5}}-\frac {\left (a d -c b \right ) b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{6}}+\frac {b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{5}}}{b}\)	\(512\)
default	\(-\frac {-\frac {b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{5}}-\frac {\left (a d -c b \right ) b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{6}}+\frac {b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{5}}}{b}\)	\(512\)

Input:

int(exp(-b*x-a)*(b*x+a+1)/(d*x+c)^5,x,method=_RETURNVERBOSE)
 

Output:

-1/4*b^4/d^5*exp(-b*x-a)/(-b*x-c*b/d)^4+1/4*b^4/d^5*exp(-b*x-a)/(-b*x-c*b/ 
d)^3+1/8*b^4/d^5*exp(-b*x-a)/(-b*x-c*b/d)^2+1/8*b^4/d^5*exp(-b*x-a)/(-b*x- 
c*b/d)+1/8*b^4/d^5*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-1/4*b^4/d^5*e 
xp(-b*x-a)/(-b*x-c*b/d)^4*a+1/4*b^5/d^6*exp(-b*x-a)/(-b*x-c*b/d)^4*c-1/12* 
b^4/d^5*exp(-b*x-a)/(-b*x-c*b/d)^3*a+1/12*b^5/d^6*exp(-b*x-a)/(-b*x-c*b/d) 
^3*c-1/24*b^4/d^5*exp(-b*x-a)/(-b*x-c*b/d)^2*a+1/24*b^5/d^6*exp(-b*x-a)/(- 
b*x-c*b/d)^2*c-1/24*b^4/d^5*exp(-b*x-a)/(-b*x-c*b/d)*a+1/24*b^5/d^6*exp(-b 
*x-a)/(-b*x-c*b/d)*c-1/24*b^4/d^5*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d 
)*a+1/24*b^5/d^6*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)*c
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (90) = 180\).

Time = 0.15 (sec) , antiderivative size = 384, normalized size of antiderivative = 3.92 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^5} \, dx=-\frac {6 \, d^{5} \Gamma \left (2, b x + a\right ) + {\left (b^{5} c^{5} - {\left (a - 3\right )} b^{4} c^{4} d + {\left (b^{5} c d^{4} - {\left (a - 3\right )} b^{4} d^{5}\right )} x^{4} + 4 \, {\left (b^{5} c^{2} d^{3} - {\left (a - 3\right )} b^{4} c d^{4}\right )} x^{3} + 6 \, {\left (b^{5} c^{3} d^{2} - {\left (a - 3\right )} b^{4} c^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{5} c^{4} d - {\left (a - 3\right )} b^{4} c^{3} d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (b^{4} c^{4} d - {\left (a - 2\right )} b^{3} c^{3} d^{2} + {\left (a - 1\right )} b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + {\left (b^{4} c d^{4} - {\left (a - 3\right )} b^{3} d^{5}\right )} x^{3} + {\left (3 \, b^{4} c^{2} d^{3} - {\left (3 \, a - 8\right )} b^{3} c d^{4} + {\left (a - 3\right )} b^{2} d^{5}\right )} x^{2} + {\left (3 \, b^{4} c^{3} d^{2} - {\left (3 \, a - 7\right )} b^{3} c^{2} d^{3} + 2 \, {\left (a - 2\right )} b^{2} c d^{4} - 2 \, a b d^{5}\right )} x\right )} e^{\left (-b x - a\right )}}{24 \, {\left (d^{10} x^{4} + 4 \, c d^{9} x^{3} + 6 \, c^{2} d^{8} x^{2} + 4 \, c^{3} d^{7} x + c^{4} d^{6}\right )}} \] Input:

integrate(gamma(2,b*x+a)/(d*x+c)^5,x, algorithm="fricas")
 

Output:

-1/24*(6*d^5*gamma(2, b*x + a) + (b^5*c^5 - (a - 3)*b^4*c^4*d + (b^5*c*d^4 
 - (a - 3)*b^4*d^5)*x^4 + 4*(b^5*c^2*d^3 - (a - 3)*b^4*c*d^4)*x^3 + 6*(b^5 
*c^3*d^2 - (a - 3)*b^4*c^2*d^3)*x^2 + 4*(b^5*c^4*d - (a - 3)*b^4*c^3*d^2)* 
x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) + (b^4*c^4*d - (a - 2)*b^3*c^3*d 
^2 + (a - 1)*b^2*c^2*d^3 - 2*a*b*c*d^4 + (b^4*c*d^4 - (a - 3)*b^3*d^5)*x^3 
 + (3*b^4*c^2*d^3 - (3*a - 8)*b^3*c*d^4 + (a - 3)*b^2*d^5)*x^2 + (3*b^4*c^ 
3*d^2 - (3*a - 7)*b^3*c^2*d^3 + 2*(a - 2)*b^2*c*d^4 - 2*a*b*d^5)*x)*e^(-b* 
x - a))/(d^10*x^4 + 4*c*d^9*x^3 + 6*c^2*d^8*x^2 + 4*c^3*d^7*x + c^4*d^6)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^5} \, dx=\text {Timed out} \] Input:

integrate(uppergamma(2,b*x+a)/(d*x+c)**5,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(gamma(2,b*x+a)/(d*x+c)^5,x, algorithm="maxima")
 

Output:

integrate(gamma(2, b*x + a)/(d*x + c)^5, x)
 

Giac [F]

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

integrate(gamma(2,b*x+a)/(d*x+c)^5,x, algorithm="giac")
 

Output:

integrate(gamma(2, b*x + a)/(d*x + c)^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^5} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,\left (a+b\,x+1\right )}{{\left (c+d\,x\right )}^5} \,d x \] Input:

int((exp(- a - b*x)*(a + b*x + 1))/(c + d*x)^5,x)
 

Output:

int((exp(- a - b*x)*(a + b*x + 1))/(c + d*x)^5, x)
 

Reduce [F]

\[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^5} \, dx=\text {too large to display} \] Input:

int(exp(-b*x-a)*(b*x+a+1)/(d*x+c)^5,x)
 

Output:

( - e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e**(b*x)* 
b*c**4*d**2*x**2 + 10*e**(b*x)*b*c**3*d**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x 
**4 + e**(b*x)*b*c*d**5*x**5 + 4*e**(b*x)*c**5*d + 20*e**(b*x)*c**4*d**2*x 
 + 40*e**(b*x)*c**3*d**3*x**2 + 40*e**(b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c 
*d**5*x**4 + 4*e**(b*x)*d**6*x**5),x)*a*b**2*c**5*d - 4*e**(b*x)*int(x/(e* 
*(b*x)*b*c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 + 10* 
e**(b*x)*b*c**3*d**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b*c*d** 
5*x**5 + 4*e**(b*x)*c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c**3*d* 
*3*x**2 + 40*e**(b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e**(b*x 
)*d**6*x**5),x)*a*b**2*c**4*d**2*x - 6*e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5 
*e**(b*x)*b*c**5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 + 10*e**(b*x)*b*c**3*d 
**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b*c*d**5*x**5 + 4*e**(b* 
x)*c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c**3*d**3*x**2 + 40*e**( 
b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e**(b*x)*d**6*x**5),x)*a 
*b**2*c**3*d**3*x**2 - 4*e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*c* 
*5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 + 10*e**(b*x)*b*c**3*d**3*x**3 + 5*e 
**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b*c*d**5*x**5 + 4*e**(b*x)*c**5*d + 20 
*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c**3*d**3*x**2 + 40*e**(b*x)*c**2*d**4 
*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e**(b*x)*d**6*x**5),x)*a*b**2*c**2*d** 
4*x**3 - e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e...