Integrand size = 15, antiderivative size = 73 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx=\frac {b e^{-a-b x}}{d^2}-\frac {b (b c-a d) e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{d^3}-\frac {\Gamma (2,a+b x)}{d (c+d x)} \] Output:
b*exp(-b*x-a)/d^2-b*(-a*d+b*c)*exp(-a+b*c/d)*Ei(1,b*(d*x+c)/d)/d^3-exp(-b* x-a)*(b*x+a+1)/d/(d*x+c)
Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx=\frac {e^{-a-b x} \left (b (b c-a d) e^{b \left (\frac {c}{d}+x\right )} (c+d x) \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+d \left (b (c+d x)-d e^{a+b x} \Gamma (2,a+b x)\right )\right )}{d^3 (c+d x)} \] Input:
Integrate[Gamma[2, a + b*x]/(c + d*x)^2,x]
Output:
(E^(-a - b*x)*(b*(b*c - a*d)*E^(b*(c/d + x))*(c + d*x)*ExpIntegralEi[-((b* (c + d*x))/d)] + d*(b*(c + d*x) - d*E^(a + b*x)*Gamma[2, a + b*x])))/(d^3* (c + d*x))
Time = 0.36 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08, 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.
| \(\displaystyle \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle -\frac {b \int \frac {e^{-a-b x} (a+b x)}{c+d x}dx}{d}-\frac {\Gamma (2,a+b x)}{d (c+d x)}\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle -\frac {b \int \left (\frac {e^{-a-b x} b}{d}+\frac {(a d-b c) e^{-a-b x}}{d (c+d x)}\right )dx}{d}-\frac {\Gamma (2,a+b x)}{d (c+d x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \left (-\frac {(b c-a d) e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^2}-\frac {e^{-a-b x}}{d}\right )}{d}-\frac {\Gamma (2,a+b x)}{d (c+d x)}\) |
Input:
Int[Gamma[2, a + b*x]/(c + d*x)^2,x]
Output:
-((b*(-(E^(-a - b*x)/d) - ((b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b *(c + d*x))/d)])/d^2))/d) - Gamma[2, a + b*x]/(d*(c + d*x))
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[$UseGamma]
Int[Gamma[n_, (a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Block[{$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]
Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(79)=158\).
Time = 0.72 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.38
| method | result | size |
| risch | \(\frac {b \,{\mathrm e}^{-b x -a}}{d^{2} \left (-b x -\frac {c b}{d}\right )}+\frac {b \,{\mathrm e}^{-b x -a} a}{d^{2} \left (-b x -\frac {c b}{d}\right )}-\frac {b^{2} {\mathrm e}^{-b x -a} c}{d^{3} \left (-b x -\frac {c b}{d}\right )}+\frac {b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a}{d^{2}}-\frac {b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c}{d^{3}}\) | \(174\) |
| derivativedivides | \(-\frac {\frac {b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{2}}+\frac {b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{2}}+\frac {b^{2} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{3}}}{b}\) | \(210\) |
| default | \(-\frac {\frac {b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{2}}+\frac {b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{2}}+\frac {b^{2} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{3}}}{b}\) | \(210\) |
Input:
int(exp(-b*x-a)*(b*x+a+1)/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
b/d^2*exp(-b*x-a)/(-b*x-c*b/d)+b/d^2*exp(-b*x-a)/(-b*x-c*b/d)*a-b^2/d^3*ex p(-b*x-a)/(-b*x-c*b/d)*c+b/d^2*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)*a -b^2/d^3*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)*c
Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.45 \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx=\frac {{\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - d^{2} \Gamma \left (2, b x + a\right ) + {\left (b d^{2} x + b c d\right )} e^{\left (-b x - a\right )}}{d^{4} x + c d^{3}} \] Input:
integrate(gamma(2,b*x+a)/(d*x+c)^2,x, algorithm="fricas")
Output:
((b^2*c^2 - a*b*c*d + (b^2*c*d - a*b*d^2)*x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) - d^2*gamma(2, b*x + a) + (b*d^2*x + b*c*d)*e^(-b*x - a))/(d^4*x + c*d^3)
\[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx=\left (\int \frac {a}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {b x}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {1}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx\right ) e^{- a} \] Input:
integrate(uppergamma(2,b*x+a)/(d*x+c)**2,x)
Output:
(Integral(a/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x) + Integral(b*x/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x) + Integral(1/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x))*e xp(-a)
\[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/(d*x+c)^2,x, algorithm="maxima")
Output:
integrate(gamma(2, b*x + a)/(d*x + c)^2, x)
\[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/(d*x+c)^2,x, algorithm="giac")
Output:
integrate(gamma(2, b*x + a)/(d*x + c)^2, x)
Timed out. \[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,\left (a+b\,x+1\right )}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((exp(- a - b*x)*(a + b*x + 1))/(c + d*x)^2,x)
Output:
int((exp(- a - b*x)*(a + b*x + 1))/(c + d*x)^2, x)
\[ \int \frac {\Gamma (2,a+b x)}{(c+d x)^2} \, dx=\frac {-e^{b x} \left (\int \frac {x}{e^{b x} b \,c^{3}+2 e^{b x} b \,c^{2} d x +e^{b x} b c \,d^{2} x^{2}+e^{b x} c^{2} d +2 e^{b x} c \,d^{2} x +e^{b x} d^{3} x^{2}}d x \right ) a \,b^{2} c^{2} d -e^{b x} \left (\int \frac {x}{e^{b x} b \,c^{3}+2 e^{b x} b \,c^{2} d x +e^{b x} b c \,d^{2} x^{2}+e^{b x} c^{2} d +2 e^{b x} c \,d^{2} x +e^{b x} d^{3} x^{2}}d x \right ) a \,b^{2} c \,d^{2} x -e^{b x} \left (\int \frac {x}{e^{b x} b \,c^{3}+2 e^{b x} b \,c^{2} d x +e^{b x} b c \,d^{2} x^{2}+e^{b x} c^{2} d +2 e^{b x} c \,d^{2} x +e^{b x} d^{3} x^{2}}d x \right ) a b c \,d^{2}-e^{b x} \left (\int \frac {x}{e^{b x} b \,c^{3}+2 e^{b x} b \,c^{2} d x +e^{b x} b c \,d^{2} x^{2}+e^{b x} c^{2} d +2 e^{b x} c \,d^{2} x +e^{b x} d^{3} x^{2}}d x \right ) a b \,d^{3} x +e^{b x} \left (\int \frac {x}{e^{b x} b \,c^{3}+2 e^{b x} b \,c^{2} d x +e^{b x} b c \,d^{2} x^{2}+e^{b x} c^{2} d +2 e^{b x} c \,d^{2} x +e^{b x} d^{3} x^{2}}d x \right ) b^{3} c^{3}+e^{b x} \left (\int \frac {x}{e^{b x} b \,c^{3}+2 e^{b x} b \,c^{2} d x +e^{b x} b c \,d^{2} x^{2}+e^{b x} c^{2} d +2 e^{b x} c \,d^{2} x +e^{b x} d^{3} x^{2}}d x \right ) b^{3} c^{2} d x +e^{b x} \left (\int \frac {x}{e^{b x} b \,c^{3}+2 e^{b x} b \,c^{2} d x +e^{b x} b c \,d^{2} x^{2}+e^{b x} c^{2} d +2 e^{b x} c \,d^{2} x +e^{b x} d^{3} x^{2}}d x \right ) b^{2} c^{2} d +e^{b x} \left (\int \frac {x}{e^{b x} b \,c^{3}+2 e^{b x} b \,c^{2} d x +e^{b x} b c \,d^{2} x^{2}+e^{b x} c^{2} d +2 e^{b x} c \,d^{2} x +e^{b x} d^{3} x^{2}}d x \right ) b^{2} c \,d^{2} x -a -1}{e^{b x +a} \left (b c d x +b \,c^{2}+d^{2} x +c d \right )} \] Input:
int(exp(-b*x-a)*(b*x+a+1)/(d*x+c)^2,x)
Output:
( - e**(b*x)*int(x/(e**(b*x)*b*c**3 + 2*e**(b*x)*b*c**2*d*x + e**(b*x)*b*c *d**2*x**2 + e**(b*x)*c**2*d + 2*e**(b*x)*c*d**2*x + e**(b*x)*d**3*x**2),x )*a*b**2*c**2*d - e**(b*x)*int(x/(e**(b*x)*b*c**3 + 2*e**(b*x)*b*c**2*d*x + e**(b*x)*b*c*d**2*x**2 + e**(b*x)*c**2*d + 2*e**(b*x)*c*d**2*x + e**(b*x )*d**3*x**2),x)*a*b**2*c*d**2*x - e**(b*x)*int(x/(e**(b*x)*b*c**3 + 2*e**( b*x)*b*c**2*d*x + e**(b*x)*b*c*d**2*x**2 + e**(b*x)*c**2*d + 2*e**(b*x)*c* d**2*x + e**(b*x)*d**3*x**2),x)*a*b*c*d**2 - e**(b*x)*int(x/(e**(b*x)*b*c* *3 + 2*e**(b*x)*b*c**2*d*x + e**(b*x)*b*c*d**2*x**2 + e**(b*x)*c**2*d + 2* e**(b*x)*c*d**2*x + e**(b*x)*d**3*x**2),x)*a*b*d**3*x + e**(b*x)*int(x/(e* *(b*x)*b*c**3 + 2*e**(b*x)*b*c**2*d*x + e**(b*x)*b*c*d**2*x**2 + e**(b*x)* c**2*d + 2*e**(b*x)*c*d**2*x + e**(b*x)*d**3*x**2),x)*b**3*c**3 + e**(b*x) *int(x/(e**(b*x)*b*c**3 + 2*e**(b*x)*b*c**2*d*x + e**(b*x)*b*c*d**2*x**2 + e**(b*x)*c**2*d + 2*e**(b*x)*c*d**2*x + e**(b*x)*d**3*x**2),x)*b**3*c**2* d*x + e**(b*x)*int(x/(e**(b*x)*b*c**3 + 2*e**(b*x)*b*c**2*d*x + e**(b*x)*b *c*d**2*x**2 + e**(b*x)*c**2*d + 2*e**(b*x)*c*d**2*x + e**(b*x)*d**3*x**2) ,x)*b**2*c**2*d + e**(b*x)*int(x/(e**(b*x)*b*c**3 + 2*e**(b*x)*b*c**2*d*x + e**(b*x)*b*c*d**2*x**2 + e**(b*x)*c**2*d + 2*e**(b*x)*c*d**2*x + e**(b*x )*d**3*x**2),x)*b**2*c*d**2*x - a - 1)/(e**(a + b*x)*(b*c**2 + b*c*d*x + c *d + d**2*x))