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

Optimal. Leaf size=215 \[ -\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^2 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^3}+\frac{b e (b c-a d) \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{d^3}-\frac{b (c+d x)^2 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^3}+\frac{(c+d x) (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{d^3}-\frac{2 \sqrt{\pi } b^2 e^{3/2} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{3 d^3}+\frac{b^2 (c+d x)^3 e^{\frac{e}{(c+d x)^2}}}{3 d^3}+\frac{2 b^2 e (c+d x) e^{\frac{e}{(c+d x)^2}}}{3 d^3} \]

[Out]

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

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Rubi [A]  time = 0.398264, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -\frac{\sqrt{\pi } \sqrt{e} (b c-a d)^2 \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{d^3}+\frac{b e (b c-a d) \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{d^3}-\frac{b (c+d x)^2 (b c-a d) e^{\frac{e}{(c+d x)^2}}}{d^3}+\frac{(c+d x) (b c-a d)^2 e^{\frac{e}{(c+d x)^2}}}{d^3}-\frac{2 \sqrt{\pi } b^2 e^{3/2} \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )}{3 d^3}+\frac{b^2 (c+d x)^3 e^{\frac{e}{(c+d x)^2}}}{3 d^3}+\frac{2 b^2 e (c+d x) e^{\frac{e}{(c+d x)^2}}}{3 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 43.9648, size = 197, normalized size = 0.92 \[ - \frac{2 \sqrt{\pi } b^{2} e^{\frac{3}{2}} \operatorname{erfi}{\left (\frac{\sqrt{e}}{c + d x} \right )}}{3 d^{3}} + \frac{2 b^{2} e \left (c + d x\right ) e^{\frac{e}{\left (c + d x\right )^{2}}}}{3 d^{3}} + \frac{b^{2} \left (c + d x\right )^{3} e^{\frac{e}{\left (c + d x\right )^{2}}}}{3 d^{3}} - \frac{b e \left (a d - b c\right ) \operatorname{Ei}{\left (\frac{e}{\left (c + d x\right )^{2}} \right )}}{d^{3}} + \frac{b \left (c + d x\right )^{2} \left (a d - b c\right ) e^{\frac{e}{\left (c + d x\right )^{2}}}}{d^{3}} - \frac{\sqrt{\pi } \sqrt{e} \left (a d - b c\right )^{2} \operatorname{erfi}{\left (\frac{\sqrt{e}}{c + d x} \right )}}{d^{3}} + \frac{\left (c + d x\right ) \left (a d - b c\right )^{2} e^{\frac{e}{\left (c + d x\right )^{2}}}}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.222596, size = 176, normalized size = 0.82 \[ \frac{-\sqrt{\pi } \sqrt{e} \left (3 a^2 d^2-6 a b c d+b^2 \left (3 c^2+2 e\right )\right ) \text{Erfi}\left (\frac{\sqrt{e}}{c+d x}\right )+d x e^{\frac{e}{(c+d x)^2}} \left (3 a^2 d^2+3 a b d^2 x+b^2 \left (d^2 x^2+2 e\right )\right )+3 b e (b c-a d) \text{ExpIntegralEi}\left (\frac{e}{(c+d x)^2}\right )}{3 d^3}+\frac{c e^{\frac{e}{(c+d x)^2}} \left (3 a^2 d^2-3 a b c d+b^2 \left (c^2+2 e\right )\right )}{3 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 313, normalized size = 1.5 \[ -{\frac{1}{d} \left ({a}^{2} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) +{\frac{{b}^{2}}{{d}^{2}} \left ( -{\frac{ \left ( dx+c \right ) ^{3}}{3}{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}}+{\frac{2\,e}{3} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) } \right ) }+{\frac{{b}^{2}{c}^{2}}{{d}^{2}} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{e\sqrt{\pi }{\it Erf} \left ({\frac{1}{dx+c}\sqrt{-e}} \right ){\frac{1}{\sqrt{-e}}}} \right ) }+2\,{\frac{ab}{d} \left ( -1/2\,{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}} \left ( dx+c \right ) ^{2}-1/2\,e{\it Ei} \left ( 1,-{\frac{e}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) }-2\,{\frac{c{b}^{2}}{{d}^{2}} \left ( -1/2\,{{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}} \left ( dx+c \right ) ^{2}-1/2\,e{\it Ei} \left ( 1,-{\frac{e}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) }-2\,{\frac{acb}{d} \left ( - \left ( dx+c \right ){{\rm e}^{{\frac{e}{ \left ( dx+c \right ) ^{2}}}}}+{\frac{e\sqrt{\pi }}{\sqrt{-e}}{\it Erf} \left ({\frac{\sqrt{-e}}{dx+c}} \right ) } \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/d*(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
)))+b^2/d^2*(-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))))+b^2/d^2*c^2*(-(d*x+c)*exp(e/(d*x+c
)^2)+e*Pi^(1/2)/(-e)^(1/2)*erf((-e)^(1/2)/(d*x+c)))+2*b/d*a*(-1/2*exp(e/(d*x+c)^
2)*(d*x+c)^2-1/2*e*Ei(1,-e/(d*x+c)^2))-2*b^2/d^2*c*(-1/2*exp(e/(d*x+c)^2)*(d*x+c
)^2-1/2*e*Ei(1,-e/(d*x+c)^2))-2*b/d*c*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. \[ \frac{{\left (b^{2} d^{2} x^{3} + 3 \, a b d^{2} x^{2} +{\left (3 \, a^{2} d^{2} + 2 \, b^{2} e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \, d^{2}} + \int -\frac{2 \,{\left (b^{2} c^{3} e + 3 \,{\left (b^{2} c d^{2} e - a b d^{3} e\right )} x^{2} -{\left (3 \, a^{2} d^{3} e -{\left (3 \, c^{2} d e - 2 \, d e^{2}\right )} b^{2}\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \,{\left (d^{5} x^{3} + 3 \, c d^{4} x^{2} + 3 \, c^{2} d^{3} x + c^{3} d^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.256208, size = 292, normalized size = 1.36 \[ -\frac{\sqrt{\pi }{\left (2 \, b^{2} e^{2} + 3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e\right )} \operatorname{erf}\left (\frac{d \sqrt{-\frac{e}{d^{2}}}}{d x + c}\right ) -{\left (3 \,{\left (b^{2} c d - a b d^{2}\right )} e{\rm Ei}\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) +{\left (b^{2} d^{4} x^{3} + 3 \, a b d^{4} x^{2} + b^{2} c^{3} d - 3 \, a b c^{2} d^{2} + 3 \, a^{2} c d^{3} + 2 \, b^{2} c d e +{\left (3 \, a^{2} d^{4} + 2 \, b^{2} d^{2} e\right )} x\right )} e^{\left (\frac{e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}\right )} \sqrt{-\frac{e}{d^{2}}}}{3 \, d^{4} \sqrt{-\frac{e}{d^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**2*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. \[ \int{\left (b x + a\right )}^{2} e^{\left (\frac{e}{{\left (d x + c\right )}^{2}}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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