Integrand size = 15, antiderivative size = 118 \[ \int (c+d x)^2 \Gamma (-1,a+b x) \, dx=-\frac {d (3 b c-2 a d) e^{-a-b x}}{3 b^3}-\frac {(b c-a d)^3 \Gamma (-1,a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \Gamma (-1,a+b x)}{3 d}-\frac {(b c-a d)^2 \Gamma (0,a+b x)}{b^3}-\frac {d^2 e^{-a} \Gamma (2,b x)}{3 b^3} \] Output:
-1/3*d*(-2*a*d+3*b*c)*exp(-b*x-a)/b^3-1/3*(-a*d+b*c)^3/(b*x+a)*Ei(2,b*x+a) /b^3/d+1/3*(d*x+c)^3/(b*x+a)*Ei(2,b*x+a)/d-(-a*d+b*c)^2*Ei(1,b*x+a)/b^3-1/ 3*d^2*exp(-b*x)*(b*x+1)/b^3/exp(a)
Time = 0.18 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.45 \[ \int (c+d x)^2 \Gamma (-1,a+b x) \, dx=\frac {\left (3 (1+a) b^2 c^2-3 a (2+a) b c d+a^2 (3+a) d^2\right ) \operatorname {ExpIntegralEi}(-a-b x)+\frac {e^{-a-b x} \left (a^3 d^2+a^2 d (-3 b c+2 d)-b d x (3 b c+d+b d x)+a \left (3 b^2 c^2-d^2+b d (-3 c+d x)\right )+b^3 e^{a+b x} x (a+b x) \left (3 c^2+3 c d x+d^2 x^2\right ) \Gamma (-1,a+b x)\right )}{a+b x}}{3 b^3} \] Input:
Integrate[(c + d*x)^2*Gamma[-1, a + b*x],x]
Output:
((3*(1 + a)*b^2*c^2 - 3*a*(2 + a)*b*c*d + a^2*(3 + a)*d^2)*ExpIntegralEi[- a - b*x] + (E^(-a - b*x)*(a^3*d^2 + a^2*d*(-3*b*c + 2*d) - b*d*x*(3*b*c + d + b*d*x) + a*(3*b^2*c^2 - d^2 + b*d*(-3*c + d*x)) + b^3*E^(a + b*x)*x*(a + b*x)*(3*c^2 + 3*c*d*x + d^2*x^2)*Gamma[-1, a + b*x]))/(a + b*x))/(3*b^3 )
Time = 0.56 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.49, 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.
\(\displaystyle \int (c+d x)^2 \Gamma (-1,a+b x) \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle \frac {b \int \frac {e^{-a-b x} (c+d x)^3}{(a+b x)^2}dx}{3 d}+\frac {(c+d x)^3 \Gamma (-1,a+b x)}{3 d}\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \frac {b \int \left (\frac {e^{-a-b x} x d^3}{b^2}+\frac {(3 b c-2 a d) e^{-a-b x} d^2}{b^3}+\frac {3 (b c-a d)^2 e^{-a-b x} d}{b^3 (a+b x)}+\frac {(b c-a d)^3 e^{-a-b x}}{b^3 (a+b x)^2}\right )dx}{3 d}+\frac {(c+d x)^3 \Gamma (-1,a+b x)}{3 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (-\frac {d^2 e^{-a-b x} (3 b c-2 a d)}{b^4}+\frac {3 d (b c-a d)^2 \operatorname {ExpIntegralEi}(-a-b x)}{b^4}-\frac {(b c-a d)^3 \operatorname {ExpIntegralEi}(-a-b x)}{b^4}-\frac {e^{-a-b x} (b c-a d)^3}{b^4 (a+b x)}-\frac {d^3 e^{-a-b x}}{b^4}-\frac {d^3 x e^{-a-b x}}{b^3}\right )}{3 d}+\frac {(c+d x)^3 \Gamma (-1,a+b x)}{3 d}\) |
Input:
Int[(c + d*x)^2*Gamma[-1, a + b*x],x]
Output:
(b*(-((d^3*E^(-a - b*x))/b^4) - (d^2*(3*b*c - 2*a*d)*E^(-a - b*x))/b^4 - ( d^3*E^(-a - b*x)*x)/b^3 - ((b*c - a*d)^3*E^(-a - b*x))/(b^4*(a + b*x)) + ( 3*d*(b*c - a*d)^2*ExpIntegralEi[-a - b*x])/b^4 - ((b*c - a*d)^3*ExpIntegra lEi[-a - b*x])/b^4))/(3*d) + ((c + d*x)^3*Gamma[-1, a + b*x])/(3*d)
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]
\[\int \frac {\left (d x +c \right )^{2} \operatorname {expIntegral}_{2}\left (b x +a \right )}{b x +a}d x\]
Input:
int((d*x+c)^2/(b*x+a)*Ei(2,b*x+a),x)
Output:
int((d*x+c)^2/(b*x+a)*Ei(2,b*x+a),x)
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (108) = 216\).
Time = 0.10 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.89 \[ \int (c+d x)^2 \Gamma (-1,a+b x) \, dx=-\frac {{\left (b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 3 \, a b c d + {\left (a^{2} + a\right )} d^{2} + {\left (3 \, b^{2} c d - {\left (a - 1\right )} b d^{2}\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b^{4} d^{2} x^{4} + 3 \, {\left (a^{2} + a\right )} b^{2} c^{2} - 3 \, {\left (a^{3} + 2 \, a^{2}\right )} b c d + {\left (3 \, b^{4} c d + a b^{3} d^{2}\right )} x^{3} + {\left (a^{4} + 3 \, a^{3}\right )} d^{2} + 3 \, {\left (b^{4} c^{2} + a b^{3} c d\right )} x^{2} + {\left (3 \, {\left (2 \, a + 1\right )} b^{3} c^{2} - 3 \, {\left (a^{2} + 2 \, a\right )} b^{2} c d + {\left (a^{3} + 3 \, a^{2}\right )} b d^{2}\right )} x\right )} \Gamma \left (-1, b x + a\right )}{3 \, {\left (b^{4} x + a b^{3}\right )}} \] Input:
integrate((d*x+c)^2*gamma(-1,b*x+a),x, algorithm="fricas")
Output:
-1/3*((b^2*d^2*x^2 + 3*b^2*c^2 - 3*a*b*c*d + (a^2 + a)*d^2 + (3*b^2*c*d - (a - 1)*b*d^2)*x)*e^(-b*x - a) - (b^4*d^2*x^4 + 3*(a^2 + a)*b^2*c^2 - 3*(a ^3 + 2*a^2)*b*c*d + (3*b^4*c*d + a*b^3*d^2)*x^3 + (a^4 + 3*a^3)*d^2 + 3*(b ^4*c^2 + a*b^3*c*d)*x^2 + (3*(2*a + 1)*b^3*c^2 - 3*(a^2 + 2*a)*b^2*c*d + ( a^3 + 3*a^2)*b*d^2)*x)*gamma(-1, b*x + a))/(b^4*x + a*b^3)
\[ \int (c+d x)^2 \Gamma (-1,a+b x) \, dx=\int \frac {\left (c + d x\right )^{2} \operatorname {E}_{2}\left (a + b x\right )}{a + b x}\, dx \] Input:
integrate((d*x+c)**2*uppergamma(-1,b*x+a),x)
Output:
Integral((c + d*x)**2*expint(2, a + b*x)/(a + b*x), x)
\[ \int (c+d x)^2 \Gamma (-1,a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \Gamma \left (-1, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^2*gamma(-1,b*x+a),x, algorithm="maxima")
Output:
((b*x + a)*gamma(-1, b*x + a) + Ei(-b*x - a))*c^2/b + integrate(d^2*x^2*ga mma(-1, b*x + a) + 2*c*d*x*gamma(-1, b*x + a), x)
\[ \int (c+d x)^2 \Gamma (-1,a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \Gamma \left (-1, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^2*gamma(-1,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)^2*gamma(-1, b*x + a), x)
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.20 \[ \int (c+d x)^2 \Gamma (-1,a+b x) \, dx=\int \frac {\mathrm {expint}\left (2,a+b\,x\right )\,{\left (c+d\,x\right )}^2}{a+b\,x} \,d x \] Input:
int((expint(2, a + b*x)*(c + d*x)^2)/(a + b*x),x)
Output:
int((expint(2, a + b*x)*(c + d*x)^2)/(a + b*x), x)
\[ \int (c+d x)^2 \Gamma (-1,a+b x) \, dx=\left (\int \frac {\mathit {ei} \left (2, b x +a \right )}{b x +a}d x \right ) c^{2}+\left (\int \frac {\mathit {ei} \left (2, b x +a \right ) x^{2}}{b x +a}d x \right ) d^{2}+2 \left (\int \frac {\mathit {ei} \left (2, b x +a \right ) x}{b x +a}d x \right ) c d \] Input:
int((d*x+c)^2/(b*x+a)*Ei(2,b*x+a),x)
Output:
int(ei(2,a + b*x)/(a + b*x),x)*c**2 + int((ei(2,a + b*x)*x**2)/(a + b*x),x )*d**2 + 2*int((ei(2,a + b*x)*x)/(a + b*x),x)*c*d