Integrand size = 15, antiderivative size = 98 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx=\frac {b \Gamma (-1,a+b x)}{d (b c-a d)}-\frac {\Gamma (-1,a+b x)}{d (c+d x)}-\frac {b \Gamma (0,a+b x)}{(b c-a d)^2}+\frac {b e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{(b c-a d)^2} \] Output:
b/(b*x+a)*Ei(2,b*x+a)/d/(-a*d+b*c)-1/(b*x+a)*Ei(2,b*x+a)/d/(d*x+c)-b*Ei(1, b*x+a)/(-a*d+b*c)^2+b*exp(-a+b*c/d)*Ei(1,b*(d*x+c)/d)/(-a*d+b*c)^2
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.23 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx=\frac {b (b c+d-a d) \operatorname {ExpIntegralEi}(-a-b x)}{d (b c-a d)^2}-\frac {b e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^2}+\frac {\frac {b e^{-a-b x}}{(b c-a d) (a+b x)}-\frac {\Gamma (-1,a+b x)}{c+d x}}{d} \] Input:
Integrate[Gamma[-1, a + b*x]/(c + d*x)^2,x]
Output:
(b*(b*c + d - a*d)*ExpIntegralEi[-a - b*x])/(d*(b*c - a*d)^2) - (b*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^2 + ((b*E^(-a - b *x))/((b*c - a*d)*(a + b*x)) - Gamma[-1, a + b*x]/(c + d*x))/d
Time = 0.94 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.38, 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.
\(\displaystyle \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle -\frac {b \int \frac {e^{-a-b x}}{(a+b x)^2 (c+d x)}dx}{d}-\frac {\Gamma (-1,a+b x)}{d (c+d x)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {b \int \left (\frac {e^{-a-b x} d^2}{(b c-a d)^2 (c+d x)}-\frac {b e^{-a-b x} d}{(b c-a d)^2 (a+b x)}+\frac {b e^{-a-b x}}{(b c-a d) (a+b x)^2}\right )dx}{d}-\frac {\Gamma (-1,a+b x)}{d (c+d x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b \left (-\frac {\operatorname {ExpIntegralEi}(-a-b x)}{b c-a d}-\frac {d \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^2}+\frac {d e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^2}-\frac {e^{-a-b x}}{(a+b x) (b c-a d)}\right )}{d}-\frac {\Gamma (-1,a+b x)}{d (c+d x)}\) |
Input:
Int[Gamma[-1, a + b*x]/(c + d*x)^2,x]
Output:
-((b*(-(E^(-a - b*x)/((b*c - a*d)*(a + b*x))) - (d*ExpIntegralEi[-a - b*x] )/(b*c - a*d)^2 - ExpIntegralEi[-a - b*x]/(b*c - a*d) + (d*E^(-a + (b*c)/d )*ExpIntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^2))/d) - Gamma[-1, a + b* x]/(d*(c + d*x))
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]
\[\int \frac {\operatorname {expIntegral}_{2}\left (b x +a \right )}{\left (b x +a \right ) \left (d x +c \right )^{2}}d x\]
Input:
int(1/(b*x+a)*Ei(2,b*x+a)/(d*x+c)^2,x)
Output:
int(1/(b*x+a)*Ei(2,b*x+a)/(d*x+c)^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (97) = 194\).
Time = 0.13 (sec) , antiderivative size = 349, normalized size of antiderivative = 3.56 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx=-\frac {{\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - {\left (a b^{2} c^{2} - {\left (a^{2} - a\right )} b c d + {\left (b^{3} c d - {\left (a - 1\right )} b^{2} d^{2}\right )} x^{2} + {\left (b^{3} c^{2} + b^{2} c d - {\left (a^{2} - a\right )} b d^{2}\right )} x\right )} {\rm Ei}\left (-b x - a\right ) - {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x\right )} e^{\left (-b x - a\right )} + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} + {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x\right )} \Gamma \left (-1, b x + a\right )}{a b^{2} c^{3} d - 2 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} + {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + {\left (b^{3} c^{3} d - a b^{2} c^{2} d^{2} - a^{2} b c d^{3} + a^{3} d^{4}\right )} x} \] Input:
integrate(gamma(-1,b*x+a)/(d*x+c)^2,x, algorithm="fricas")
Output:
-((b^2*d^2*x^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) - (a*b^2*c^2 - (a^2 - a)*b*c*d + (b^3*c*d - (a - 1)*b^2*d^2 )*x^2 + (b^3*c^2 + b^2*c*d - (a^2 - a)*b*d^2)*x)*Ei(-b*x - a) - (b^2*c^2 - a*b*c*d + (b^2*c*d - a*b*d^2)*x)*e^(-b*x - a) + (a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2 + (b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*x)*gamma(-1, b*x + a))/(a* b^2*c^3*d - 2*a^2*b*c^2*d^2 + a^3*c*d^3 + (b^3*c^2*d^2 - 2*a*b^2*c*d^3 + a ^2*b*d^4)*x^2 + (b^3*c^3*d - a*b^2*c^2*d^2 - a^2*b*c*d^3 + a^3*d^4)*x)
\[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx=\int \frac {\operatorname {E}_{2}\left (a + b x\right )}{\left (a + b x\right ) \left (c + d x\right )^{2}}\, dx \] Input:
integrate(uppergamma(-1,b*x+a)/(d*x+c)**2,x)
Output:
Integral(expint(2, a + b*x)/((a + b*x)*(c + d*x)**2), x)
\[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx=\int { \frac {\Gamma \left (-1, b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate(gamma(-1,b*x+a)/(d*x+c)^2,x, algorithm="maxima")
Output:
integrate(gamma(-1, b*x + a)/(d*x + c)^2, x)
\[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx=\int { \frac {\Gamma \left (-1, b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \] Input:
integrate(gamma(-1,b*x+a)/(d*x+c)^2,x, algorithm="giac")
Output:
integrate(gamma(-1, b*x + a)/(d*x + c)^2, x)
Time = 0.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.24 \[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {expint}\left (2,a+b\,x\right )}{\left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \] Input:
int(expint(2, a + b*x)/((a + b*x)*(c + d*x)^2),x)
Output:
int(expint(2, a + b*x)/((a + b*x)*(c + d*x)^2), x)
\[ \int \frac {\Gamma (-1,a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathit {ei} \left (2, b x +a \right )}{b \,d^{2} x^{3}+a \,d^{2} x^{2}+2 b c d \,x^{2}+2 a c d x +b \,c^{2} x +a \,c^{2}}d x \] Input:
int(1/(b*x+a)*Ei(2,b*x+a)/(d*x+c)^2,x)
Output:
int(ei(2,a + b*x)/(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b*c**2*x + 2*b*c*d*x **2 + b*d**2*x**3),x)