Integrand size = 13, antiderivative size = 84 \[ \int (c+d x) \Gamma (-3,a+b x) \, dx=-\frac {(b c-a d)^2 \Gamma (-3,a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \Gamma (-3,a+b x)}{2 d}-\frac {(b c-a d) \Gamma (-2,a+b x)}{b^2}-\frac {d \Gamma (-1,a+b x)}{2 b^2} \] Output:
-1/2*(-a*d+b*c)^2/(b*x+a)^3*Ei(4,b*x+a)/b^2/d+1/2*(d*x+c)^2/(b*x+a)^3*Ei(4 ,b*x+a)/d-(-a*d+b*c)/(b*x+a)^2*Ei(3,b*x+a)/b^2-1/2*d/(b*x+a)*Ei(2,b*x+a)/b ^2
Leaf count is larger than twice the leaf count of optimal. \(270\) vs. \(2(84)=168\).
Time = 0.13 (sec) , antiderivative size = 270, normalized size of antiderivative = 3.21 \[ \int (c+d x) \Gamma (-3,a+b x) \, dx=d e^{-b x} \left (-\frac {a^2 e^{-a}}{6 b^2 (a+b x)^3}+\frac {a (6+a) e^{-a}}{12 b^2 (a+b x)^2}-\frac {\left (6+6 a+a^2\right ) e^{-a}}{12 b^2 (a+b x)}\right )+c e^{-b x} \left (\frac {a e^{-a}}{3 b (a+b x)^3}-\frac {(3+a) e^{-a}}{6 b (a+b x)^2}+\frac {(3+a) e^{-a}}{6 b (a+b x)}\right )+\frac {c \operatorname {ExpIntegralEi}(-a-b x)}{2 b}+\frac {a c \operatorname {ExpIntegralEi}(-a-b x)}{6 b}-\frac {d \operatorname {ExpIntegralEi}(-a-b x)}{2 b^2}-\frac {a d \operatorname {ExpIntegralEi}(-a-b x)}{2 b^2}-\frac {a^2 d \operatorname {ExpIntegralEi}(-a-b x)}{12 b^2}+c x \Gamma (-3,a+b x)+\frac {1}{2} d x^2 \Gamma (-3,a+b x) \] Input:
Integrate[(c + d*x)*Gamma[-3, a + b*x],x]
Output:
(d*(-1/6*a^2/(b^2*E^a*(a + b*x)^3) + (a*(6 + a))/(12*b^2*E^a*(a + b*x)^2) - (6 + 6*a + a^2)/(12*b^2*E^a*(a + b*x))))/E^(b*x) + (c*(a/(3*b*E^a*(a + b *x)^3) - (3 + a)/(6*b*E^a*(a + b*x)^2) + (3 + a)/(6*b*E^a*(a + b*x))))/E^( b*x) + (c*ExpIntegralEi[-a - b*x])/(2*b) + (a*c*ExpIntegralEi[-a - b*x])/( 6*b) - (d*ExpIntegralEi[-a - b*x])/(2*b^2) - (a*d*ExpIntegralEi[-a - b*x]) /(2*b^2) - (a^2*d*ExpIntegralEi[-a - b*x])/(12*b^2) + c*x*Gamma[-3, a + b* x] + (d*x^2*Gamma[-3, a + b*x])/2
Leaf count is larger than twice the leaf count of optimal. \(284\) vs. \(2(84)=168\).
Time = 0.71 (sec) , antiderivative size = 284, normalized size of antiderivative = 3.38, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, 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.
\(\displaystyle \int (c+d x) \Gamma (-3,a+b x) \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle \frac {b \int \frac {e^{-a-b x} (c+d x)^2}{(a+b x)^4}dx}{2 d}+\frac {(c+d x)^2 \Gamma (-3,a+b x)}{2 d}\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \frac {b \int \left (\frac {e^{-a-b x} d^2}{b^2 (a+b x)^2}+\frac {2 (b c-a d) e^{-a-b x} d}{b^2 (a+b x)^3}+\frac {(b c-a d)^2 e^{-a-b x}}{b^2 (a+b x)^4}\right )dx}{2 d}+\frac {(c+d x)^2 \Gamma (-3,a+b x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (\frac {d (b c-a d) \operatorname {ExpIntegralEi}(-a-b x)}{b^3}-\frac {(b c-a d)^2 \operatorname {ExpIntegralEi}(-a-b x)}{6 b^3}+\frac {d e^{-a-b x} (b c-a d)}{b^3 (a+b x)}-\frac {d e^{-a-b x} (b c-a d)}{b^3 (a+b x)^2}-\frac {e^{-a-b x} (b c-a d)^2}{6 b^3 (a+b x)}+\frac {e^{-a-b x} (b c-a d)^2}{6 b^3 (a+b x)^2}-\frac {e^{-a-b x} (b c-a d)^2}{3 b^3 (a+b x)^3}-\frac {d^2 \operatorname {ExpIntegralEi}(-a-b x)}{b^3}-\frac {d^2 e^{-a-b x}}{b^3 (a+b x)}\right )}{2 d}+\frac {(c+d x)^2 \Gamma (-3,a+b x)}{2 d}\) |
Input:
Int[(c + d*x)*Gamma[-3, a + b*x],x]
Output:
(b*(-1/3*((b*c - a*d)^2*E^(-a - b*x))/(b^3*(a + b*x)^3) - (d*(b*c - a*d)*E ^(-a - b*x))/(b^3*(a + b*x)^2) + ((b*c - a*d)^2*E^(-a - b*x))/(6*b^3*(a + b*x)^2) - (d^2*E^(-a - b*x))/(b^3*(a + b*x)) + (d*(b*c - a*d)*E^(-a - b*x) )/(b^3*(a + b*x)) - ((b*c - a*d)^2*E^(-a - b*x))/(6*b^3*(a + b*x)) - (d^2* ExpIntegralEi[-a - b*x])/b^3 + (d*(b*c - a*d)*ExpIntegralEi[-a - b*x])/b^3 - ((b*c - a*d)^2*ExpIntegralEi[-a - b*x])/(6*b^3)))/(2*d) + ((c + d*x)^2* Gamma[-3, a + b*x])/(2*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]
\[\int \frac {\left (d x +c \right ) \operatorname {expIntegral}_{4}\left (b x +a \right )}{\left (b x +a \right )^{3}}d x\]
Input:
int((d*x+c)/(b*x+a)^3*Ei(4,b*x+a),x)
Output:
int((d*x+c)/(b*x+a)^3*Ei(4,b*x+a),x)
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (78) = 156\).
Time = 0.15 (sec) , antiderivative size = 235, normalized size of antiderivative = 2.80 \[ \int (c+d x) \Gamma (-3,a+b x) \, dx=-\frac {{\left (b d x + 2 \, b c - {\left (a + 2\right )} d\right )} e^{\left (-b x - a\right )} - {\left (b^{5} d x^{5} + {\left (2 \, b^{5} c + 3 \, a b^{4} d\right )} x^{4} + 2 \, {\left ({\left (4 \, a + 3\right )} b^{4} c + {\left (a^{2} - 3 \, a - 3\right )} b^{3} d\right )} x^{3} + 2 \, {\left (a^{4} + 3 \, a^{3}\right )} b c + 2 \, {\left (3 \, {\left (2 \, a^{2} + 3 \, a\right )} b^{3} c - {\left (a^{3} + 9 \, a^{2} + 9 \, a\right )} b^{2} d\right )} x^{2} - {\left (a^{5} + 6 \, a^{4} + 6 \, a^{3}\right )} d + {\left (2 \, {\left (4 \, a^{3} + 9 \, a^{2}\right )} b^{2} c - 3 \, {\left (a^{4} + 6 \, a^{3} + 6 \, a^{2}\right )} b d\right )} x\right )} \Gamma \left (-3, b x + a\right )}{2 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \] Input:
integrate((d*x+c)*gamma(-3,b*x+a),x, algorithm="fricas")
Output:
-1/2*((b*d*x + 2*b*c - (a + 2)*d)*e^(-b*x - a) - (b^5*d*x^5 + (2*b^5*c + 3 *a*b^4*d)*x^4 + 2*((4*a + 3)*b^4*c + (a^2 - 3*a - 3)*b^3*d)*x^3 + 2*(a^4 + 3*a^3)*b*c + 2*(3*(2*a^2 + 3*a)*b^3*c - (a^3 + 9*a^2 + 9*a)*b^2*d)*x^2 - (a^5 + 6*a^4 + 6*a^3)*d + (2*(4*a^3 + 9*a^2)*b^2*c - 3*(a^4 + 6*a^3 + 6*a^ 2)*b*d)*x)*gamma(-3, b*x + a))/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3* b^2)
\[ \int (c+d x) \Gamma (-3,a+b x) \, dx=\int \frac {\left (c + d x\right ) \operatorname {E}_{4}\left (a + b x\right )}{\left (a + b x\right )^{3}}\, dx \] Input:
integrate((d*x+c)*uppergamma(-3,b*x+a),x)
Output:
Integral((c + d*x)*expint(4, a + b*x)/(a + b*x)**3, x)
\[ \int (c+d x) \Gamma (-3,a+b x) \, dx=\int { {\left (d x + c\right )} \Gamma \left (-3, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)*gamma(-3,b*x+a),x, algorithm="maxima")
Output:
d*integrate(x*gamma(-3, b*x + a), x) + ((b*x + a)*gamma(-3, b*x + a) - gam ma(-2, b*x + a))*c/b
\[ \int (c+d x) \Gamma (-3,a+b x) \, dx=\int { {\left (d x + c\right )} \Gamma \left (-3, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)*gamma(-3,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)*gamma(-3, b*x + a), x)
Time = 0.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.26 \[ \int (c+d x) \Gamma (-3,a+b x) \, dx=\int \frac {\mathrm {expint}\left (4,a+b\,x\right )\,\left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \] Input:
int((expint(4, a + b*x)*(c + d*x))/(a + b*x)^3,x)
Output:
int((expint(4, a + b*x)*(c + d*x))/(a + b*x)^3, x)
\[ \int (c+d x) \Gamma (-3,a+b x) \, dx=\left (\int \frac {\mathit {ei} \left (4, b x +a \right )}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) c +\left (\int \frac {\mathit {ei} \left (4, b x +a \right ) x}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}d x \right ) d \] Input:
int((d*x+c)/(b*x+a)^3*Ei(4,b*x+a),x)
Output:
int(ei(4,a + b*x)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*c + i nt((ei(4,a + b*x)*x)/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3),x)*d