Integrand size = 10, antiderivative size = 124 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx=-\frac {2 e^{a+b x}}{3 b^3}-\frac {a e^{a+b x}}{3 b^3}-\frac {a^2 e^{a+b x}}{3 b^3}+\frac {2 e^{a+b x} x}{3 b^2}+\frac {a e^{a+b x} x}{3 b^2}-\frac {e^{a+b x} x^2}{3 b}+\frac {a^3 \operatorname {ExpIntegralEi}(a+b x)}{3 b^3}+\frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x) \] Output:
-2/3*exp(b*x+a)/b^3-1/3*a*exp(b*x+a)/b^3-1/3*a^2*exp(b*x+a)/b^3+2/3*exp(b* x+a)*x/b^2+1/3*a*exp(b*x+a)*x/b^2-1/3*exp(b*x+a)*x^2/b+1/3*a^3*Ei(b*x+a)/b ^3+1/3*x^3*Ei(b*x+a)
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.46 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx=\frac {-e^{a+b x} \left (2+a+a^2-2 b x-a b x+b^2 x^2\right )+\left (a^3+b^3 x^3\right ) \operatorname {ExpIntegralEi}(a+b x)}{3 b^3} \] Input:
Integrate[x^2*ExpIntegralEi[a + b*x],x]
Output:
(-(E^(a + b*x)*(2 + a + a^2 - 2*b*x - a*b*x + b^2*x^2)) + (a^3 + b^3*x^3)* ExpIntegralEi[a + b*x])/(3*b^3)
Time = 0.37 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {7039, 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.
\(\displaystyle \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx\) |
\(\Big \downarrow \) 7039 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)-\frac {1}{3} b \int \frac {e^{a+b x} x^3}{a+b x}dx\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)-\frac {1}{3} b \int \left (-\frac {e^{a+b x} a^3}{b^3 (a+b x)}+\frac {e^{a+b x} a^2}{b^3}-\frac {e^{a+b x} x a}{b^2}+\frac {e^{a+b x} x^2}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \operatorname {ExpIntegralEi}(a+b x)-\frac {1}{3} b \left (-\frac {a^3 \operatorname {ExpIntegralEi}(a+b x)}{b^4}+\frac {a^2 e^{a+b x}}{b^4}+\frac {a e^{a+b x}}{b^4}+\frac {2 e^{a+b x}}{b^4}-\frac {a x e^{a+b x}}{b^3}-\frac {2 x e^{a+b x}}{b^3}+\frac {x^2 e^{a+b x}}{b^2}\right )\) |
Input:
Int[x^2*ExpIntegralEi[a + b*x],x]
Output:
(x^3*ExpIntegralEi[a + b*x])/3 - (b*((2*E^(a + b*x))/b^4 + (a*E^(a + b*x)) /b^4 + (a^2*E^(a + b*x))/b^4 - (2*E^(a + b*x)*x)/b^3 - (a*E^(a + b*x)*x)/b ^3 + (E^(a + b*x)*x^2)/b^2 - (a^3*ExpIntegralEi[a + b*x])/b^4))/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[ExpIntegralEi[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(ExpIntegralEi[a + b*x]/(d*(m + 1))), x] - Simp[ b/(d*(m + 1)) Int[(c + d*x)^(m + 1)*(E^(a + b*x)/(a + b*x)), x], x] /; Fr eeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.83
method | result | size |
parallelrisch | \(\frac {x^{3} \operatorname {expIntegral}\left (b x +a \right ) a \,b^{3}-{\mathrm e}^{b x +a} x^{2} b^{2} a +x \,{\mathrm e}^{b x +a} a^{2} b +\operatorname {expIntegral}\left (b x +a \right ) a^{4}+2 x \,{\mathrm e}^{b x +a} a b -{\mathrm e}^{b x +a} a^{3}-{\mathrm e}^{b x +a} a^{2}-2 a \,{\mathrm e}^{b x +a}}{3 b^{3} a}\) | \(103\) |
parts | \(\frac {x^{3} \operatorname {expIntegral}\left (b x +a \right )}{3}-\frac {\left (b x +a \right )^{2} {\mathrm e}^{b x +a}-2 \left (b x +a \right ) {\mathrm e}^{b x +a}+2 \,{\mathrm e}^{b x +a}+3 \,{\mathrm e}^{b x +a} a^{2}+a^{3} \operatorname {expIntegral}_{1}\left (-b x -a \right )-3 a \left (\left (b x +a \right ) {\mathrm e}^{b x +a}-{\mathrm e}^{b x +a}\right )}{3 b^{3}}\) | \(103\) |
derivativedivides | \(\frac {\frac {b^{3} x^{3} \operatorname {expIntegral}\left (b x +a \right )}{3}-\frac {a^{3} \operatorname {expIntegral}_{1}\left (-b x -a \right )}{3}-\frac {\left (b x +a \right )^{2} {\mathrm e}^{b x +a}}{3}+\frac {2 \left (b x +a \right ) {\mathrm e}^{b x +a}}{3}-\frac {2 \,{\mathrm e}^{b x +a}}{3}-{\mathrm e}^{b x +a} a^{2}+a \left (\left (b x +a \right ) {\mathrm e}^{b x +a}-{\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(105\) |
default | \(\frac {\frac {b^{3} x^{3} \operatorname {expIntegral}\left (b x +a \right )}{3}-\frac {a^{3} \operatorname {expIntegral}_{1}\left (-b x -a \right )}{3}-\frac {\left (b x +a \right )^{2} {\mathrm e}^{b x +a}}{3}+\frac {2 \left (b x +a \right ) {\mathrm e}^{b x +a}}{3}-\frac {2 \,{\mathrm e}^{b x +a}}{3}-{\mathrm e}^{b x +a} a^{2}+a \left (\left (b x +a \right ) {\mathrm e}^{b x +a}-{\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(105\) |
Input:
int(x^2*Ei(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/3*(x^3*Ei(b*x+a)*a*b^3-exp(b*x+a)*x^2*b^2*a+x*exp(b*x+a)*a^2*b+Ei(b*x+a) *a^4+2*x*exp(b*x+a)*a*b-exp(b*x+a)*a^3-exp(b*x+a)*a^2-2*a*exp(b*x+a))/b^3/ a
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.42 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx=\frac {{\left (b^{3} x^{3} + a^{3}\right )} {\rm Ei}\left (b x + a\right ) - {\left (b^{2} x^{2} - {\left (a + 2\right )} b x + a^{2} + a + 2\right )} e^{\left (b x + a\right )}}{3 \, b^{3}} \] Input:
integrate(x^2*Ei(b*x+a),x, algorithm="fricas")
Output:
1/3*((b^3*x^3 + a^3)*Ei(b*x + a) - (b^2*x^2 - (a + 2)*b*x + a^2 + a + 2)*e ^(b*x + a))/b^3
Time = 5.62 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.06 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx=\begin {cases} \frac {a^{3} \operatorname {Ei}{\left (a + b x \right )}}{3 b^{3}} - \frac {a^{2} e^{a} e^{b x}}{3 b^{3}} + \frac {a x e^{a} e^{b x}}{3 b^{2}} - \frac {a e^{a} e^{b x}}{3 b^{3}} + \frac {x^{3} \operatorname {Ei}{\left (a + b x \right )}}{3} - \frac {x^{2} e^{a} e^{b x}}{3 b} + \frac {2 x e^{a} e^{b x}}{3 b^{2}} - \frac {2 e^{a} e^{b x}}{3 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {Ei}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \] Input:
integrate(x**2*Ei(b*x+a),x)
Output:
Piecewise((a**3*Ei(a + b*x)/(3*b**3) - a**2*exp(a)*exp(b*x)/(3*b**3) + a*x *exp(a)*exp(b*x)/(3*b**2) - a*exp(a)*exp(b*x)/(3*b**3) + x**3*Ei(a + b*x)/ 3 - x**2*exp(a)*exp(b*x)/(3*b) + 2*x*exp(a)*exp(b*x)/(3*b**2) - 2*exp(a)*e xp(b*x)/(3*b**3), Ne(b, 0)), (x**3*Ei(a)/3, True))
\[ \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx=\int { x^{2} {\rm Ei}\left (b x + a\right ) \,d x } \] Input:
integrate(x^2*Ei(b*x+a),x, algorithm="maxima")
Output:
integrate(x^2*Ei(b*x + a), x)
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.72 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx=\frac {1}{3} \, x^{3} {\rm Ei}\left (b x + a\right ) - \frac {b^{2} x^{2} e^{\left (b x + a\right )} - a^{3} {\rm Ei}\left (b x + a\right ) - a b x e^{\left (b x + a\right )} + a^{2} e^{\left (b x + a\right )} - 2 \, b x e^{\left (b x + a\right )} + a e^{\left (b x + a\right )} + 2 \, e^{\left (b x + a\right )}}{3 \, b^{3}} \] Input:
integrate(x^2*Ei(b*x+a),x, algorithm="giac")
Output:
1/3*x^3*Ei(b*x + a) - 1/3*(b^2*x^2*e^(b*x + a) - a^3*Ei(b*x + a) - a*b*x*e ^(b*x + a) + a^2*e^(b*x + a) - 2*b*x*e^(b*x + a) + a*e^(b*x + a) + 2*e^(b* x + a))/b^3
Time = 0.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx=\frac {x^3\,\mathrm {ei}\left (a+b\,x\right )}{3}-\frac {\frac {2\,{\mathrm {e}}^{a+b\,x}}{3}+\frac {a\,{\mathrm {e}}^{a+b\,x}}{3}-\frac {a^3\,\mathrm {ei}\left (a+b\,x\right )}{3}+\frac {a^2\,{\mathrm {e}}^{a+b\,x}}{3}-b\,\left (\frac {2\,x\,{\mathrm {e}}^{a+b\,x}}{3}+\frac {a\,x\,{\mathrm {e}}^{a+b\,x}}{3}\right )+\frac {b^2\,x^2\,{\mathrm {e}}^{a+b\,x}}{3}}{b^3} \] Input:
int(x^2*ei(a + b*x),x)
Output:
(x^3*ei(a + b*x))/3 - ((2*exp(a + b*x))/3 + (a*exp(a + b*x))/3 - (a^3*ei(a + b*x))/3 + (a^2*exp(a + b*x))/3 - b*((2*x*exp(a + b*x))/3 + (a*x*exp(a + b*x))/3) + (b^2*x^2*exp(a + b*x))/3)/b^3
\[ \int x^2 \operatorname {ExpIntegralEi}(a+b x) \, dx=\int \mathit {ei} \left (b x +a \right ) x^{2}d x \] Input:
int(x^2*Ei(b*x+a),x)
Output:
int(ei(a + b*x)*x**2,x)