\(\int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx\) [19]

Optimal result

Integrand size = 10, antiderivative size = 194 \[ \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx=\frac {3 e^{a+b x}}{2 b^4}+\frac {a e^{a+b x}}{2 b^4}+\frac {a^2 e^{a+b x}}{4 b^4}+\frac {a^3 e^{a+b x}}{4 b^4}-\frac {3 e^{a+b x} x}{2 b^3}-\frac {a e^{a+b x} x}{2 b^3}-\frac {a^2 e^{a+b x} x}{4 b^3}+\frac {3 e^{a+b x} x^2}{4 b^2}+\frac {a e^{a+b x} x^2}{4 b^2}-\frac {e^{a+b x} x^3}{4 b}-\frac {a^4 \operatorname {ExpIntegralEi}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \operatorname {ExpIntegralEi}(a+b x) \] Output:

3/2*exp(b*x+a)/b^4+1/2*a*exp(b*x+a)/b^4+1/4*a^2*exp(b*x+a)/b^4+1/4*a^3*exp 
(b*x+a)/b^4-3/2*exp(b*x+a)*x/b^3-1/2*a*exp(b*x+a)*x/b^3-1/4*a^2*exp(b*x+a) 
*x/b^3+3/4*exp(b*x+a)*x^2/b^2+1/4*a*exp(b*x+a)*x^2/b^2-1/4*exp(b*x+a)*x^3/ 
b-1/4*a^4*Ei(b*x+a)/b^4+1/4*x^4*Ei(b*x+a)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.44 \[ \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx=\frac {e^{a+b x} \left (6+a^3-6 b x+3 b^2 x^2-b^3 x^3+a^2 (1-b x)+a \left (2-2 b x+b^2 x^2\right )\right )+\left (-a^4+b^4 x^4\right ) \operatorname {ExpIntegralEi}(a+b x)}{4 b^4} \] Input:

Integrate[x^3*ExpIntegralEi[a + b*x],x]
 

Output:

(E^(a + b*x)*(6 + a^3 - 6*b*x + 3*b^2*x^2 - b^3*x^3 + a^2*(1 - b*x) + a*(2 
 - 2*b*x + b^2*x^2)) + (-a^4 + b^4*x^4)*ExpIntegralEi[a + b*x])/(4*b^4)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.90, 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.

\(\displaystyle \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx\)

\(\Big \downarrow \) 7039

\(\displaystyle \frac {1}{4} x^4 \operatorname {ExpIntegralEi}(a+b x)-\frac {1}{4} b \int \frac {e^{a+b x} x^4}{a+b x}dx\)

\(\Big \downarrow \) 2629

\(\displaystyle \frac {1}{4} x^4 \operatorname {ExpIntegralEi}(a+b x)-\frac {1}{4} b \int \left (\frac {e^{a+b x} a^4}{b^4 (a+b x)}-\frac {e^{a+b x} a^3}{b^4}+\frac {e^{a+b x} x a^2}{b^3}-\frac {e^{a+b x} x^2 a}{b^2}+\frac {e^{a+b x} x^3}{b}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} x^4 \operatorname {ExpIntegralEi}(a+b x)-\frac {1}{4} b \left (\frac {a^4 \operatorname {ExpIntegralEi}(a+b x)}{b^5}-\frac {a^3 e^{a+b x}}{b^5}-\frac {a^2 e^{a+b x}}{b^5}+\frac {a^2 x e^{a+b x}}{b^4}-\frac {2 a e^{a+b x}}{b^5}-\frac {6 e^{a+b x}}{b^5}+\frac {2 a x e^{a+b x}}{b^4}+\frac {6 x e^{a+b x}}{b^4}-\frac {a x^2 e^{a+b x}}{b^3}-\frac {3 x^2 e^{a+b x}}{b^3}+\frac {x^3 e^{a+b x}}{b^2}\right )\)

Input:

Int[x^3*ExpIntegralEi[a + b*x],x]
 

Output:

(x^4*ExpIntegralEi[a + b*x])/4 - (b*((-6*E^(a + b*x))/b^5 - (2*a*E^(a + b* 
x))/b^5 - (a^2*E^(a + b*x))/b^5 - (a^3*E^(a + b*x))/b^5 + (6*E^(a + b*x)*x 
)/b^4 + (2*a*E^(a + b*x)*x)/b^4 + (a^2*E^(a + b*x)*x)/b^4 - (3*E^(a + b*x) 
*x^2)/b^3 - (a*E^(a + b*x)*x^2)/b^3 + (E^(a + b*x)*x^3)/b^2 + (a^4*ExpInte 
gralEi[a + b*x])/b^5))/4
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2629

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]
 

rule 7039

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]
 

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(\frac {\operatorname {expIntegral}\left (b x +a \right ) b^{4} x^{4}-{\mathrm e}^{b x +a} x^{3} b^{3}+{\mathrm e}^{b x +a} x^{2} b^{2} a +3 \,{\mathrm e}^{b x +a} x^{2} b^{2}-x \,{\mathrm e}^{b x +a} a^{2} b -\operatorname {expIntegral}\left (b x +a \right ) a^{4}-2 x \,{\mathrm e}^{b x +a} a b +{\mathrm e}^{b x +a} a^{3}-6 x b \,{\mathrm e}^{b x +a}+{\mathrm e}^{b x +a} a^{2}+2 a \,{\mathrm e}^{b x +a}+6 \,{\mathrm e}^{b x +a}}{4 b^{4}}\)	\(144\)
derivativedivides	\(\frac {\frac {\operatorname {expIntegral}\left (b x +a \right ) b^{4} x^{4}}{4}-\frac {{\mathrm e}^{b x +a} \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} {\mathrm e}^{b x +a}}{4}-\frac {3 \left (b x +a \right ) {\mathrm e}^{b x +a}}{2}+\frac {3 \,{\mathrm e}^{b x +a}}{2}+\frac {a^{4} \operatorname {expIntegral}_{1}\left (-b x -a \right )}{4}+{\mathrm e}^{b x +a} a^{3}-\frac {3 a^{2} \left (\left (b x +a \right ) {\mathrm e}^{b x +a}-{\mathrm e}^{b x +a}\right )}{2}+a \left (\left (b x +a \right )^{2} {\mathrm e}^{b x +a}-2 \left (b x +a \right ) {\mathrm e}^{b x +a}+2 \,{\mathrm e}^{b x +a}\right )}{b^{4}}\)	\(160\)
default	\(\frac {\frac {\operatorname {expIntegral}\left (b x +a \right ) b^{4} x^{4}}{4}-\frac {{\mathrm e}^{b x +a} \left (b x +a \right )^{3}}{4}+\frac {3 \left (b x +a \right )^{2} {\mathrm e}^{b x +a}}{4}-\frac {3 \left (b x +a \right ) {\mathrm e}^{b x +a}}{2}+\frac {3 \,{\mathrm e}^{b x +a}}{2}+\frac {a^{4} \operatorname {expIntegral}_{1}\left (-b x -a \right )}{4}+{\mathrm e}^{b x +a} a^{3}-\frac {3 a^{2} \left (\left (b x +a \right ) {\mathrm e}^{b x +a}-{\mathrm e}^{b x +a}\right )}{2}+a \left (\left (b x +a \right )^{2} {\mathrm e}^{b x +a}-2 \left (b x +a \right ) {\mathrm e}^{b x +a}+2 \,{\mathrm e}^{b x +a}\right )}{b^{4}}\)	\(160\)
parts	\(\frac {x^{4} \operatorname {expIntegral}\left (b x +a \right )}{4}-\frac {{\mathrm e}^{b x +a} \left (b x +a \right )^{3}-3 \left (b x +a \right )^{2} {\mathrm e}^{b x +a}+6 \left (b x +a \right ) {\mathrm e}^{b x +a}-6 \,{\mathrm e}^{b x +a}-a^{4} \operatorname {expIntegral}_{1}\left (-b x -a \right )-4 \,{\mathrm e}^{b x +a} a^{3}+6 a^{2} \left (\left (b x +a \right ) {\mathrm e}^{b x +a}-{\mathrm e}^{b x +a}\right )-4 a \left (\left (b x +a \right )^{2} {\mathrm e}^{b x +a}-2 \left (b x +a \right ) {\mathrm e}^{b x +a}+2 \,{\mathrm e}^{b x +a}\right )}{4 b^{4}}\)	\(160\)

Input:

int(x^3*Ei(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/4*(Ei(b*x+a)*b^4*x^4-exp(b*x+a)*x^3*b^3+exp(b*x+a)*x^2*b^2*a+3*exp(b*x+a 
)*x^2*b^2-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 
^3-6*x*b*exp(b*x+a)+exp(b*x+a)*a^2+2*a*exp(b*x+a)+6*exp(b*x+a))/b^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.40 \[ \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx=\frac {{\left (b^{4} x^{4} - a^{4}\right )} {\rm Ei}\left (b x + a\right ) - {\left (b^{3} x^{3} - {\left (a + 3\right )} b^{2} x^{2} - a^{3} + {\left (a^{2} + 2 \, a + 6\right )} b x - a^{2} - 2 \, a - 6\right )} e^{\left (b x + a\right )}}{4 \, b^{4}} \] Input:

integrate(x^3*Ei(b*x+a),x, algorithm="fricas")
 

Output:

1/4*((b^4*x^4 - a^4)*Ei(b*x + a) - (b^3*x^3 - (a + 3)*b^2*x^2 - a^3 + (a^2 
 + 2*a + 6)*b*x - a^2 - 2*a - 6)*e^(b*x + a))/b^4
 

Sympy [A] (verification not implemented)

Time = 12.39 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.05 \[ \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx=\begin {cases} - \frac {a^{4} \operatorname {Ei}{\left (a + b x \right )}}{4 b^{4}} + \frac {a^{3} e^{a} e^{b x}}{4 b^{4}} - \frac {a^{2} x e^{a} e^{b x}}{4 b^{3}} + \frac {a^{2} e^{a} e^{b x}}{4 b^{4}} + \frac {a x^{2} e^{a} e^{b x}}{4 b^{2}} - \frac {a x e^{a} e^{b x}}{2 b^{3}} + \frac {a e^{a} e^{b x}}{2 b^{4}} + \frac {x^{4} \operatorname {Ei}{\left (a + b x \right )}}{4} - \frac {x^{3} e^{a} e^{b x}}{4 b} + \frac {3 x^{2} e^{a} e^{b x}}{4 b^{2}} - \frac {3 x e^{a} e^{b x}}{2 b^{3}} + \frac {3 e^{a} e^{b x}}{2 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {Ei}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \] Input:

integrate(x**3*Ei(b*x+a),x)
 

Output:

Piecewise((-a**4*Ei(a + b*x)/(4*b**4) + a**3*exp(a)*exp(b*x)/(4*b**4) - a* 
*2*x*exp(a)*exp(b*x)/(4*b**3) + a**2*exp(a)*exp(b*x)/(4*b**4) + a*x**2*exp 
(a)*exp(b*x)/(4*b**2) - a*x*exp(a)*exp(b*x)/(2*b**3) + a*exp(a)*exp(b*x)/( 
2*b**4) + x**4*Ei(a + b*x)/4 - x**3*exp(a)*exp(b*x)/(4*b) + 3*x**2*exp(a)* 
exp(b*x)/(4*b**2) - 3*x*exp(a)*exp(b*x)/(2*b**3) + 3*exp(a)*exp(b*x)/(2*b* 
*4), Ne(b, 0)), (x**4*Ei(a)/4, True))
 

Maxima [F]

\[ \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx=\int { x^{3} {\rm Ei}\left (b x + a\right ) \,d x } \] Input:

integrate(x^3*Ei(b*x+a),x, algorithm="maxima")
 

Output:

integrate(x^3*Ei(b*x + a), x)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.73 \[ \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx=\frac {1}{4} \, x^{4} {\rm Ei}\left (b x + a\right ) - \frac {b^{3} x^{3} e^{\left (b x + a\right )} - a b^{2} x^{2} e^{\left (b x + a\right )} + a^{4} {\rm Ei}\left (b x + a\right ) + a^{2} b x e^{\left (b x + a\right )} - 3 \, b^{2} x^{2} e^{\left (b x + a\right )} - a^{3} e^{\left (b x + a\right )} + 2 \, a b x e^{\left (b x + a\right )} - a^{2} e^{\left (b x + a\right )} + 6 \, b x e^{\left (b x + a\right )} - 2 \, a e^{\left (b x + a\right )} - 6 \, e^{\left (b x + a\right )}}{4 \, b^{4}} \] Input:

integrate(x^3*Ei(b*x+a),x, algorithm="giac")
 

Output:

1/4*x^4*Ei(b*x + a) - 1/4*(b^3*x^3*e^(b*x + a) - a*b^2*x^2*e^(b*x + a) + a 
^4*Ei(b*x + a) + a^2*b*x*e^(b*x + a) - 3*b^2*x^2*e^(b*x + a) - a^3*e^(b*x 
+ a) + 2*a*b*x*e^(b*x + a) - a^2*e^(b*x + a) + 6*b*x*e^(b*x + a) - 2*a*e^( 
b*x + a) - 6*e^(b*x + a))/b^4
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx=\int x^3\,\mathrm {ei}\left (a+b\,x\right ) \,d x \] Input:

int(x^3*ei(a + b*x),x)
 

Output:

int(x^3*ei(a + b*x), x)
 

Reduce [F]

\[ \int x^3 \operatorname {ExpIntegralEi}(a+b x) \, dx=\int \mathit {ei} \left (b x +a \right ) x^{3}d x \] Input:

int(x^3*Ei(b*x+a),x)
 

Output:

int(ei(a + b*x)*x**3,x)