3.404 \(\int e^{\frac{e}{c+d x}} (a+b x) \, dx\)

Optimal. Leaf size=125 \[ \frac{e (b c-a d) \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d^2}-\frac{(c+d x) (b c-a d) e^{\frac{e}{c+d x}}}{d^2}-\frac{b e^2 \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{2 d^2}+\frac{b e (c+d x) e^{\frac{e}{c+d x}}}{2 d^2}+\frac{b (c+d x)^2 e^{\frac{e}{c+d x}}}{2 d^2} \]

[Out]

-(((b*c - a*d)*E^(e/(c + d*x))*(c + d*x))/d^2) + (b*e*E^(e/(c + d*x))*(c + d*x))
/(2*d^2) + (b*E^(e/(c + d*x))*(c + d*x)^2)/(2*d^2) + ((b*c - a*d)*e*ExpIntegralE
i[e/(c + d*x)])/d^2 - (b*e^2*ExpIntegralEi[e/(c + d*x)])/(2*d^2)

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Rubi [A]  time = 0.220712, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{e (b c-a d) \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d^2}-\frac{(c+d x) (b c-a d) e^{\frac{e}{c+d x}}}{d^2}-\frac{b e^2 \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{2 d^2}+\frac{b e (c+d x) e^{\frac{e}{c+d x}}}{2 d^2}+\frac{b (c+d x)^2 e^{\frac{e}{c+d x}}}{2 d^2} \]

Antiderivative was successfully verified.

[In]  Int[E^(e/(c + d*x))*(a + b*x),x]

[Out]

-(((b*c - a*d)*E^(e/(c + d*x))*(c + d*x))/d^2) + (b*e*E^(e/(c + d*x))*(c + d*x))
/(2*d^2) + (b*E^(e/(c + d*x))*(c + d*x)^2)/(2*d^2) + ((b*c - a*d)*e*ExpIntegralE
i[e/(c + d*x)])/d^2 - (b*e^2*ExpIntegralEi[e/(c + d*x)])/(2*d^2)

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Rubi in Sympy [A]  time = 19.8111, size = 105, normalized size = 0.84 \[ - \frac{b e^{2} \operatorname{Ei}{\left (\frac{e}{c + d x} \right )}}{2 d^{2}} + \frac{b e \left (c + d x\right ) e^{\frac{e}{c + d x}}}{2 d^{2}} + \frac{b \left (c + d x\right )^{2} e^{\frac{e}{c + d x}}}{2 d^{2}} - \frac{e \left (a d - b c\right ) \operatorname{Ei}{\left (\frac{e}{c + d x} \right )}}{d^{2}} + \frac{\left (c + d x\right ) \left (a d - b c\right ) e^{\frac{e}{c + d x}}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e/(d*x+c))*(b*x+a),x)

[Out]

-b*e**2*Ei(e/(c + d*x))/(2*d**2) + b*e*(c + d*x)*exp(e/(c + d*x))/(2*d**2) + b*(
c + d*x)**2*exp(e/(c + d*x))/(2*d**2) - e*(a*d - b*c)*Ei(e/(c + d*x))/d**2 + (c
+ d*x)*(a*d - b*c)*exp(e/(c + d*x))/d**2

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Mathematica [A]  time = 0.111291, size = 91, normalized size = 0.73 \[ \frac{d x e^{\frac{e}{c+d x}} (2 a d+b (d x+e))-e (2 a d+b (e-2 c)) \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{2 d^2}+\frac{c e^{\frac{e}{c+d x}} (2 a d+b (e-c))}{2 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(e/(c + d*x))*(a + b*x),x]

[Out]

(c*(2*a*d + b*(-c + e))*E^(e/(c + d*x)))/(2*d^2) + (d*E^(e/(c + d*x))*x*(2*a*d +
b*(e + d*x)) - e*(2*a*d + b*(-2*c + e))*ExpIntegralEi[e/(c + d*x)])/(2*d^2)

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Maple [A]  time = 0.007, size = 150, normalized size = 1.2 \[ -{\frac{e}{d} \left ( a \left ( -{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) +{\frac{be}{d} \left ( -{\frac{ \left ( dx+c \right ) ^{2}}{2\,{e}^{2}}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\frac{dx+c}{2\,e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\frac{1}{2}{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) } \right ) }-{\frac{cb}{d} \left ( -{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e/(d*x+c))*(b*x+a),x)

[Out]

-1/d*e*(a*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+b/d*e*(-1/2*exp(e/(d*x+c)
)*(d*x+c)^2/e^2-1/2*(d*x+c)/e*exp(e/(d*x+c))-1/2*Ei(1,-e/(d*x+c)))-b/d*c*(-(d*x+
c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (b d x^{2} +{\left (2 \, a d + b e\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{2 \, d} + \int -\frac{{\left (b c^{2} e -{\left (2 \, a d^{2} e -{\left (2 \, c d e - d e^{2}\right )} b\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*e^(e/(d*x + c)),x, algorithm="maxima")

[Out]

1/2*(b*d*x^2 + (2*a*d + b*e)*x)*e^(e/(d*x + c))/d + integrate(-1/2*(b*c^2*e - (2
*a*d^2*e - (2*c*d*e - d*e^2)*b)*x)*e^(e/(d*x + c))/(d^3*x^2 + 2*c*d^2*x + c^2*d)
, x)

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Fricas [A]  time = 0.272701, size = 112, normalized size = 0.9 \[ -\frac{{\left (b e^{2} - 2 \,{\left (b c - a d\right )} e\right )}{\rm Ei}\left (\frac{e}{d x + c}\right ) -{\left (b d^{2} x^{2} - b c^{2} + 2 \, a c d + b c e +{\left (2 \, a d^{2} + b d e\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{2 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*e^(e/(d*x + c)),x, algorithm="fricas")

[Out]

-1/2*((b*e^2 - 2*(b*c - a*d)*e)*Ei(e/(d*x + c)) - (b*d^2*x^2 - b*c^2 + 2*a*c*d +
b*c*e + (2*a*d^2 + b*d*e)*x)*e^(e/(d*x + c)))/d^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right ) e^{\frac{e}{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(e/(d*x+c))*(b*x+a),x)

[Out]

Integral((a + b*x)*exp(e/(c + d*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )} e^{\left (\frac{e}{d x + c}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*e^(e/(d*x + c)),x, algorithm="giac")

[Out]

integrate((b*x + a)*e^(e/(d*x + c)), x)