3.409 \(\int e^{\frac{e}{(c+d x)^2}} (a+b x)^3 \, dx\)

Optimal. Leaf size=322 \[ \frac{2 \sqrt{\pi } b^2 e^{3/2} (b c-a d) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{2 b^2 e (c+d x) (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}+\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^3 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{3 b e (b c-a d)^2 \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}+\frac{3 b (c+d x)^2 (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{2 d^4}-\frac{(c+d x) (b c-a d)^3 e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{b^3 e^2 \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{4 d^4}+\frac{b^3 (c+d x)^4 e^{\frac{e}{(c+d x)^2}}}{4 d^4}+\frac{b^3 e (c+d x)^2 e^{\frac{e}{(c+d x)^2}}}{4 d^4} \]

[Out]

-(((b*c - a*d)^3*E^(e/(c + d*x)^2)*(c + d*x))/d^4) - (2*b^2*(b*c - a*d)*e*E^(e/(
c + d*x)^2)*(c + d*x))/d^4 + (3*b*(b*c - a*d)^2*E^(e/(c + d*x)^2)*(c + d*x)^2)/(
2*d^4) + (b^3*e*E^(e/(c + d*x)^2)*(c + d*x)^2)/(4*d^4) - (b^2*(b*c - a*d)*E^(e/(
c + d*x)^2)*(c + d*x)^3)/d^4 + (b^3*E^(e/(c + d*x)^2)*(c + d*x)^4)/(4*d^4) + ((b
*c - a*d)^3*Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^4 + (2*b^2*(b*c - a*d)*e
^(3/2)*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^4 - (3*b*(b*c - a*d)^2*e*ExpIntegralE
i[e/(c + d*x)^2])/(2*d^4) - (b^3*e^2*ExpIntegralEi[e/(c + d*x)^2])/(4*d^4)

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Rubi [A]  time = 0.601621, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{2 \sqrt{\pi } b^2 e^{3/2} (b c-a d) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{2 b^2 e (c+d x) (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^4}+\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^3 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^4}-\frac{3 b e (b c-a d)^2 \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{2 d^4}+\frac{3 b (c+d x)^2 (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{2 d^4}-\frac{(c+d x) (b c-a d)^3 e^{\frac{e}{(c+d x)^2}}}{d^4}-\frac{b^3 e^2 \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{4 d^4}+\frac{b^3 (c+d x)^4 e^{\frac{e}{(c+d x)^2}}}{4 d^4}+\frac{b^3 e (c+d x)^2 e^{\frac{e}{(c+d x)^2}}}{4 d^4} \]

Antiderivative was successfully verified.

[In]  Int[E^(e/(c + d*x)^2)*(a + b*x)^3,x]

[Out]

-(((b*c - a*d)^3*E^(e/(c + d*x)^2)*(c + d*x))/d^4) - (2*b^2*(b*c - a*d)*e*E^(e/(
c + d*x)^2)*(c + d*x))/d^4 + (3*b*(b*c - a*d)^2*E^(e/(c + d*x)^2)*(c + d*x)^2)/(
2*d^4) + (b^3*e*E^(e/(c + d*x)^2)*(c + d*x)^2)/(4*d^4) - (b^2*(b*c - a*d)*E^(e/(
c + d*x)^2)*(c + d*x)^3)/d^4 + (b^3*E^(e/(c + d*x)^2)*(c + d*x)^4)/(4*d^4) + ((b
*c - a*d)^3*Sqrt[e]*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^4 + (2*b^2*(b*c - a*d)*e
^(3/2)*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)])/d^4 - (3*b*(b*c - a*d)^2*e*ExpIntegralE
i[e/(c + d*x)^2])/(2*d^4) - (b^3*e^2*ExpIntegralEi[e/(c + d*x)^2])/(4*d^4)

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Rubi in Sympy [A]  time = 66.527, size = 298, normalized size = 0.93 \[ - \frac{b^{3} e^{2} \operatorname{Ei}{\left (\frac{e}{\left (c + d x\right )^{2}} \right )}}{4 d^{4}} + \frac{b^{3} e \left (c + d x\right )^{2} e^{\frac{e}{\left (c + d x\right )^{2}}}}{4 d^{4}} + \frac{b^{3} \left (c + d x\right )^{4} e^{\frac{e}{\left (c + d x\right )^{2}}}}{4 d^{4}} - \frac{2 \sqrt{\pi } b^{2} e^{\frac{3}{2}} \left (a d - b c\right ) \operatorname{erfi}{\left (\frac{\sqrt{e}}{c + d x} \right )}}{d^{4}} + \frac{2 b^{2} e \left (c + d x\right ) \left (a d - b c\right ) e^{\frac{e}{\left (c + d x\right )^{2}}}}{d^{4}} + \frac{b^{2} \left (c + d x\right )^{3} \left (a d - b c\right ) e^{\frac{e}{\left (c + d x\right )^{2}}}}{d^{4}} - \frac{3 b e \left (a d - b c\right )^{2} \operatorname{Ei}{\left (\frac{e}{\left (c + d x\right )^{2}} \right )}}{2 d^{4}} + \frac{3 b \left (c + d x\right )^{2} \left (a d - b c\right )^{2} e^{\frac{e}{\left (c + d x\right )^{2}}}}{2 d^{4}} - \frac{\sqrt{\pi } \sqrt{e} \left (a d - b c\right )^{3} \operatorname{erfi}{\left (\frac{\sqrt{e}}{c + d x} \right )}}{d^{4}} + \frac{\left (c + d x\right ) \left (a d - b c\right )^{3} e^{\frac{e}{\left (c + d x\right )^{2}}}}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(e/(d*x+c)**2)*(b*x+a)**3,x)

[Out]

-b**3*e**2*Ei(e/(c + d*x)**2)/(4*d**4) + b**3*e*(c + d*x)**2*exp(e/(c + d*x)**2)
/(4*d**4) + b**3*(c + d*x)**4*exp(e/(c + d*x)**2)/(4*d**4) - 2*sqrt(pi)*b**2*e**
(3/2)*(a*d - b*c)*erfi(sqrt(e)/(c + d*x))/d**4 + 2*b**2*e*(c + d*x)*(a*d - b*c)*
exp(e/(c + d*x)**2)/d**4 + b**2*(c + d*x)**3*(a*d - b*c)*exp(e/(c + d*x)**2)/d**
4 - 3*b*e*(a*d - b*c)**2*Ei(e/(c + d*x)**2)/(2*d**4) + 3*b*(c + d*x)**2*(a*d - b
*c)**2*exp(e/(c + d*x)**2)/(2*d**4) - sqrt(pi)*sqrt(e)*(a*d - b*c)**3*erfi(sqrt(
e)/(c + d*x))/d**4 + (c + d*x)*(a*d - b*c)**3*exp(e/(c + d*x)**2)/d**4

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Mathematica [A]  time = 0.355645, size = 243, normalized size = 0.75 \[ \frac{4 \sqrt{\pi } \sqrt{e} (b c-a d) \left (a^2 d^2-2 a b c d+b^2 \left (c^2+2 e\right )\right ) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )-b e \left (6 a^2 d^2-12 a b c d+b^2 \left (6 c^2+e\right )\right ) \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )+d x e^{\frac{e}{(c+d x)^2}} \left (4 a^3 d^3+6 a^2 b d^3 x+4 a b^2 d \left (d^2 x^2+2 e\right )+b^3 \left (-6 c e+d^3 x^3+d e x\right )\right )}{4 d^4}-\frac{c e^{\frac{e}{(c+d x)^2}} \left (-4 a^3 d^3+6 a^2 b c d^2-4 a b^2 d \left (c^2+2 e\right )+b^3 \left (c^3+7 c e\right )\right )}{4 d^4} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(e/(c + d*x)^2)*(a + b*x)^3,x]

[Out]

-(c*(6*a^2*b*c*d^2 - 4*a^3*d^3 - 4*a*b^2*d*(c^2 + 2*e) + b^3*(c^3 + 7*c*e))*E^(e
/(c + d*x)^2))/(4*d^4) + (d*E^(e/(c + d*x)^2)*x*(4*a^3*d^3 + 6*a^2*b*d^3*x + 4*a
*b^2*d*(2*e + d^2*x^2) + b^3*(-6*c*e + d*e*x + d^3*x^3)) + 4*(b*c - a*d)*Sqrt[e]
*(-2*a*b*c*d + a^2*d^2 + b^2*(c^2 + 2*e))*Sqrt[Pi]*Erfi[Sqrt[e]/(c + d*x)] - b*e
*(-12*a*b*c*d + 6*a^2*d^2 + b^2*(6*c^2 + e))*ExpIntegralEi[e/(c + d*x)^2])/(4*d^
4)

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Maple [A]  time = 0.023, size = 560, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e/(d*x+c)^2)*(b*x+a)^3,x)

[Out]

-1/d*(a^3*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c
)))+b^3/d^3*(-1/4*(d*x+c)^4*exp(e/(d*x+c)^2)+1/2*e*(-1/2*exp(e/(d*x+c)^2)*(d*x+c
)^2-1/2*e*Ei(1,-e/(d*x+c)^2)))+3*b^2/d^2*a*(-1/3*(d*x+c)^3*exp(e/(d*x+c)^2)+2/3*
e*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c))))-3*b
^3/d^3*c*(-1/3*(d*x+c)^3*exp(e/(d*x+c)^2)+2/3*e*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^
(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c))))+3*b/d*a^2*(-1/2*exp(e/(d*x+c)^2)*(d*x
+c)^2-1/2*e*Ei(1,-e/(d*x+c)^2))+3*b^3/d^3*c^2*(-1/2*exp(e/(d*x+c)^2)*(d*x+c)^2-1
/2*e*Ei(1,-e/(d*x+c)^2))-b^3/d^3*c^3*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^
(1/2)*erf((-e)^(1/2)/(d*x+c)))-6*b^2/d^2*c*a*(-1/2*exp(e/(d*x+c)^2)*(d*x+c)^2-1/
2*e*Ei(1,-e/(d*x+c)^2))-3*b/d*c*a^2*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1/2)/(-e)^(
1/2)*erf((-e)^(1/2)/(d*x+c)))+3*b^2/d^2*c^2*a*(-(d*x+c)*exp(e/(d*x+c)^2)+e*Pi^(1
/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (b^{3} d^{3} x^{4} + 4 \, a b^{2} d^{3} x^{3} +{\left (6 \, a^{2} b d^{3} + b^{3} d e\right )} x^{2} + 2 \,{\left (2 \, a^{3} d^{3} - 3 \, b^{3} c e + 4 \, a b^{2} d e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{4 \, d^{3}} + \int \frac{{\left (3 \, b^{3} c^{4} e - 4 \, a b^{2} c^{3} d e -{\left (12 \, a b^{2} c d^{3} e - 6 \, a^{2} b d^{4} e -{\left (6 \, c^{2} d^{2} e + d^{2} e^{2}\right )} b^{3}\right )} x^{2} + 2 \,{\left (2 \, a^{3} d^{4} e - 2 \,{\left (3 \, c^{2} d^{2} e - 2 \, d^{2} e^{2}\right )} a b^{2} +{\left (4 \, c^{3} d e - 3 \, c d e^{2}\right )} b^{3}\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{2 \,{\left (d^{6} x^{3} + 3 \, c d^{5} x^{2} + 3 \, c^{2} d^{4} x + c^{3} d^{3}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(e/(d*x + c)^2),x, algorithm="maxima")

[Out]

1/4*(b^3*d^3*x^4 + 4*a*b^2*d^3*x^3 + (6*a^2*b*d^3 + b^3*d*e)*x^2 + 2*(2*a^3*d^3
- 3*b^3*c*e + 4*a*b^2*d*e)*x)*e^(e/(d^2*x^2 + 2*c*d*x + c^2))/d^3 + integrate(1/
2*(3*b^3*c^4*e - 4*a*b^2*c^3*d*e - (12*a*b^2*c*d^3*e - 6*a^2*b*d^4*e - (6*c^2*d^
2*e + d^2*e^2)*b^3)*x^2 + 2*(2*a^3*d^4*e - 2*(3*c^2*d^2*e - 2*d^2*e^2)*a*b^2 + (
4*c^3*d*e - 3*c*d*e^2)*b^3)*x)*e^(e/(d^2*x^2 + 2*c*d*x + c^2))/(d^6*x^3 + 3*c*d^
5*x^2 + 3*c^2*d^4*x + c^3*d^3), x)

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Fricas [A]  time = 0.283888, size = 450, normalized size = 1.4 \[ \frac{4 \, \sqrt{\pi }{\left (2 \,{\left (b^{3} c - a b^{2} d\right )} e^{2} +{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e\right )} \operatorname{erf}\left (\frac{d \sqrt{-\frac{e}{d^{2}}}}{d x + c}\right ) -{\left ({\left (b^{3} d e^{2} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} e\right )}{\rm Ei}\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) -{\left (b^{3} d^{5} x^{4} + 4 \, a b^{2} d^{5} x^{3} - b^{3} c^{4} d + 4 \, a b^{2} c^{3} d^{2} - 6 \, a^{2} b c^{2} d^{3} + 4 \, a^{3} c d^{4} +{\left (6 \, a^{2} b d^{5} + b^{3} d^{3} e\right )} x^{2} -{\left (7 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2}\right )} e + 2 \,{\left (2 \, a^{3} d^{5} -{\left (3 \, b^{3} c d^{2} - 4 \, a b^{2} d^{3}\right )} e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}\right )} \sqrt{-\frac{e}{d^{2}}}}{4 \, d^{5} \sqrt{-\frac{e}{d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(e/(d*x + c)^2),x, algorithm="fricas")

[Out]

1/4*(4*sqrt(pi)*(2*(b^3*c - a*b^2*d)*e^2 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*
d^2 - a^3*d^3)*e)*erf(d*sqrt(-e/d^2)/(d*x + c)) - ((b^3*d*e^2 + 6*(b^3*c^2*d - 2
*a*b^2*c*d^2 + a^2*b*d^3)*e)*Ei(e/(d^2*x^2 + 2*c*d*x + c^2)) - (b^3*d^5*x^4 + 4*
a*b^2*d^5*x^3 - b^3*c^4*d + 4*a*b^2*c^3*d^2 - 6*a^2*b*c^2*d^3 + 4*a^3*c*d^4 + (6
*a^2*b*d^5 + b^3*d^3*e)*x^2 - (7*b^3*c^2*d - 8*a*b^2*c*d^2)*e + 2*(2*a^3*d^5 - (
3*b^3*c*d^2 - 4*a*b^2*d^3)*e)*x)*e^(e/(d^2*x^2 + 2*c*d*x + c^2)))*sqrt(-e/d^2))/
(d^5*sqrt(-e/d^2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(e/(d*x+c)**2)*(b*x+a)**3,x)

[Out]

Integral((a + b*x)**3*exp(e/(c**2 + 2*c*d*x + d**2*x**2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^3*e^(e/(d*x + c)^2),x, algorithm="giac")

[Out]

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