Integrand size = 15, antiderivative size = 112 \[ \int (c+d x)^2 \Gamma (-2,a+b x) \, dx=-\frac {d^2 e^{-a-b x}}{3 b^3}-\frac {(b c-a d)^3 \Gamma (-2,a+b x)}{3 b^3 d}+\frac {(c+d x)^3 \Gamma (-2,a+b x)}{3 d}-\frac {(b c-a d)^2 \Gamma (-1,a+b x)}{b^3}-\frac {d (b c-a d) \Gamma (0,a+b x)}{b^3} \] Output:
-1/3*d^2*exp(-b*x-a)/b^3-1/3*(-a*d+b*c)^3/(b*x+a)^2*Ei(3,b*x+a)/b^3/d+1/3* (d*x+c)^3/(b*x+a)^2*Ei(3,b*x+a)/d-(-a*d+b*c)^2/(b*x+a)*Ei(2,b*x+a)/b^3-d*( -a*d+b*c)*Ei(1,b*x+a)/b^3
Leaf count is larger than twice the leaf count of optimal. \(229\) vs. \(2(112)=224\).
Time = 0.24 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.04 \[ \int (c+d x)^2 \Gamma (-2,a+b x) \, dx=\frac {-\left (\left (3 (2+a) b^2 c^2-3 \left (2+4 a+a^2\right ) b c d+a \left (6+6 a+a^2\right ) d^2\right ) \operatorname {ExpIntegralEi}(-a-b x)\right )+\frac {e^{-a-b x} \left (-a^4 d^2-a^3 d (-3 b c+5 d+b d x)-2 b^2 x \left (3 b c^2+d^2 x\right )-a b \left (3 b^2 c^2 x+4 d^2 x+3 b c (c-4 d x)\right )+a^2 \left (-2 d^2+3 b d (3 c-2 d x)-3 b^2 c (c-d x)\right )+2 b^3 e^{a+b x} x (a+b x)^2 \left (3 c^2+3 c d x+d^2 x^2\right ) \Gamma (-2,a+b x)\right )}{(a+b x)^2}}{6 b^3} \] Input:
Integrate[(c + d*x)^2*Gamma[-2, a + b*x],x]
Output:
(-((3*(2 + a)*b^2*c^2 - 3*(2 + 4*a + a^2)*b*c*d + a*(6 + 6*a + a^2)*d^2)*E xpIntegralEi[-a - b*x]) + (E^(-a - b*x)*(-(a^4*d^2) - a^3*d*(-3*b*c + 5*d + b*d*x) - 2*b^2*x*(3*b*c^2 + d^2*x) - a*b*(3*b^2*c^2*x + 4*d^2*x + 3*b*c* (c - 4*d*x)) + a^2*(-2*d^2 + 3*b*d*(3*c - 2*d*x) - 3*b^2*c*(c - d*x)) + 2* b^3*E^(a + b*x)*x*(a + b*x)^2*(3*c^2 + 3*c*d*x + d^2*x^2)*Gamma[-2, a + b* x]))/(a + b*x)^2)/(6*b^3)
Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(112)=224\).
Time = 0.65 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.02, 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.
\(\displaystyle \int (c+d x)^2 \Gamma (-2,a+b x) \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle \frac {b \int \frac {e^{-a-b x} (c+d x)^3}{(a+b x)^3}dx}{3 d}+\frac {(c+d x)^3 \Gamma (-2,a+b x)}{3 d}\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \frac {b \int \left (\frac {e^{-a-b x} d^3}{b^3}+\frac {3 (b c-a d) e^{-a-b x} d^2}{b^3 (a+b x)}+\frac {3 (b c-a d)^2 e^{-a-b x} d}{b^3 (a+b x)^2}+\frac {(b c-a d)^3 e^{-a-b x}}{b^3 (a+b x)^3}\right )dx}{3 d}+\frac {(c+d x)^3 \Gamma (-2,a+b x)}{3 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (\frac {3 d^2 (b c-a d) \operatorname {ExpIntegralEi}(-a-b x)}{b^4}-\frac {3 d (b c-a d)^2 \operatorname {ExpIntegralEi}(-a-b x)}{b^4}+\frac {(b c-a d)^3 \operatorname {ExpIntegralEi}(-a-b x)}{2 b^4}-\frac {3 d e^{-a-b x} (b c-a d)^2}{b^4 (a+b x)}+\frac {e^{-a-b x} (b c-a d)^3}{2 b^4 (a+b x)}-\frac {e^{-a-b x} (b c-a d)^3}{2 b^4 (a+b x)^2}-\frac {d^3 e^{-a-b x}}{b^4}\right )}{3 d}+\frac {(c+d x)^3 \Gamma (-2,a+b x)}{3 d}\) |
Input:
Int[(c + d*x)^2*Gamma[-2, a + b*x],x]
Output:
(b*(-((d^3*E^(-a - b*x))/b^4) - ((b*c - a*d)^3*E^(-a - b*x))/(2*b^4*(a + b *x)^2) - (3*d*(b*c - a*d)^2*E^(-a - b*x))/(b^4*(a + b*x)) + ((b*c - a*d)^3 *E^(-a - b*x))/(2*b^4*(a + b*x)) + (3*d^2*(b*c - a*d)*ExpIntegralEi[-a - b *x])/b^4 - (3*d*(b*c - a*d)^2*ExpIntegralEi[-a - b*x])/b^4 + ((b*c - a*d)^ 3*ExpIntegralEi[-a - b*x])/(2*b^4)))/(3*d) + ((c + d*x)^3*Gamma[-2, 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]
\[\int \frac {\left (d x +c \right )^{2} \operatorname {expIntegral}_{3}\left (b x +a \right )}{\left (b x +a \right )^{2}}d x\]
Input:
int((d*x+c)^2/(b*x+a)^2*Ei(3,b*x+a),x)
Output:
int((d*x+c)^2/(b*x+a)^2*Ei(3,b*x+a),x)
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (105) = 210\).
Time = 0.10 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.91 \[ \int (c+d x)^2 \Gamma (-2,a+b x) \, dx=-\frac {{\left (b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} - 3 \, {\left (a + 1\right )} b c d + {\left (a^{2} + 3 \, a\right )} d^{2} + {\left (3 \, b^{2} c d - a b d^{2}\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b^{5} d^{2} x^{5} + 3 \, {\left (a^{3} + 2 \, a^{2}\right )} b^{2} c^{2} + {\left (3 \, b^{5} c d + 2 \, a b^{4} d^{2}\right )} x^{4} - 3 \, {\left (a^{4} + 4 \, a^{3} + 2 \, a^{2}\right )} b c d + {\left (3 \, b^{5} c^{2} + 6 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} x^{3} + {\left (a^{5} + 6 \, a^{4} + 6 \, a^{3}\right )} d^{2} + {\left (3 \, {\left (3 \, a + 2\right )} b^{4} c^{2} - 6 \, {\left (2 \, a + 1\right )} b^{3} c d + {\left (a^{3} + 6 \, a^{2} + 6 \, a\right )} b^{2} d^{2}\right )} x^{2} + {\left (3 \, {\left (3 \, a^{2} + 4 \, a\right )} b^{3} c^{2} - 6 \, {\left (a^{3} + 4 \, a^{2} + 2 \, a\right )} b^{2} c d + 2 \, {\left (a^{4} + 6 \, a^{3} + 6 \, a^{2}\right )} b d^{2}\right )} x\right )} \Gamma \left (-2, b x + a\right )}{3 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \] Input:
integrate((d*x+c)^2*gamma(-2,b*x+a),x, algorithm="fricas")
Output:
-1/3*((b^2*d^2*x^2 + 3*b^2*c^2 - 3*(a + 1)*b*c*d + (a^2 + 3*a)*d^2 + (3*b^ 2*c*d - a*b*d^2)*x)*e^(-b*x - a) - (b^5*d^2*x^5 + 3*(a^3 + 2*a^2)*b^2*c^2 + (3*b^5*c*d + 2*a*b^4*d^2)*x^4 - 3*(a^4 + 4*a^3 + 2*a^2)*b*c*d + (3*b^5*c ^2 + 6*a*b^4*c*d + a^2*b^3*d^2)*x^3 + (a^5 + 6*a^4 + 6*a^3)*d^2 + (3*(3*a + 2)*b^4*c^2 - 6*(2*a + 1)*b^3*c*d + (a^3 + 6*a^2 + 6*a)*b^2*d^2)*x^2 + (3 *(3*a^2 + 4*a)*b^3*c^2 - 6*(a^3 + 4*a^2 + 2*a)*b^2*c*d + 2*(a^4 + 6*a^3 + 6*a^2)*b*d^2)*x)*gamma(-2, b*x + a))/(b^5*x^2 + 2*a*b^4*x + a^2*b^3)
\[ \int (c+d x)^2 \Gamma (-2,a+b x) \, dx=\int \frac {\left (c + d x\right )^{2} \operatorname {E}_{3}\left (a + b x\right )}{\left (a + b x\right )^{2}}\, dx \] Input:
integrate((d*x+c)**2*uppergamma(-2,b*x+a),x)
Output:
Integral((c + d*x)**2*expint(3, a + b*x)/(a + b*x)**2, x)
\[ \int (c+d x)^2 \Gamma (-2,a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \Gamma \left (-2, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^2*gamma(-2,b*x+a),x, algorithm="maxima")
Output:
((b*x + a)*gamma(-2, b*x + a) - gamma(-1, b*x + a))*c^2/b + integrate(d^2* x^2*gamma(-2, b*x + a) + 2*c*d*x*gamma(-2, b*x + a), x)
\[ \int (c+d x)^2 \Gamma (-2,a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \Gamma \left (-2, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^2*gamma(-2,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)^2*gamma(-2, b*x + a), x)
Time = 1.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.21 \[ \int (c+d x)^2 \Gamma (-2,a+b x) \, dx=\int \frac {\mathrm {expint}\left (3,a+b\,x\right )\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2} \,d x \] Input:
int((expint(3, a + b*x)*(c + d*x)^2)/(a + b*x)^2,x)
Output:
int((expint(3, a + b*x)*(c + d*x)^2)/(a + b*x)^2, x)
\[ \int (c+d x)^2 \Gamma (-2,a+b x) \, dx=\left (\int \frac {\mathit {ei} \left (3, b x +a \right )}{b^{2} x^{2}+2 a b x +a^{2}}d x \right ) c^{2}+\left (\int \frac {\mathit {ei} \left (3, b x +a \right ) x^{2}}{b^{2} x^{2}+2 a b x +a^{2}}d x \right ) d^{2}+2 \left (\int \frac {\mathit {ei} \left (3, b x +a \right ) x}{b^{2} x^{2}+2 a b x +a^{2}}d x \right ) c d \] Input:
int((d*x+c)^2/(b*x+a)^2*Ei(3,b*x+a),x)
Output:
int(ei(3,a + b*x)/(a**2 + 2*a*b*x + b**2*x**2),x)*c**2 + int((ei(3,a + b*x )*x**2)/(a**2 + 2*a*b*x + b**2*x**2),x)*d**2 + 2*int((ei(3,a + b*x)*x)/(a* *2 + 2*a*b*x + b**2*x**2),x)*c*d