\(\int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx\) [403]

Optimal result

Integrand size = 19, antiderivative size = 255 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}+\frac {b^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{6 d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b^2 e^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{6 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2258, 2237, 2241, 2245} \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\frac {b e^2 (b c-a d) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {e (b c-a d)^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b e (c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{d^3}-\frac {b (c+d x)^2 (b c-a d) e^{\frac {e}{c+d x}}}{d^3}+\frac {(c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^3}-\frac {b^2 e^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{6 d^3}+\frac {b^2 e^2 (c+d x) e^{\frac {e}{c+d x}}}{6 d^3}+\frac {b^2 e (c+d x)^2 e^{\frac {e}{c+d x}}}{6 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{c+d x}}}{3 d^3} \]

[In]

Int[E^(e/(c + d*x))*(a + b*x)^2,x]

[Out]

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

Rule 2237

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(c + d*x)*(F^(a + b*(c + d*x)^n)/d), x]
- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]
&& ILtQ[n, 0]

Rule 2241

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[
b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2245

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), Int[(c + d*x)^(m + n)*F^(a + b*(c +
d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n
] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Rule 2258

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^2 e^{\frac {e}{c+d x}}}{d^2}-\frac {2 b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)}{d^2}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^2}{d^2}\right ) \, dx \\ & = \frac {b^2 \int e^{\frac {e}{c+d x}} (c+d x)^2 \, dx}{d^2}-\frac {(2 b (b c-a d)) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^2}+\frac {(b c-a d)^2 \int e^{\frac {e}{c+d x}} \, dx}{d^2} \\ & = \frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}+\frac {\left (b^2 e\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{3 d^2}-\frac {(b (b c-a d) e) \int e^{\frac {e}{c+d x}} \, dx}{d^2}+\frac {\left ((b c-a d)^2 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^2} \\ & = \frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {\left (b^2 e^2\right ) \int e^{\frac {e}{c+d x}} \, dx}{6 d^2}-\frac {\left (b (b c-a d) e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^2} \\ & = \frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}+\frac {b^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{6 d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {\left (b^2 e^3\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{6 d^2} \\ & = \frac {(b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {b (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)}{d^3}+\frac {b^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{6 d^3}-\frac {b (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{6 d^3}+\frac {b^2 e^{\frac {e}{c+d x}} (c+d x)^3}{3 d^3}-\frac {(b c-a d)^2 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}+\frac {b (b c-a d) e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^3}-\frac {b^2 e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{6 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.67 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\frac {c \left (6 a^2 d^2+6 a b d (-c+e)+b^2 \left (2 c^2-5 c e+e^2\right )\right ) e^{\frac {e}{c+d x}}}{6 d^3}+\frac {d e^{\frac {e}{c+d x}} x \left (6 a^2 d^2+6 a b d (e+d x)+b^2 \left (-4 c e+e^2+d e x+2 d^2 x^2\right )\right )-e \left (6 a^2 d^2+6 a b d (-2 c+e)+b^2 \left (6 c^2-6 c e+e^2\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{6 d^3} \]

[In]

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

[Out]

(c*(6*a^2*d^2 + 6*a*b*d*(-c + e) + b^2*(2*c^2 - 5*c*e + e^2))*E^(e/(c + d*x)))/(6*d^3) + (d*E^(e/(c + d*x))*x*
(6*a^2*d^2 + 6*a*b*d*(e + d*x) + b^2*(-4*c*e + e^2 + d*e*x + 2*d^2*x^2)) - e*(6*a^2*d^2 + 6*a*b*d*(-2*c + e) +
b^2*(6*c^2 - 6*c*e + e^2))*ExpIntegralEi[e/(c + d*x)])/(6*d^3)

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.40

method	result	size
derivativedivides	\(-\frac {e \left (a^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )+\frac {b^{2} e^{2} \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{6 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d^{2}}+\frac {2 b e a \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d}-\frac {2 b^{2} e c \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d^{2}}-\frac {2 b c a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}\right )}{d}\)	\(356\)
default	\(-\frac {e \left (a^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )+\frac {b^{2} e^{2} \left (-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{3 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{6 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{6 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d^{2}}+\frac {2 b e a \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d}-\frac {2 b^{2} e c \left (-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{2 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{2 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{2}\right )}{d^{2}}-\frac {2 b c a \left (-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{e}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )\right )}{d}\right )}{d}\)	\(356\)
risch	\(a b \,{\mathrm e}^{\frac {e}{d x +c}} x^{2}-\frac {a b \,{\mathrm e}^{\frac {e}{d x +c}} c^{2}}{d^{2}}+\frac {a^{2} {\mathrm e}^{\frac {e}{d x +c}} c}{d}+\frac {e \,b^{2} {\mathrm e}^{\frac {e}{d x +c}} x^{2}}{6 d}+a^{2} {\mathrm e}^{\frac {e}{d x +c}} x +\frac {b^{2} {\mathrm e}^{\frac {e}{d x +c}} x^{3}}{3}+\frac {b^{2} {\mathrm e}^{\frac {e}{d x +c}} c^{3}}{3 d^{3}}-\frac {2 e \,b^{2} {\mathrm e}^{\frac {e}{d x +c}} c x}{3 d^{2}}+\frac {e^{2} b^{2} {\mathrm e}^{\frac {e}{d x +c}} x}{6 d^{2}}+\frac {e^{2} b^{2} {\mathrm e}^{\frac {e}{d x +c}} c}{6 d^{3}}+\frac {e a b \,{\mathrm e}^{\frac {e}{d x +c}} c}{d^{2}}+\frac {e^{2} a b \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{2}}-\frac {e^{2} c \,b^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{3}}-\frac {2 e a b c \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{2}}+\frac {e a b \,{\mathrm e}^{\frac {e}{d x +c}} x}{d}+\frac {e \,b^{2} c^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d^{3}}+\frac {e \,a^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d}+\frac {e^{3} b^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{6 d^{3}}-\frac {5 e \,b^{2} {\mathrm e}^{\frac {e}{d x +c}} c^{2}}{6 d^{3}}\)	\(387\)
parts	\(b^{2} {\mathrm e}^{\frac {e}{d x +c}} x^{3}+2 a b \,{\mathrm e}^{\frac {e}{d x +c}} x^{2}+a^{2} {\mathrm e}^{\frac {e}{d x +c}} x +\frac {b^{2} {\mathrm e}^{\frac {e}{d x +c}} c \,x^{2}}{d}+\frac {2 a b \,{\mathrm e}^{\frac {e}{d x +c}} c x}{d}+\frac {a^{2} {\mathrm e}^{\frac {e}{d x +c}} c}{d}+\frac {e \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} x^{2}}{d}+\frac {2 e \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a b x}{d}+\frac {e \,a^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{d}-\frac {4 x^{3} {\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{2} d^{3}+6 x^{2} {\mathrm e}^{\frac {e}{d x +c}} a b \,c^{2} d^{3}+6 x^{2} {\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{3} d^{2}-x^{2} {\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{2} d^{2} e +6 x^{2} \operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} c^{2} d^{2} e +12 x \,{\mathrm e}^{\frac {e}{d x +c}} a b \,c^{3} d^{2}-6 x \,{\mathrm e}^{\frac {e}{d x +c}} a b \,c^{2} d^{2} e +4 x \,{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{3} d e -x \,{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{2} d \,e^{2}+12 x \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a b \,c^{2} d^{2} e +6 \,{\mathrm e}^{\frac {e}{d x +c}} a b \,c^{4} d -6 \,{\mathrm e}^{\frac {e}{d x +c}} a b \,c^{3} d e -2 \,{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{5}+5 \,{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{4} e -{\mathrm e}^{\frac {e}{d x +c}} b^{2} c^{3} e^{2}+12 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a b \,c^{3} d e -6 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a b \,c^{2} d \,e^{2}-6 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} c^{4} e +6 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} c^{3} e^{2}-\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{2} c^{2} e^{3}}{6 c^{2} d^{3}}\)	\(622\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.77 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=-\frac {{\left (b^{2} e^{3} - 6 \, {\left (b^{2} c - a b d\right )} e^{2} + 6 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (2 \, b^{2} d^{3} x^{3} + 2 \, b^{2} c^{3} - 6 \, a b c^{2} d + 6 \, a^{2} c d^{2} + b^{2} c e^{2} + {\left (6 \, a b d^{3} + b^{2} d^{2} e\right )} x^{2} - {\left (5 \, b^{2} c^{2} - 6 \, a b c d\right )} e + {\left (6 \, a^{2} d^{3} + b^{2} d e^{2} - 2 \, {\left (2 \, b^{2} c d - 3 \, a b d^{2}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{6 \, d^{3}} \]

[In]

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

[Out]

-1/6*((b^2*e^3 - 6*(b^2*c - a*b*d)*e^2 + 6*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e)*Ei(e/(d*x + c)) - (2*b^2*d^3*x^3
+ 2*b^2*c^3 - 6*a*b*c^2*d + 6*a^2*c*d^2 + b^2*c*e^2 + (6*a*b*d^3 + b^2*d^2*e)*x^2 - (5*b^2*c^2 - 6*a*b*c*d)*e
+ (6*a^2*d^3 + b^2*d*e^2 - 2*(2*b^2*c*d - 3*a*b*d^2)*e)*x)*e^(e/(d*x + c)))/d^3

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.67 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=-\frac {{\left (\frac {6 \, b^{2} c^{2} e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} - \frac {12 \, a b c d e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} + \frac {6 \, a^{2} d^{2} e^{5} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} - \frac {6 \, b^{2} c e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} + \frac {6 \, a b d e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} + \frac {b^{2} e^{7} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} - 2 \, b^{2} e^{4} e^{\left (\frac {e}{d x + c}\right )} + \frac {6 \, b^{2} c e^{4} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} - \frac {6 \, b^{2} c^{2} e^{4} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a b d e^{4} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {12 \, a b c d e^{4} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a^{2} d^{2} e^{4} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {b^{2} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {6 \, b^{2} c e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {6 \, a b d e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {b^{2} e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}}\right )} {\left (d x + c\right )}^{3}}{6 \, d^{3} e^{4}} \]

[In]

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

[Out]

-1/6*(6*b^2*c^2*e^5*Ei(e/(d*x + c))/(d*x + c)^3 - 12*a*b*c*d*e^5*Ei(e/(d*x + c))/(d*x + c)^3 + 6*a^2*d^2*e^5*E
i(e/(d*x + c))/(d*x + c)^3 - 6*b^2*c*e^6*Ei(e/(d*x + c))/(d*x + c)^3 + 6*a*b*d*e^6*Ei(e/(d*x + c))/(d*x + c)^3
+ b^2*e^7*Ei(e/(d*x + c))/(d*x + c)^3 - 2*b^2*e^4*e^(e/(d*x + c)) + 6*b^2*c*e^4*e^(e/(d*x + c))/(d*x + c) - 6
*b^2*c^2*e^4*e^(e/(d*x + c))/(d*x + c)^2 - 6*a*b*d*e^4*e^(e/(d*x + c))/(d*x + c) + 12*a*b*c*d*e^4*e^(e/(d*x +
c))/(d*x + c)^2 - 6*a^2*d^2*e^4*e^(e/(d*x + c))/(d*x + c)^2 - b^2*e^5*e^(e/(d*x + c))/(d*x + c) + 6*b^2*c*e^5*
e^(e/(d*x + c))/(d*x + c)^2 - 6*a*b*d*e^5*e^(e/(d*x + c))/(d*x + c)^2 - b^2*e^6*e^(e/(d*x + c))/(d*x + c)^2)*(
d*x + c)^3/(d^3*e^4)

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.20 \[ \int e^{\frac {e}{c+d x}} (a+b x)^2 \, dx=\frac {x\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (2\,a^2\,c+\frac {\frac {b^2\,c^3}{3}-d\,\left (a\,b\,c^2-2\,a\,b\,c\,e\right )+\frac {b^2\,c\,e^2}{3}-\frac {3\,b^2\,c^2\,e}{2}}{d^2}\right )+\frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (\frac {b^2\,c^4}{3}-d\,\left (a\,b\,c^3-a\,b\,c^2\,e\right )-\frac {5\,b^2\,c^3\,e}{6}+a^2\,c^2\,d^2+\frac {b^2\,c^2\,e^2}{6}\right )}{d^3}+x^2\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (\frac {\frac {b^2\,e^2}{6}-\frac {b^2\,c\,e}{2}}{d}+a^2\,d+a\,b\,c+a\,b\,e\right )+\frac {b^2\,d\,x^4\,{\mathrm {e}}^{\frac {e}{c+d\,x}}}{3}+\frac {b\,x^3\,{\mathrm {e}}^{\frac {e}{c+d\,x}}\,\left (6\,a\,d+2\,b\,c+b\,e\right )}{6}}{c+d\,x}-\frac {\mathrm {ei}\left (\frac {e}{c+d\,x}\right )\,\left (\frac {b^2\,e^3}{6}+d\,\left (a\,b\,e^2-2\,a\,b\,c\,e\right )+a^2\,d^2\,e-b^2\,c\,e^2+b^2\,c^2\,e\right )}{d^3} \]

[In]

int(exp(e/(c + d*x))*(a + b*x)^2,x)

[Out]

(x*exp(e/(c + d*x))*(2*a^2*c + ((b^2*c^3)/3 - d*(a*b*c^2 - 2*a*b*c*e) + (b^2*c*e^2)/3 - (3*b^2*c^2*e)/2)/d^2)
+ (exp(e/(c + d*x))*((b^2*c^4)/3 - d*(a*b*c^3 - a*b*c^2*e) - (5*b^2*c^3*e)/6 + a^2*c^2*d^2 + (b^2*c^2*e^2)/6))
/d^3 + x^2*exp(e/(c + d*x))*(((b^2*e^2)/6 - (b^2*c*e)/2)/d + a^2*d + a*b*c + a*b*e) + (b^2*d*x^4*exp(e/(c + d*
x)))/3 + (b*x^3*exp(e/(c + d*x))*(6*a*d + 2*b*c + b*e))/6)/(c + d*x) - (ei(e/(c + d*x))*((b^2*e^3)/6 + d*(a*b*
e^2 - 2*a*b*c*e) + a^2*d^2*e - b^2*c*e^2 + b^2*c^2*e))/d^3