3.5.10 \(\int e^{\frac {e}{(c+d x)^2}} (a+b x)^2 \, dx\) [410]

3.5.10.1 Optimal result

Integrand size = 19, antiderivative size = 215 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^2 \, dx=\frac {(b c-a d)^2 e^{\frac {e}{(c+d x)^2}} (c+d x)}{d^3}+\frac {2 b^2 e e^{\frac {e}{(c+d x)^2}} (c+d x)}{3 d^3}-\frac {b (b c-a d) e^{\frac {e}{(c+d x)^2}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{(c+d x)^2}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 \sqrt {e} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^3}-\frac {2 b^2 e^{3/2} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{3 d^3}+\frac {b (b c-a d) e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{d^3} \]

output

(-a*d+b*c)^2*exp(e/(d*x+c)^2)*(d*x+c)/d^3+2/3*b^2*e*exp(e/(d*x+c)^2)*(d*x+ 
c)/d^3-b*(-a*d+b*c)*exp(e/(d*x+c)^2)*(d*x+c)^2/d^3+1/3*b^2*exp(e/(d*x+c)^2 
)*(d*x+c)^3/d^3+b*(-a*d+b*c)*e*Ei(e/(d*x+c)^2)/d^3-2/3*b^2*e^(3/2)*erfi(e^ 
(1/2)/(d*x+c))*Pi^(1/2)/d^3-(-a*d+b*c)^2*erfi(e^(1/2)/(d*x+c))*e^(1/2)*Pi^ 
(1/2)/d^3
 

3.5.10.2 Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.82 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^2 \, dx=\frac {c \left (-3 a b c d+3 a^2 d^2+b^2 \left (c^2+2 e\right )\right ) e^{\frac {e}{(c+d x)^2}}}{3 d^3}+\frac {d e^{\frac {e}{(c+d x)^2}} x \left (3 a^2 d^2+3 a b d^2 x+b^2 \left (2 e+d^2 x^2\right )\right )-\sqrt {e} \left (-6 a b c d+3 a^2 d^2+b^2 \left (3 c^2+2 e\right )\right ) \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )+3 b (b c-a d) e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{3 d^3} \]

input

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

output

(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)
 

3.5.10.3 Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2656, 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.

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

\(\Big \downarrow \) 2656

\(\displaystyle \int \left (\frac {(a d-b c)^2 e^{\frac {e}{(c+d x)^2}}}{d^2}-\frac {2 b (c+d x) (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^2}+\frac {b^2 (c+d x)^2 e^{\frac {e}{(c+d x)^2}}}{d^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {\pi } \sqrt {e} (b c-a d)^2 \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{d^3}+\frac {b e (b c-a d) \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^2}\right )}{d^3}-\frac {b (c+d x)^2 (b c-a d) e^{\frac {e}{(c+d x)^2}}}{d^3}+\frac {(c+d x) (b c-a d)^2 e^{\frac {e}{(c+d x)^2}}}{d^3}-\frac {2 \sqrt {\pi } b^2 e^{3/2} \text {erfi}\left (\frac {\sqrt {e}}{c+d x}\right )}{3 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{(c+d x)^2}}}{3 d^3}+\frac {2 b^2 e (c+d x) e^{\frac {e}{(c+d x)^2}}}{3 d^3}\)

input

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

output

((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]*Sqr 
t[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
 

3.5.10.3.1 Defintions of rubi rules used

rule 2009

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

rule 2656

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ 
ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, 
 b, c, d, n}, x] && PolynomialQ[Px, x]
 

3.5.10.4 Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.46

method	result	size
derivativedivides	\(-\frac {a^{2} \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )+\frac {b^{2} \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{3}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d^{2}}+\frac {2 b a \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}-\frac {2 b^{2} c \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d^{2}}-\frac {2 b c a \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d}}{d}\)	\(313\)
default	\(-\frac {a^{2} \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )+\frac {b^{2} \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{3}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d^{2}}+\frac {2 b a \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}-\frac {2 b^{2} c \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d^{2}}-\frac {2 b c a \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{d}}{d}\)	\(313\)
risch	\(a^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {a^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {a^{2} e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}+\frac {b^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x^{3}}{3}+\frac {b^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c^{3}}{3 d^{3}}+\frac {2 b^{2} e \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x}{3 d^{2}}+\frac {2 b^{2} e \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{3 d^{3}}-\frac {2 b^{2} e^{2} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{3 d^{3} \sqrt {-e}}-\frac {c^{2} b^{2} e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d^{3} \sqrt {-e}}+a b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x^{2}-\frac {a b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c^{2}}{d^{2}}+\frac {a b e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{d^{2}}-\frac {b^{2} c e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{d^{3}}+\frac {2 a b c e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d^{2} \sqrt {-e}}\)	\(320\)
parts	\(b^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x^{3}+2 a b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x^{2}+a^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} x +\frac {b^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c \,x^{2}}{d}+\frac {2 a b \,{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c x}{d}+\frac {a^{2} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} c}{d}-\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) b^{2} x^{2}}{d \sqrt {-e}}-\frac {2 e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) a b x}{d \sqrt {-e}}-\frac {a^{2} e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{d \sqrt {-e}}+\frac {2 b \left (-\frac {e \sqrt {\pi }\, \left (\frac {\operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right ) \left (-\frac {\left (a d -c b \right ) \left (d x +c \right )}{\sqrt {-e}}+\frac {\sqrt {-e}\, b \left (d x +c \right )^{2}}{2 e}\right )}{d}-\frac {a d \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )-c b \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )-\sqrt {-e}\, b \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )}{\sqrt {-e}}-\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )\right )}{\sqrt {\pi }\, d}\right )}{d}+\frac {a \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )+\frac {b \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}}{3}+\frac {2 e \left (-\left (d x +c \right ) {\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}}+\frac {e \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-e}}{d x +c}\right )}{\sqrt {-e}}\right )}{3}\right )}{d}-\frac {b c \left (-\frac {{\mathrm e}^{\frac {e}{\left (d x +c \right )^{2}}} \left (d x +c \right )^{2}}{2}-\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{\left (d x +c \right )^{2}}\right )}{2}\right )}{d}}{d}\right )}{d}\)	\(526\)

input

int(exp(e/(d*x+c)^2)*(b*x+a)^2,x,method=_RETURNVERBOSE)
 

output

-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))))
 

3.5.10.5 Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.91 \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^2 \, dx=\frac {3 \, {\left (b^{2} c - a b d\right )} e {\rm Ei}\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \sqrt {\pi } {\left (3 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 3 \, a^{2} d^{3} + 2 \, b^{2} d e\right )} \sqrt {-\frac {e}{d^{2}}} \operatorname {erf}\left (\frac {d \sqrt {-\frac {e}{d^{2}}}}{d x + c}\right ) + {\left (b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2} + 2 \, b^{2} c e + {\left (3 \, a^{2} d^{3} + 2 \, b^{2} d e\right )} x\right )} e^{\left (\frac {e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}}{3 \, d^{3}} \]

input

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

output

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

3.5.10.6 Sympy [F]

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

input

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

output

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

3.5.10.7 Maxima [F]

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

input

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

output

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)
 

3.5.10.8 Giac [F]

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

input

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

output

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

3.5.10.9 Mupad [F(-1)]

Timed out. \[ \int e^{\frac {e}{(c+d x)^2}} (a+b x)^2 \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^2}}\,{\left (a+b\,x\right )}^2 \,d x \]

input

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

output

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