Integrand size = 15, antiderivative size = 120 \[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^2} \, dx=\frac {b \Gamma (-2,a+b x)}{d (b c-a d)}-\frac {\Gamma (-2,a+b x)}{d (c+d x)}-\frac {b \Gamma (-1,a+b x)}{(b c-a d)^2}+\frac {b d \Gamma (0,a+b x)}{(b c-a d)^3}-\frac {b d e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{(b c-a d)^3} \] Output:
b/(b*x+a)^2*Ei(3,b*x+a)/d/(-a*d+b*c)-1/(b*x+a)^2*Ei(3,b*x+a)/d/(d*x+c)-b/( b*x+a)*Ei(2,b*x+a)/(-a*d+b*c)^2+b*d*Ei(1,b*x+a)/(-a*d+b*c)^3-b*d*exp(-a+b* c/d)*Ei(1,b*(d*x+c)/d)/(-a*d+b*c)^3
Time = 0.28 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.65 \[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^2} \, dx=-\frac {b e^{-a-b x} (-1+a+b x)}{2 d (b c-a d) (a+b x)^2}-\frac {b e^{-a-b x}}{(b c-a d)^2 (a+b x)}-\frac {b \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^2}+\frac {b d \operatorname {ExpIntegralEi}(-a-b x)}{(-b c+a d)^3}-\frac {b \operatorname {ExpIntegralEi}(-a-b x)}{2 b c d-2 a d^2}+\frac {b d e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^3}-\frac {\Gamma (-2,a+b x)}{d (c+d x)} \] Input:
Integrate[Gamma[-2, a + b*x]/(c + d*x)^2,x]
Output:
-1/2*(b*E^(-a - b*x)*(-1 + a + b*x))/(d*(b*c - a*d)*(a + b*x)^2) - (b*E^(- a - b*x))/((b*c - a*d)^2*(a + b*x)) - (b*ExpIntegralEi[-a - b*x])/(b*c - a *d)^2 + (b*d*ExpIntegralEi[-a - b*x])/(-(b*c) + a*d)^3 - (b*ExpIntegralEi[ -a - b*x])/(2*b*c*d - 2*a*d^2) + (b*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b* (c + d*x))/d)])/(b*c - a*d)^3 - Gamma[-2, a + b*x]/(d*(c + d*x))
Time = 1.18 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.87, 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.
\(\displaystyle \int \frac {\Gamma (-2,a+b x)}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle -\frac {b \int \frac {e^{-a-b x}}{(a+b x)^3 (c+d x)}dx}{d}-\frac {\Gamma (-2,a+b x)}{d (c+d x)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {b \int \left (-\frac {e^{-a-b x} d^3}{(b c-a d)^3 (c+d x)}+\frac {b e^{-a-b x} d^2}{(b c-a d)^3 (a+b x)}-\frac {b e^{-a-b x} d}{(b c-a d)^2 (a+b x)^2}+\frac {b e^{-a-b x}}{(b c-a d) (a+b x)^3}\right )dx}{d}-\frac {\Gamma (-2,a+b x)}{d (c+d x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b \left (\frac {d^2 \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^3}-\frac {d^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{(b c-a d)^3}+\frac {d \operatorname {ExpIntegralEi}(-a-b x)}{(b c-a d)^2}+\frac {\operatorname {ExpIntegralEi}(-a-b x)}{2 (b c-a d)}+\frac {d e^{-a-b x}}{(a+b x) (b c-a d)^2}+\frac {e^{-a-b x}}{2 (a+b x) (b c-a d)}-\frac {e^{-a-b x}}{2 (a+b x)^2 (b c-a d)}\right )}{d}-\frac {\Gamma (-2,a+b x)}{d (c+d x)}\) |
Input:
Int[Gamma[-2, a + b*x]/(c + d*x)^2,x]
Output:
-((b*(-1/2*E^(-a - b*x)/((b*c - a*d)*(a + b*x)^2) + (d*E^(-a - b*x))/((b*c - a*d)^2*(a + b*x)) + E^(-a - b*x)/(2*(b*c - a*d)*(a + b*x)) + (d^2*ExpIn tegralEi[-a - b*x])/(b*c - a*d)^3 + (d*ExpIntegralEi[-a - b*x])/(b*c - a*d )^2 + ExpIntegralEi[-a - b*x]/(2*(b*c - a*d)) - (d^2*E^(-a + (b*c)/d)*ExpI ntegralEi[-((b*(c + d*x))/d)])/(b*c - a*d)^3))/d) - Gamma[-2, 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]
\[\int \frac {\operatorname {expIntegral}_{3}\left (b x +a \right )}{\left (b x +a \right )^{2} \left (d x +c \right )^{2}}d x\]
Input:
int(1/(b*x+a)^2*Ei(3,b*x+a)/(d*x+c)^2,x)
Output:
int(1/(b*x+a)^2*Ei(3,b*x+a)/(d*x+c)^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (119) = 238\).
Time = 0.11 (sec) , antiderivative size = 821, normalized size of antiderivative = 6.84 \[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:
integrate(gamma(-2,b*x+a)/(d*x+c)^2,x, algorithm="fricas")
Output:
1/2*(2*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (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) - (a^2*b^3 *c^3 - 2*(a^3 - a^2)*b^2*c^2*d + (a^4 - 2*a^3 + 2*a^2)*b*c*d^2 + (b^5*c^2* d - 2*(a - 1)*b^4*c*d^2 + (a^2 - 2*a + 2)*b^3*d^3)*x^3 + (b^5*c^3 + 2*b^4* c^2*d - (3*a^2 - 2*a - 2)*b^3*c*d^2 + 2*(a^3 - 2*a^2 + 2*a)*b^2*d^3)*x^2 + (2*a*b^4*c^3 - (3*a^2 - 4*a)*b^3*c^2*d - 2*(a^2 - 2*a)*b^2*c*d^2 + (a^4 - 2*a^3 + 2*a^2)*b*d^3)*x)*Ei(-b*x - a) - ((a - 1)*b^3*c^3 - 2*(a^2 - 2*a)* b^2*c^2*d + (a^3 - 3*a^2)*b*c*d^2 + (b^4*c^2*d - 2*(a - 1)*b^3*c*d^2 + (a^ 2 - 2*a)*b^2*d^3)*x^2 + (b^4*c^3 - (a - 1)*b^3*c^2*d - (a^2 - 2*a)*b^2*c*d ^2 + (a^3 - 3*a^2)*b*d^3)*x)*e^(-b*x - a) - 2*(a^2*b^3*c^3 - 3*a^3*b^2*c^2 *d + 3*a^4*b*c*d^2 - a^5*d^3 + (b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*x^2 + 2*(a*b^4*c^3 - 3*a^2*b^3*c^2*d + 3*a^3*b^2*c*d^2 - a^ 4*b*d^3)*x)*gamma(-2, b*x + a))/(a^2*b^3*c^4*d - 3*a^3*b^2*c^3*d^2 + 3*a^4 *b*c^2*d^3 - a^5*c*d^4 + (b^5*c^3*d^2 - 3*a*b^4*c^2*d^3 + 3*a^2*b^3*c*d^4 - a^3*b^2*d^5)*x^3 + (b^5*c^4*d - a*b^4*c^3*d^2 - 3*a^2*b^3*c^2*d^3 + 5*a^ 3*b^2*c*d^4 - 2*a^4*b*d^5)*x^2 + (2*a*b^4*c^4*d - 5*a^2*b^3*c^3*d^2 + 3*a^ 3*b^2*c^2*d^3 + a^4*b*c*d^4 - a^5*d^5)*x)
\[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^2} \, dx=\int \frac {\operatorname {E}_{3}\left (a + b x\right )}{\left (a + b x\right )^{2} \left (c + d x\right )^{2}}\, dx \] Input:
integrate(uppergamma(-2,b*x+a)/(d*x+c)**2,x)
Output:
Integral(expint(3, a + b*x)/((a + b*x)**2*(c + d*x)**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="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)
Time = 5.71 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.20 \[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {expint}\left (3,a+b\,x\right )}{{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \] Input:
int(expint(3, a + b*x)/((a + b*x)^2*(c + d*x)^2),x)
Output:
int(expint(3, a + b*x)/((a + b*x)^2*(c + d*x)^2), x)
\[ \int \frac {\Gamma (-2,a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathit {ei} \left (3, b x +a \right )}{b^{2} d^{2} x^{4}+2 a b \,d^{2} x^{3}+2 b^{2} c d \,x^{3}+a^{2} d^{2} x^{2}+4 a b c d \,x^{2}+b^{2} c^{2} x^{2}+2 a^{2} c d x +2 a b \,c^{2} x +a^{2} c^{2}}d x \] Input:
int(1/(b*x+a)^2*Ei(3,b*x+a)/(d*x+c)^2,x)
Output:
int(ei(3,a + b*x)/(a**2*c**2 + 2*a**2*c*d*x + a**2*d**2*x**2 + 2*a*b*c**2* x + 4*a*b*c*d*x**2 + 2*a*b*d**2*x**3 + b**2*c**2*x**2 + 2*b**2*c*d*x**3 + b**2*d**2*x**4),x)