Integrand size = 15, antiderivative size = 142 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx=\frac {b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \Gamma \left (-2,\frac {b (c+d x)}{d}\right )}{3 d^6}-\frac {2 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \Gamma \left (-1,\frac {b (c+d x)}{d}\right )}{3 d^5}+\frac {b^3 e^{-a+\frac {b c}{d}} \Gamma \left (0,\frac {b (c+d x)}{d}\right )}{3 d^4}-\frac {\Gamma (3,a+b x)}{3 d (c+d x)^3} \] Output:
1/3*b*(-a*d+b*c)^2*exp(-a+b*c/d)/(d*x+c)^2/d^4*Ei(3,b*(d*x+c)/d)-2/3*b^2*( -a*d+b*c)*exp(-a+b*c/d)/(d*x+c)/d^4*Ei(2,b*(d*x+c)/d)+1/3*b^3*exp(-a+b*c/d )*Ei(1,b*(d*x+c)/d)/d^4-2/3*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/d/(d*x+c)^ 3
Time = 0.23 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.02 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx=\frac {-b^3 \left (b^2 c^2-2 (-2+a) b c d+\left (2-4 a+a^2\right ) d^2\right ) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+\frac {d \left (-b (b c-a d) e^{-a-b x} (c+d x) \left (a d^2+b^2 c (c+d x)-b d ((-3+a) c+(-4+a) d x)\right )-2 d^4 \Gamma (3,a+b x)\right )}{(c+d x)^3}}{6 d^6} \] Input:
Integrate[Gamma[3, a + b*x]/(c + d*x)^4,x]
Output:
(-(b^3*(b^2*c^2 - 2*(-2 + a)*b*c*d + (2 - 4*a + a^2)*d^2)*E^(-a + (b*c)/d) *ExpIntegralEi[-((b*(c + d*x))/d)]) + (d*(-(b*(b*c - a*d)*E^(-a - b*x)*(c + d*x)*(a*d^2 + b^2*c*(c + d*x) - b*d*((-3 + a)*c + (-4 + a)*d*x))) - 2*d^ 4*Gamma[3, a + b*x]))/(c + d*x)^3)/(6*d^6)
Time = 0.64 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.73, 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.
| \(\displaystyle \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle -\frac {b \int \frac {e^{-a-b x} (a+b x)^2}{(c+d x)^3}dx}{3 d}-\frac {\Gamma (3,a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle -\frac {b \int \left (\frac {e^{-a-b x} b^2}{d^2 (c+d x)}-\frac {2 (b c-a d) e^{-a-b x} b}{d^2 (c+d x)^2}+\frac {(a d-b c)^2 e^{-a-b x}}{d^2 (c+d x)^3}\right )dx}{3 d}-\frac {\Gamma (3,a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b \left (\frac {b^2 (b c-a d)^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{2 d^5}+\frac {2 b^2 (b c-a d) e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^4}+\frac {b^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^3}+\frac {b e^{-a-b x} (b c-a d)^2}{2 d^4 (c+d x)}+\frac {2 b e^{-a-b x} (b c-a d)}{d^3 (c+d x)}-\frac {e^{-a-b x} (b c-a d)^2}{2 d^3 (c+d x)^2}\right )}{3 d}-\frac {\Gamma (3,a+b x)}{3 d (c+d x)^3}\) |
Input:
Int[Gamma[3, a + b*x]/(c + d*x)^4,x]
Output:
-1/3*(b*(-1/2*((b*c - a*d)^2*E^(-a - b*x))/(d^3*(c + d*x)^2) + (2*b*(b*c - a*d)*E^(-a - b*x))/(d^3*(c + d*x)) + (b*(b*c - a*d)^2*E^(-a - b*x))/(2*d^ 4*(c + d*x)) + (b^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^ 3 + (2*b^2*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)]) /d^4 + (b^2*(b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d )])/(2*d^5)))/d - Gamma[3, a + b*x]/(3*d*(c + d*x)^3)
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. \(762\) vs. \(2(160)=320\).
Time = 0.88 (sec) , antiderivative size = 763, normalized size of antiderivative = 5.37
| method | result | size |
| derivativedivides | \(-\frac {\frac {b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{4}}-\frac {2 b^{4} \left (a d -c b \right ) \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 {b^{4} \left (a d -c b \right )^{2} \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 {2 b^{4} \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^{4}}-\frac {2 b^{4} \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^{4}}+\frac {2 b^{4} \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^{5}}}{b}\) | \(763\) |
| default | \(-\frac {\frac {b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{4}}-\frac {2 b^{4} \left (a d -c b \right ) \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 {b^{4} \left (a d -c b \right )^{2} \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 {2 b^{4} \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^{4}}-\frac {2 b^{4} \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^{4}}+\frac {2 b^{4} \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^{5}}}{b}\) | \(763\) |
| risch | \(-\frac {2 b^{4} {\mathrm e}^{-b x -a} a c}{3 d^{5} \left (-b x -\frac {c b}{d}\right )^{3}}-\frac {b^{4} {\mathrm e}^{-b x -a} a c}{3 d^{5} \left (-b x -\frac {c b}{d}\right )^{2}}-\frac {b^{4} {\mathrm e}^{-b x -a} a c}{3 d^{5} \left (-b x -\frac {c b}{d}\right )}-\frac {b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a c}{3 d^{5}}+\frac {2 b^{3} {\mathrm e}^{-b x -a} a}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{3}}-\frac {2 b^{4} {\mathrm e}^{-b x -a} c}{3 d^{5} \left (-b x -\frac {c b}{d}\right )^{3}}+\frac {b^{3} {\mathrm e}^{-b x -a}}{3 d^{4} \left (-b x -\frac {c b}{d}\right )}+\frac {b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{3 d^{4}}-\frac {2 b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a}{3 d^{4}}+\frac {2 b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c}{3 d^{5}}-\frac {2 b^{3} {\mathrm e}^{-b x -a} a}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{2}}+\frac {2 b^{4} {\mathrm e}^{-b x -a} c}{3 d^{5} \left (-b x -\frac {c b}{d}\right )^{2}}-\frac {2 b^{3} {\mathrm e}^{-b x -a} a}{3 d^{4} \left (-b x -\frac {c b}{d}\right )}+\frac {2 b^{4} {\mathrm e}^{-b x -a} c}{3 d^{5} \left (-b x -\frac {c b}{d}\right )}+\frac {2 b^{3} {\mathrm e}^{-b x -a}}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{3}}-\frac {2 b^{3} {\mathrm e}^{-b x -a}}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{2}}+\frac {b^{3} {\mathrm e}^{-b x -a} a^{2}}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{3}}+\frac {b^{5} {\mathrm e}^{-b x -a} c^{2}}{3 d^{6} \left (-b x -\frac {c b}{d}\right )^{3}}+\frac {b^{3} {\mathrm e}^{-b x -a} a^{2}}{6 d^{4} \left (-b x -\frac {c b}{d}\right )^{2}}+\frac {b^{5} {\mathrm e}^{-b x -a} c^{2}}{6 d^{6} \left (-b x -\frac {c b}{d}\right )^{2}}+\frac {b^{3} {\mathrm e}^{-b x -a} a^{2}}{6 d^{4} \left (-b x -\frac {c b}{d}\right )}+\frac {b^{5} {\mathrm e}^{-b x -a} c^{2}}{6 d^{6} \left (-b x -\frac {c b}{d}\right )}+\frac {b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{2}}{6 d^{4}}+\frac {b^{5} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c^{2}}{6 d^{6}}\) | \(852\) |
Input:
int(2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/(d*x+c)^4,x,method=_RETURNVERBOS E)
Output:
-1/b*(b^4/d^4*(-exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)-exp(-(a*d-b*c)/d)*Ei(1,b* x+a-(a*d-b*c)/d))-2*b^4*(a*d-b*c)/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))+b^4*(a*d-b*c)^2/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))+2*b^4/d^4*(-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*b^4/d^4*(-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^4*(a*d -b*c)/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)))
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (131) = 262\).
Time = 0.11 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.86 \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx=-\frac {2 \, d^{5} \Gamma \left (3, b x + a\right ) + {\left (b^{5} c^{5} - 2 \, {\left (a - 2\right )} b^{4} c^{4} d + {\left (a^{2} - 4 \, a + 2\right )} b^{3} c^{3} d^{2} + {\left (b^{5} c^{2} d^{3} - 2 \, {\left (a - 2\right )} b^{4} c d^{4} + {\left (a^{2} - 4 \, a + 2\right )} b^{3} d^{5}\right )} x^{3} + 3 \, {\left (b^{5} c^{3} d^{2} - 2 \, {\left (a - 2\right )} b^{4} c^{2} d^{3} + {\left (a^{2} - 4 \, a + 2\right )} b^{3} c d^{4}\right )} x^{2} + 3 \, {\left (b^{5} c^{4} d - 2 \, {\left (a - 2\right )} b^{4} c^{3} d^{2} + {\left (a^{2} - 4 \, a + 2\right )} b^{3} c^{2} d^{3}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (b^{4} c^{4} d - {\left (2 \, a - 3\right )} b^{3} c^{3} d^{2} + {\left (a^{2} - 2 \, a\right )} b^{2} c^{2} d^{3} - a^{2} b c d^{4} + {\left (b^{4} c^{2} d^{3} - 2 \, {\left (a - 2\right )} b^{3} c d^{4} + {\left (a^{2} - 4 \, a\right )} b^{2} d^{5}\right )} x^{2} + {\left (2 \, b^{4} c^{3} d^{2} - {\left (4 \, a - 7\right )} b^{3} c^{2} d^{3} + 2 \, {\left (a^{2} - 3 \, a\right )} b^{2} c d^{4} - a^{2} b d^{5}\right )} x\right )} e^{\left (-b x - a\right )}}{6 \, {\left (d^{9} x^{3} + 3 \, c d^{8} x^{2} + 3 \, c^{2} d^{7} x + c^{3} d^{6}\right )}} \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^4,x, algorithm="fricas")
Output:
-1/6*(2*d^5*gamma(3, b*x + a) + (b^5*c^5 - 2*(a - 2)*b^4*c^4*d + (a^2 - 4* a + 2)*b^3*c^3*d^2 + (b^5*c^2*d^3 - 2*(a - 2)*b^4*c*d^4 + (a^2 - 4*a + 2)* b^3*d^5)*x^3 + 3*(b^5*c^3*d^2 - 2*(a - 2)*b^4*c^2*d^3 + (a^2 - 4*a + 2)*b^ 3*c*d^4)*x^2 + 3*(b^5*c^4*d - 2*(a - 2)*b^4*c^3*d^2 + (a^2 - 4*a + 2)*b^3* c^2*d^3)*x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) + (b^4*c^4*d - (2*a - 3 )*b^3*c^3*d^2 + (a^2 - 2*a)*b^2*c^2*d^3 - a^2*b*c*d^4 + (b^4*c^2*d^3 - 2*( a - 2)*b^3*c*d^4 + (a^2 - 4*a)*b^2*d^5)*x^2 + (2*b^4*c^3*d^2 - (4*a - 7)*b ^3*c^2*d^3 + 2*(a^2 - 3*a)*b^2*c*d^4 - a^2*b*d^5)*x)*e^(-b*x - a))/(d^9*x^ 3 + 3*c*d^8*x^2 + 3*c^2*d^7*x + c^3*d^6)
Timed out. \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx=\text {Timed out} \] Input:
integrate(uppergamma(3,b*x+a)/(d*x+c)**4,x)
Output:
Timed out
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{{\left (d x + c\right )}^{4}} \,d x } \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^4,x, algorithm="maxima")
Output:
integrate(gamma(3, b*x + a)/(d*x + c)^4, x)
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx=\int { \frac {\Gamma \left (3, b x + a\right )}{{\left (d x + c\right )}^{4}} \,d x } \] Input:
integrate(gamma(3,b*x+a)/(d*x+c)^4,x, algorithm="giac")
Output:
integrate(gamma(3, b*x + a)/(d*x + c)^4, x)
Timed out. \[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx=\int \frac {2\,{\mathrm {e}}^{-a-b\,x}\,\left (a+b\,x+\frac {{\left (a+b\,x\right )}^2}{2}+1\right )}{{\left (c+d\,x\right )}^4} \,d x \] Input:
int((2*exp(- a - b*x)*(a + b*x + (a + b*x)^2/2 + 1))/(c + d*x)^4,x)
Output:
int((2*exp(- a - b*x)*(a + b*x + (a + b*x)^2/2 + 1))/(c + d*x)^4, x)
\[ \int \frac {\Gamma (3,a+b x)}{(c+d x)^4} \, dx=\text {too large to display} \] Input:
int(2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/(d*x+c)^4,x)
Output:
( - e**(b*x)*int(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b *c**3*d**2*x**2 + 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3 *e**(b*x)*c**4*d + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12*e**(b*x)*c*d**4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a**2*b**2*c**4*d**2 - 3 *e**(b*x)*int(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b*c* *3*d**2*x**2 + 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3*e* *(b*x)*c**4*d + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12* e**(b*x)*c*d**4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a**2*b**2*c**3*d**3*x - 3* e**(b*x)*int(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b*c** 3*d**2*x**2 + 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3*e** (b*x)*c**4*d + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12*e **(b*x)*c*d**4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a**2*b**2*c**2*d**4*x**2 - e**(b*x)*int(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b*c** 3*d**2*x**2 + 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3*e** (b*x)*c**4*d + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12*e **(b*x)*c*d**4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a**2*b**2*c*d**5*x**3 - 3*e **(b*x)*int(x/(e**(b*x)*b*c**5 + 4*e**(b*x)*b*c**4*d*x + 6*e**(b*x)*b*c**3 *d**2*x**2 + 4*e**(b*x)*b*c**2*d**3*x**3 + e**(b*x)*b*c*d**4*x**4 + 3*e**( b*x)*c**4*d + 12*e**(b*x)*c**3*d**2*x + 18*e**(b*x)*c**2*d**3*x**2 + 12*e* *(b*x)*c*d**4*x**3 + 3*e**(b*x)*d**5*x**4),x)*a**2*b*c**3*d**3 - 9*e**(...