Integrand size = 15, antiderivative size = 95 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx=-\frac {b (b c-a d) e^{-a-b x}}{d^3}+\frac {b (b c-a d)^2 e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{d^4}+\frac {b \Gamma (2,a+b x)}{d^2}-\frac {\Gamma (3,a+b x)}{d (c+d x)} \] Output:
-b*(-a*d+b*c)*exp(-b*x-a)/d^3+b*(-a*d+b*c)^2*exp(-a+b*c/d)*Ei(1,b*(d*x+c)/ d)/d^4+b*exp(-b*x-a)*(b*x+a+1)/d^2-2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/d /(d*x+c)
Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx=\frac {b d e^{-a-b x} (-b c+d+2 a d+b d x)-b (b c-a d)^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-\frac {d^3 \Gamma (3,a+b x)}{c+d x}}{d^4} \] Input:
Integrate[Gamma[3, a + b*x]/(c + d*x)^2,x]
Output:
(b*d*E^(-a - b*x)*(-(b*c) + d + 2*a*d + b*d*x) - b*(b*c - a*d)^2*E^(-a + ( b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] - (d^3*Gamma[3, a + b*x])/(c + d *x))/d^4
Time = 0.44 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.28, 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.
| \(\displaystyle \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle -\frac {b \int \frac {e^{-a-b x} (a+b x)^2}{c+d x}dx}{d}-\frac {\Gamma (3,a+b x)}{d (c+d x)}\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle -\frac {b \int \left (\frac {e^{-a-b x} (a d-b c)^2}{d^2 (c+d x)}-\frac {b (b c-a d) e^{-a-b x}}{d^2}+\frac {b e^{-a-b x} (a+b x)}{d}\right )dx}{d}-\frac {\Gamma (3,a+b x)}{d (c+d x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \left (\frac {e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3}+\frac {e^{-a-b x} (b c-a d)}{d^2}-\frac {e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)}{d}\right )}{d}-\frac {\Gamma (3,a+b x)}{d (c+d x)}\) |
Input:
Int[Gamma[3, a + b*x]/(c + d*x)^2,x]
Output:
-((b*(-(E^(-a - b*x)/d) + ((b*c - a*d)*E^(-a - b*x))/d^2 - (E^(-a - b*x)*( a + b*x))/d + ((b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x) )/d)])/d^3))/d) - Gamma[3, 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. \(334\) vs. \(2(118)=236\).
Time = 0.85 (sec) , antiderivative size = 335, normalized size of antiderivative = 3.53
| method | result | size |
| risch | \(-\frac {b \,{\mathrm e}^{-b x -a}}{d^{2}}+\frac {b \,{\mathrm e}^{-b x -a} a^{2}}{d^{2} \left (-b x -\frac {c b}{d}\right )}-\frac {2 b^{2} {\mathrm e}^{-b x -a} a c}{d^{3} \left (-b x -\frac {c b}{d}\right )}+\frac {b^{3} {\mathrm e}^{-b x -a} c^{2}}{d^{4} \left (-b x -\frac {c b}{d}\right )}+\frac {b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{2}}{d^{2}}-\frac {2 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a c}{d^{3}}+\frac {b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c^{2}}{d^{4}}+\frac {2 b \,{\mathrm e}^{-b x -a}}{d^{2} \left (-b x -\frac {c b}{d}\right )}+\frac {2 b \,{\mathrm e}^{-b x -a} a}{d^{2} \left (-b x -\frac {c b}{d}\right )}-\frac {2 b^{2} {\mathrm e}^{-b x -a} c}{d^{3} \left (-b x -\frac {c b}{d}\right )}\) | \(335\) |
| derivativedivides | \(-\frac {\frac {b^{2} {\mathrm e}^{-b x -a}}{d^{2}}+\frac {2 \left (a d -c b \right ) b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{4}}+\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{2}}+\frac {2 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{2}}+\frac {2 b^{2} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{3}}}{b}\) | \(377\) |
| default | \(-\frac {\frac {b^{2} {\mathrm e}^{-b x -a}}{d^{2}}+\frac {2 \left (a d -c b \right ) b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{3}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{4}}+\frac {2 b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{2}}+\frac {2 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{2}}+\frac {2 b^{2} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{3}}}{b}\) | \(377\) |
Input:
int(2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/(d*x+c)^2,x,method=_RETURNVERBOS E)
Output:
-b*exp(-b*x-a)/d^2+b/d^2*exp(-b*x-a)/(-b*x-c*b/d)*a^2-2*b^2/d^3*exp(-b*x-a )/(-b*x-c*b/d)*a*c+b^3/d^4*exp(-b*x-a)/(-b*x-c*b/d)*c^2+b/d^2*exp(-(a*d-b* c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)*a^2-2*b^2/d^3*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-( a*d-b*c)/d)*a*c+b^3/d^4*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)*c^2+2*b/ d^2*exp(-b*x-a)/(-b*x-c*b/d)+2*b/d^2*exp(-b*x-a)/(-b*x-c*b/d)*a-2*b^2/d^3* exp(-b*x-a)/(-b*x-c*b/d)*c
Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.73 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx=-\frac {d^{3} \Gamma \left (3, b x + a\right ) + {\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} + {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - {\left (b^{2} d^{3} x^{2} + {\left (2 \, a + 1\right )} b d^{3} x - b^{2} c^{2} d + {\left (2 \, a + 1\right )} b c d^{2}\right )} e^{\left (-b x - a\right )}}{d^{5} x + c d^{4}} \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^2,x, algorithm="fricas")
Output:
-(d^3*gamma(3, b*x + a) + (b^3*c^3 - 2*a*b^2*c^2*d + a^2*b*c*d^2 + (b^3*c^ 2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) - (b^2*d^3*x^2 + (2*a + 1)*b*d^3*x - b^2*c^2*d + (2*a + 1)*b*c*d^2)*e^(-b *x - a))/(d^5*x + c*d^4)
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx=\left (\int \frac {2 a}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {a^{2}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {2 b x}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {b^{2} x^{2}}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {2 a b x}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx + \int \frac {2}{c^{2} e^{b x} + 2 c d x e^{b x} + d^{2} x^{2} e^{b x}}\, dx\right ) e^{- a} \] Input:
integrate(uppergamma(3,b*x+a)/(d*x+c)**2,x)
Output:
(Integral(2*a/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x) + Integral(a**2/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x ) + Integral(2*b*x/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)) , x) + Integral(b**2*x**2/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*ex p(b*x)), x) + Integral(2*a*b*x/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x* *2*exp(b*x)), x) + Integral(2/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x** 2*exp(b*x)), x))*exp(-a)
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^2,x, algorithm="maxima")
Output:
integrate(gamma(3, b*x + a)/(d*x + c)^2, x)
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^2,x, algorithm="giac")
Output:
integrate(gamma(3, b*x + a)/(d*x + c)^2, x)
Timed out. \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx=\int \frac {2\,{\mathrm {e}}^{-a-b\,x}\,\left (a+b\,x+\frac {{\left (a+b\,x\right )}^2}{2}+1\right )}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((2*exp(- a - b*x)*(a + b*x + (a + b*x)^2/2 + 1))/(c + d*x)^2,x)
Output:
int((2*exp(- a - b*x)*(a + b*x + (a + b*x)^2/2 + 1))/(c + d*x)^2, x)
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:
int(2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/(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**2*b**2*c**2*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**2*b**2*c*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)*a**2*b*c*d**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)*a**2*b*d**4*x + 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 **3*c**3*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**3*c**2*d**2*x + 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**2*c**2*d**2 + 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**2*c*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*...