Integrand size = 15, antiderivative size = 142 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx=\frac {b^4 (b c-a d)^2 e^{-a+\frac {b c}{d}} \Gamma \left (-3,\frac {b (c+d x)}{d}\right )}{4 d^7}-\frac {b^4 (b c-a d) e^{-a+\frac {b c}{d}} \Gamma \left (-2,\frac {b (c+d x)}{d}\right )}{2 d^6}+\frac {b^4 e^{-a+\frac {b c}{d}} \Gamma \left (-1,\frac {b (c+d x)}{d}\right )}{4 d^5}-\frac {\Gamma (3,a+b x)}{4 d (c+d x)^4} \] Output:
1/4*b*(-a*d+b*c)^2*exp(-a+b*c/d)/(d*x+c)^3/d^4*Ei(4,b*(d*x+c)/d)-1/2*b^2*( -a*d+b*c)*exp(-a+b*c/d)/(d*x+c)^2/d^4*Ei(3,b*(d*x+c)/d)+1/4*b^3*exp(-a+b*c /d)/(d*x+c)/d^4*Ei(2,b*(d*x+c)/d)-1/2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/ d/(d*x+c)^4
Leaf count is larger than twice the leaf count of optimal. \(328\) vs. \(2(142)=284\).
Time = 0.18 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.31 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx=\frac {\frac {b d e^{-a-b x} \left (2 d^2 (b c-a d)^2-b d \left (b^2 c^2-2 (-3+a) b c d+(-6+a) a d^2\right ) (c+d x)+b^2 \left (b^2 c^2-2 (-3+a) b c d+\left (6-6 a+a^2\right ) d^2\right ) (c+d x)^2\right )}{(c+d x)^3}+b^6 c^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+6 b^5 c d e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-2 a b^5 c d e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+6 b^4 d^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-6 a b^4 d^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+a^2 b^4 d^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-\frac {6 d^6 \Gamma (3,a+b x)}{(c+d x)^4}}{24 d^7} \] Input:
Integrate[Gamma[3, a + b*x]/(c + d*x)^5,x]
Output:
((b*d*E^(-a - b*x)*(2*d^2*(b*c - a*d)^2 - b*d*(b^2*c^2 - 2*(-3 + a)*b*c*d + (-6 + a)*a*d^2)*(c + d*x) + b^2*(b^2*c^2 - 2*(-3 + a)*b*c*d + (6 - 6*a + a^2)*d^2)*(c + d*x)^2))/(c + d*x)^3 + b^6*c^2*E^(-a + (b*c)/d)*ExpIntegra lEi[-((b*(c + d*x))/d)] + 6*b^5*c*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] - 2*a*b^5*c*d*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/ d)] + 6*b^4*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] - 6*a*b ^4*d^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] + a^2*b^4*d^2*E^ (-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] - (6*d^6*Gamma[3, a + b*x ])/(c + d*x)^4)/(24*d^7)
Leaf count is larger than twice the leaf count of optimal. \(341\) vs. \(2(142)=284\).
Time = 0.82 (sec) , antiderivative size = 341, normalized size of antiderivative = 2.40, 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.
| \(\displaystyle \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle -\frac {b \int \frac {e^{-a-b x} (a+b x)^2}{(c+d x)^4}dx}{4 d}-\frac {\Gamma (3,a+b x)}{4 d (c+d x)^4}\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle -\frac {b \int \left (\frac {e^{-a-b x} b^2}{d^2 (c+d x)^2}-\frac {2 (b c-a d) e^{-a-b x} b}{d^2 (c+d x)^3}+\frac {(a d-b c)^2 e^{-a-b x}}{d^2 (c+d x)^4}\right )dx}{4 d}-\frac {\Gamma (3,a+b x)}{4 d (c+d x)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \left (-\frac {b^3 (b c-a d)^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{6 d^6}-\frac {b^3 (b c-a d) e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b^3 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^4}-\frac {b^2 e^{-a-b x} (b c-a d)^2}{6 d^5 (c+d x)}-\frac {b^2 e^{-a-b x} (b c-a d)}{d^4 (c+d x)}-\frac {b^2 e^{-a-b x}}{d^3 (c+d x)}+\frac {b e^{-a-b x} (b c-a d)^2}{6 d^4 (c+d x)^2}+\frac {b e^{-a-b x} (b c-a d)}{d^3 (c+d x)^2}-\frac {e^{-a-b x} (b c-a d)^2}{3 d^3 (c+d x)^3}\right )}{4 d}-\frac {\Gamma (3,a+b x)}{4 d (c+d x)^4}\) |
Input:
Int[Gamma[3, a + b*x]/(c + d*x)^5,x]
Output:
-1/4*(b*(-1/3*((b*c - a*d)^2*E^(-a - b*x))/(d^3*(c + d*x)^3) + (b*(b*c - a *d)*E^(-a - b*x))/(d^3*(c + d*x)^2) + (b*(b*c - a*d)^2*E^(-a - b*x))/(6*d^ 4*(c + d*x)^2) - (b^2*E^(-a - b*x))/(d^3*(c + d*x)) - (b^2*(b*c - a*d)*E^( -a - b*x))/(d^4*(c + d*x)) - (b^2*(b*c - a*d)^2*E^(-a - b*x))/(6*d^5*(c + d*x)) - (b^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^4 - (b^ 3*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5 - (b ^3*(b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(6*d^ 6)))/d - Gamma[3, a + b*x]/(4*d*(c + d*x)^4)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(962\) vs. \(2(167)=334\).
Time = 0.89 (sec) , antiderivative size = 963, normalized size of antiderivative = 6.78
| method | result | size |
| derivativedivides | \(-\frac {-\frac {b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{5}}+\frac {2 b^{5} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{6}}-\frac {b^{5} \left (a d -c b \right )^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{7}}-\frac {2 b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{5}}+\frac {2 b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{5}}-\frac {2 \left (a d -c b \right ) b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{6}}}{b}\) | \(963\) |
| default | \(-\frac {-\frac {b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{5}}+\frac {2 b^{5} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{6}}-\frac {b^{5} \left (a d -c b \right )^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{7}}-\frac {2 b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{5}}+\frac {2 b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{5}}-\frac {2 \left (a d -c b \right ) b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{6}}}{b}\) | \(963\) |
| risch | \(\text {Expression too large to display}\) | \(1048\) |
Input:
int(2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/(d*x+c)^5,x,method=_RETURNVERBOS E)
Output:
-1/b*(-b^5/d^5*(-1/2*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/2*exp(-b*x-a)/(- b*x-a+(a*d-b*c)/d)-1/2*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))+2*b^5*(a *d-b*c)/d^6*(-1/3*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^3-1/6*exp(-b*x-a)/(-b*x -a+(a*d-b*c)/d)^2-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)-1/6*exp(-(a*d-b*c)/ d)*Ei(1,b*x+a-(a*d-b*c)/d))-b^5*(a*d-b*c)^2/d^7*(-1/4*exp(-b*x-a)/(-b*x-a+ (a*d-b*c)/d)^4-1/12*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^3-1/24*exp(-b*x-a)/(- b*x-a+(a*d-b*c)/d)^2-1/24*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)-1/24*exp(-(a*d- b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))-2*b^5/d^5*(-1/4*exp(-b*x-a)/(-b*x-a+(a*d- b*c)/d)^4-1/12*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^3-1/24*exp(-b*x-a)/(-b*x-a +(a*d-b*c)/d)^2-1/24*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)-1/24*exp(-(a*d-b*c)/ d)*Ei(1,b*x+a-(a*d-b*c)/d))+2*b^5/d^5*(-1/3*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/ d)^3-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b *c)/d)-1/6*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))-2*(a*d-b*c)/d^6*b^5* (-1/4*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^4-1/12*exp(-b*x-a)/(-b*x-a+(a*d-b*c )/d)^3-1/24*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/24*exp(-b*x-a)/(-b*x-a+(a *d-b*c)/d)-1/24*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)))
Leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (131) = 262\).
Time = 0.15 (sec) , antiderivative size = 572, normalized size of antiderivative = 4.03 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx=-\frac {6 \, d^{6} \Gamma \left (3, b x + a\right ) - {\left (b^{6} c^{6} - 2 \, {\left (a - 3\right )} b^{5} c^{5} d + {\left (a^{2} - 6 \, a + 6\right )} b^{4} c^{4} d^{2} + {\left (b^{6} c^{2} d^{4} - 2 \, {\left (a - 3\right )} b^{5} c d^{5} + {\left (a^{2} - 6 \, a + 6\right )} b^{4} d^{6}\right )} x^{4} + 4 \, {\left (b^{6} c^{3} d^{3} - 2 \, {\left (a - 3\right )} b^{5} c^{2} d^{4} + {\left (a^{2} - 6 \, a + 6\right )} b^{4} c d^{5}\right )} x^{3} + 6 \, {\left (b^{6} c^{4} d^{2} - 2 \, {\left (a - 3\right )} b^{5} c^{3} d^{3} + {\left (a^{2} - 6 \, a + 6\right )} b^{4} c^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{6} c^{5} d - 2 \, {\left (a - 3\right )} b^{5} c^{4} d^{2} + {\left (a^{2} - 6 \, a + 6\right )} b^{4} c^{3} d^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - {\left (b^{5} c^{5} d - {\left (2 \, a - 5\right )} b^{4} c^{4} d^{2} + {\left (a^{2} - 4 \, a + 2\right )} b^{3} c^{3} d^{3} - {\left (a^{2} - 2 \, a\right )} b^{2} c^{2} d^{4} + 2 \, a^{2} b c d^{5} + {\left (b^{5} c^{2} d^{4} - 2 \, {\left (a - 3\right )} b^{4} c d^{5} + {\left (a^{2} - 6 \, a + 6\right )} b^{3} d^{6}\right )} x^{3} + {\left (3 \, b^{5} c^{3} d^{3} - {\left (6 \, a - 17\right )} b^{4} c^{2} d^{4} + {\left (3 \, a^{2} - 16 \, a + 12\right )} b^{3} c d^{5} - {\left (a^{2} - 6 \, a\right )} b^{2} d^{6}\right )} x^{2} + {\left (3 \, b^{5} c^{4} d^{2} - 2 \, {\left (3 \, a - 8\right )} b^{4} c^{3} d^{3} + {\left (3 \, a^{2} - 14 \, a + 8\right )} b^{3} c^{2} d^{4} - 2 \, {\left (a^{2} - 4 \, a\right )} b^{2} c d^{5} + 2 \, a^{2} b d^{6}\right )} x\right )} e^{\left (-b x - a\right )}}{24 \, {\left (d^{11} x^{4} + 4 \, c d^{10} x^{3} + 6 \, c^{2} d^{9} x^{2} + 4 \, c^{3} d^{8} x + c^{4} d^{7}\right )}} \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^5,x, algorithm="fricas")
Output:
-1/24*(6*d^6*gamma(3, b*x + a) - (b^6*c^6 - 2*(a - 3)*b^5*c^5*d + (a^2 - 6 *a + 6)*b^4*c^4*d^2 + (b^6*c^2*d^4 - 2*(a - 3)*b^5*c*d^5 + (a^2 - 6*a + 6) *b^4*d^6)*x^4 + 4*(b^6*c^3*d^3 - 2*(a - 3)*b^5*c^2*d^4 + (a^2 - 6*a + 6)*b ^4*c*d^5)*x^3 + 6*(b^6*c^4*d^2 - 2*(a - 3)*b^5*c^3*d^3 + (a^2 - 6*a + 6)*b ^4*c^2*d^4)*x^2 + 4*(b^6*c^5*d - 2*(a - 3)*b^5*c^4*d^2 + (a^2 - 6*a + 6)*b ^4*c^3*d^3)*x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) - (b^5*c^5*d - (2*a - 5)*b^4*c^4*d^2 + (a^2 - 4*a + 2)*b^3*c^3*d^3 - (a^2 - 2*a)*b^2*c^2*d^4 + 2*a^2*b*c*d^5 + (b^5*c^2*d^4 - 2*(a - 3)*b^4*c*d^5 + (a^2 - 6*a + 6)*b^3* d^6)*x^3 + (3*b^5*c^3*d^3 - (6*a - 17)*b^4*c^2*d^4 + (3*a^2 - 16*a + 12)*b ^3*c*d^5 - (a^2 - 6*a)*b^2*d^6)*x^2 + (3*b^5*c^4*d^2 - 2*(3*a - 8)*b^4*c^3 *d^3 + (3*a^2 - 14*a + 8)*b^3*c^2*d^4 - 2*(a^2 - 4*a)*b^2*c*d^5 + 2*a^2*b* d^6)*x)*e^(-b*x - a))/(d^11*x^4 + 4*c*d^10*x^3 + 6*c^2*d^9*x^2 + 4*c^3*d^8 *x + c^4*d^7)
Timed out. \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx=\text {Timed out} \] Input:
integrate(uppergamma(3,b*x+a)/(d*x+c)**5,x)
Output:
Timed out
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{{\left (d x + c\right )}^{5}} \,d x } \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^5,x, algorithm="maxima")
Output:
integrate(gamma(3, b*x + a)/(d*x + c)^5, x)
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{{\left (d x + c\right )}^{5}} \,d x } \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^5,x, algorithm="giac")
Output:
integrate(gamma(3, b*x + a)/(d*x + c)^5, x)
Timed out. \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx=\int \frac {2\,{\mathrm {e}}^{-a-b\,x}\,\left (a+b\,x+\frac {{\left (a+b\,x\right )}^2}{2}+1\right )}{{\left (c+d\,x\right )}^5} \,d x \] Input:
int((2*exp(- a - b*x)*(a + b*x + (a + b*x)^2/2 + 1))/(c + d*x)^5,x)
Output:
int((2*exp(- a - b*x)*(a + b*x + (a + b*x)^2/2 + 1))/(c + d*x)^5, x)
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^5} \, dx=\text {too large to display} \] Input:
int(2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/(d*x+c)^5,x)
Output:
( - e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e**(b*x)* b*c**4*d**2*x**2 + 10*e**(b*x)*b*c**3*d**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x **4 + e**(b*x)*b*c*d**5*x**5 + 4*e**(b*x)*c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c**3*d**3*x**2 + 40*e**(b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c *d**5*x**4 + 4*e**(b*x)*d**6*x**5),x)*a**2*b**2*c**5*d**2 - 4*e**(b*x)*int (x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 + 10*e**(b*x)*b*c**3*d**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b *c*d**5*x**5 + 4*e**(b*x)*c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c **3*d**3*x**2 + 40*e**(b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e **(b*x)*d**6*x**5),x)*a**2*b**2*c**4*d**3*x - 6*e**(b*x)*int(x/(e**(b*x)*b *c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 + 10*e**(b*x) *b*c**3*d**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b*c*d**5*x**5 + 4*e**(b*x)*c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c**3*d**3*x**2 + 40*e**(b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e**(b*x)*d**6*x **5),x)*a**2*b**2*c**3*d**4*x**2 - 4*e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e **(b*x)*b*c**5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 + 10*e**(b*x)*b*c**3*d** 3*x**3 + 5*e**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b*c*d**5*x**5 + 4*e**(b*x) *c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c**3*d**3*x**2 + 40*e**(b* x)*c**2*d**4*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e**(b*x)*d**6*x**5),x)*a** 2*b**2*c**2*d**5*x**3 - e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*...