\(\int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx\) [402]

Optimal result

Integrand size = 19, antiderivative size = 320 \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=-\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^2 (b c-a d) e^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2258, 2237, 2241, 2245, 2250} \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\frac {b^2 e^3 (b c-a d) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^4}-\frac {b^2 e^2 (c+d x) (b c-a d) e^{\frac {e}{c+d x}}}{2 d^4}-\frac {b^2 e (c+d x)^2 (b c-a d) e^{\frac {e}{c+d x}}}{2 d^4}-\frac {b^2 (c+d x)^3 (b c-a d) e^{\frac {e}{c+d x}}}{d^4}-\frac {3 b e^2 (b c-a d)^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {e (b c-a d)^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^4}+\frac {3 b e (c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{2 d^4}+\frac {3 b (c+d x)^2 (b c-a d)^2 e^{\frac {e}{c+d x}}}{2 d^4}-\frac {(c+d x) (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4} \]

[In]

Int[E^(e/(c + d*x))*(a + b*x)^3,x]

[Out]

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

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 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

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)^3 e^{\frac {e}{c+d x}}}{d^3}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)}{d^3}-\frac {3 b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^2}{d^3}+\frac {b^3 e^{\frac {e}{c+d x}} (c+d x)^3}{d^3}\right ) \, dx \\ & = \frac {b^3 \int e^{\frac {e}{c+d x}} (c+d x)^3 \, dx}{d^3}-\frac {\left (3 b^2 (b c-a d)\right ) \int e^{\frac {e}{c+d x}} (c+d x)^2 \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^3}-\frac {(b c-a d)^3 \int e^{\frac {e}{c+d x}} \, dx}{d^3} \\ & = -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^3}+\frac {\left (3 b (b c-a d)^2 e\right ) \int e^{\frac {e}{c+d x}} \, dx}{2 d^3}-\frac {\left ((b c-a d)^3 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^3} \\ & = -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e^2\right ) \int e^{\frac {e}{c+d x}} \, dx}{2 d^3}+\frac {\left (3 b (b c-a d)^2 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 d^3} \\ & = -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4}-\frac {\left (b^2 (b c-a d) e^3\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{2 d^3} \\ & = -\frac {(b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {3 b (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}-\frac {b^2 (b c-a d) e^2 e^{\frac {e}{c+d x}} (c+d x)}{2 d^4}+\frac {3 b (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e e^{\frac {e}{c+d x}} (c+d x)^2}{2 d^4}-\frac {b^2 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {(b c-a d)^3 e \text {Ei}\left (\frac {e}{c+d x}\right )}{d^4}-\frac {3 b (b c-a d)^2 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^2 (b c-a d) e^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{2 d^4}+\frac {b^3 e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.91 \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=-\frac {c \left (-24 a^3 d^3+36 a^2 b d^2 (c-e)-12 a b^2 d \left (2 c^2-5 c e+e^2\right )+b^3 \left (6 c^3-26 c^2 e+11 c e^2-e^3\right )\right ) e^{\frac {e}{c+d x}}}{24 d^4}+\frac {d e^{\frac {e}{c+d x}} x \left (24 a^3 d^3+36 a^2 b d^2 (e+d x)+12 a b^2 d \left (-4 c e+e^2+d e x+2 d^2 x^2\right )+b^3 \left (18 c^2 e+e^3+d e^2 x+2 d^2 e x^2+6 d^3 x^3-2 c e (5 e+3 d x)\right )\right )-e \left (24 a^3 d^3+36 a^2 b d^2 (-2 c+e)+12 a b^2 d \left (6 c^2-6 c e+e^2\right )+b^3 \left (-24 c^3+36 c^2 e-12 c e^2+e^3\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{24 d^4} \]

[In]

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

[Out]

-1/24*(c*(-24*a^3*d^3 + 36*a^2*b*d^2*(c - e) - 12*a*b^2*d*(2*c^2 - 5*c*e + e^2) + b^3*(6*c^3 - 26*c^2*e + 11*c
*e^2 - e^3))*E^(e/(c + d*x)))/d^4 + (d*E^(e/(c + d*x))*x*(24*a^3*d^3 + 36*a^2*b*d^2*(e + d*x) + 12*a*b^2*d*(-4
*c*e + e^2 + d*e*x + 2*d^2*x^2) + b^3*(18*c^2*e + e^3 + d*e^2*x + 2*d^2*e*x^2 + 6*d^3*x^3 - 2*c*e*(5*e + 3*d*x
))) - e*(24*a^3*d^3 + 36*a^2*b*d^2*(-2*c + e) + 12*a*b^2*d*(6*c^2 - 6*c*e + e^2) + b^3*(-24*c^3 + 36*c^2*e - 1
2*c*e^2 + e^3))*ExpIntegralEi[e/(c + d*x)])/(24*d^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(681\) vs. \(2(306)=612\).

Time = 0.36 (sec) , antiderivative size = 682, normalized size of antiderivative = 2.13

method	result	size
derivativedivides	\(-\frac {e \left (a^{3} \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^{3} e^{3} \left (-\frac {\left (d x +c \right )^{4} {\mathrm e}^{\frac {e}{d x +c}}}{4 e^{4}}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{12 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{24 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{24 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{24}\right )}{d^{3}}-\frac {b^{3} c^{3} \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^{3}}+\frac {3 b^{2} e^{2} a \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 {3 b^{3} e^{2} c \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^{3}}+\frac {3 b e \,a^{2} \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 {3 b^{3} e \,c^{2} \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^{3}}-\frac {3 b c \,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 )}{d}+\frac {3 b^{2} c^{2} 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^{2}}-\frac {6 b^{2} e c 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^{2}}\right )}{d}\)	\(682\)
default	\(-\frac {e \left (a^{3} \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^{3} e^{3} \left (-\frac {\left (d x +c \right )^{4} {\mathrm e}^{\frac {e}{d x +c}}}{4 e^{4}}-\frac {\left (d x +c \right )^{3} {\mathrm e}^{\frac {e}{d x +c}}}{12 e^{3}}-\frac {{\mathrm e}^{\frac {e}{d x +c}} \left (d x +c \right )^{2}}{24 e^{2}}-\frac {\left (d x +c \right ) {\mathrm e}^{\frac {e}{d x +c}}}{24 e}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right )}{24}\right )}{d^{3}}-\frac {b^{3} c^{3} \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^{3}}+\frac {3 b^{2} e^{2} a \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 {3 b^{3} e^{2} c \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^{3}}+\frac {3 b e \,a^{2} \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 {3 b^{3} e \,c^{2} \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^{3}}-\frac {3 b c \,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 )}{d}+\frac {3 b^{2} c^{2} 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^{2}}-\frac {6 b^{2} e c 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^{2}}\right )}{d}\)	\(682\)
risch	\(\frac {{\mathrm e}^{\frac {e}{d x +c}} a^{3} c}{d}-\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} c^{4}}{4 d^{4}}+\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a^{3} e}{d}+\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{3} e^{4}}{24 d^{4}}+{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} x^{3}+\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} a^{2} b \,x^{2}}{2}-\frac {2 \,{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} c e x}{d^{2}}+\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a \,b^{2} e^{3}}{2 d^{3}}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{3} c^{3} e}{d^{4}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{3} c^{2} e^{2}}{2 d^{4}}-\frac {\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) b^{3} c \,e^{3}}{2 d^{4}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} e \,x^{3}}{12 d}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} e^{2} x^{2}}{24 d^{2}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} e^{3} x}{24 d^{3}}-\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} a^{2} b \,c^{2}}{2 d^{2}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} c^{3}}{d^{3}}+\frac {13 \,{\mathrm e}^{\frac {e}{d x +c}} b^{3} c^{3} e}{12 d^{4}}-\frac {11 \,{\mathrm e}^{\frac {e}{d x +c}} b^{3} c^{2} e^{2}}{24 d^{4}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} c \,e^{3}}{24 d^{4}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a^{2} b \,e^{2}}{2 d^{2}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} x^{4}}{4}+{\mathrm e}^{\frac {e}{d x +c}} a^{3} x -\frac {5 \,{\mathrm e}^{\frac {e}{d x +c}} b^{3} c \,e^{2} x}{12 d^{3}}+\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} a^{2} b c e}{2 d^{2}}-\frac {5 \,{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} c^{2} e}{2 d^{3}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} c \,e^{2}}{2 d^{3}}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a^{2} b c e}{d^{2}}+\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a \,b^{2} c^{2} e}{d^{3}}-\frac {3 \,\operatorname {Ei}_{1}\left (-\frac {e}{d x +c}\right ) a \,b^{2} c \,e^{2}}{d^{3}}+\frac {{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} e \,x^{2}}{2 d}-\frac {{\mathrm e}^{\frac {e}{d x +c}} b^{3} c e \,x^{2}}{4 d^{2}}+\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} a^{2} b e x}{2 d}+\frac {{\mathrm e}^{\frac {e}{d x +c}} a \,b^{2} e^{2} x}{2 d^{2}}+\frac {3 \,{\mathrm e}^{\frac {e}{d x +c}} b^{3} c^{2} e x}{4 d^{3}}\)	\(753\)
parts	\(\text {Expression too large to display}\)	\(1129\)

[In]

int(exp(e/(d*x+c))*(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/d*e*(a^3*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+b^3/d^3*e^3*(-1/4*(d*x+c)^4/e^4*exp(e/(d*x+c))-1/12*(
d*x+c)^3/e^3*exp(e/(d*x+c))-1/24*exp(e/(d*x+c))*(d*x+c)^2/e^2-1/24*(d*x+c)/e*exp(e/(d*x+c))-1/24*Ei(1,-e/(d*x+
c)))-b^3/d^3*c^3*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+3*b^2/d^2*e^2*a*(-1/3*(d*x+c)^3/e^3*exp(e/(d*x+c
))-1/6*exp(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)))-3*b^3/d^3*e^2*c*(-1/3*(
d*x+c)^3/e^3*exp(e/(d*x+c))-1/6*exp(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))
)+3*b/d*e*a^2*(-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)))+3*b^3/d^3*
e*c^2*(-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)))-3*b/d*c*a^2*(-(d*x
+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+3*b^2/d^2*c^2*a*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))-6*b^2/d^2*
e*c*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))))

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.18 \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=-\frac {{\left (b^{3} e^{4} - 12 \, {\left (b^{3} c - a b^{2} d\right )} e^{3} + 36 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} e^{2} - 24 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (6 \, b^{3} d^{4} x^{4} - 6 \, b^{3} c^{4} + 24 \, a b^{2} c^{3} d - 36 \, a^{2} b c^{2} d^{2} + 24 \, a^{3} c d^{3} + b^{3} c e^{3} + 2 \, {\left (12 \, a b^{2} d^{4} + b^{3} d^{3} e\right )} x^{3} - {\left (11 \, b^{3} c^{2} - 12 \, a b^{2} c d\right )} e^{2} + {\left (36 \, a^{2} b d^{4} + b^{3} d^{2} e^{2} - 6 \, {\left (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \, {\left (13 \, b^{3} c^{3} - 30 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2}\right )} e + {\left (24 \, a^{3} d^{4} + b^{3} d e^{3} - 2 \, {\left (5 \, b^{3} c d - 6 \, a b^{2} d^{2}\right )} e^{2} + 6 \, {\left (3 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + 6 \, a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{24 \, d^{4}} \]

[In]

integrate(exp(e/(d*x+c))*(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/24*((b^3*e^4 - 12*(b^3*c - a*b^2*d)*e^3 + 36*(b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*e^2 - 24*(b^3*c^3 - 3*a*b^
2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e)*Ei(e/(d*x + c)) - (6*b^3*d^4*x^4 - 6*b^3*c^4 + 24*a*b^2*c^3*d - 36*a^2*b
*c^2*d^2 + 24*a^3*c*d^3 + b^3*c*e^3 + 2*(12*a*b^2*d^4 + b^3*d^3*e)*x^3 - (11*b^3*c^2 - 12*a*b^2*c*d)*e^2 + (36
*a^2*b*d^4 + b^3*d^2*e^2 - 6*(b^3*c*d^2 - 2*a*b^2*d^3)*e)*x^2 + 2*(13*b^3*c^3 - 30*a*b^2*c^2*d + 18*a^2*b*c*d^
2)*e + (24*a^3*d^4 + b^3*d*e^3 - 2*(5*b^3*c*d - 6*a*b^2*d^2)*e^2 + 6*(3*b^3*c^2*d - 8*a*b^2*c*d^2 + 6*a^2*b*d^
3)*e)*x)*e^(e/(d*x + c)))/d^4

Sympy [F]

\[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\int \left (a + b x\right )^{3} e^{\frac {e}{c + d x}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\int { {\left (b x + a\right )}^{3} e^{\left (\frac {e}{d x + c}\right )} \,d x } \]

[In]

integrate(exp(e/(d*x+c))*(b*x+a)^3,x, algorithm="maxima")

[Out]

1/24*(6*b^3*d^3*x^4 + 2*(12*a*b^2*d^3 + b^3*d^2*e)*x^3 + (36*a^2*b*d^3 + 12*a*b^2*d^2*e - (6*c*d*e - d*e^2)*b^
3)*x^2 + (24*a^3*d^3 + 36*a^2*b*d^2*e - 12*(4*c*d*e - d*e^2)*a*b^2 + (18*c^2*e - 10*c*e^2 + e^3)*b^3)*x)*e^(e/
(d*x + c))/d^3 + integrate(-1/24*(36*a^2*b*c^2*d^2*e - 12*(4*c^3*d*e - c^2*d*e^2)*a*b^2 + (18*c^4*e - 10*c^3*e
^2 + c^2*e^3)*b^3 - (24*a^3*d^4*e - 36*(2*c*d^3*e - d^3*e^2)*a^2*b + 12*(6*c^2*d^2*e - 6*c*d^2*e^2 + d^2*e^3)*
a*b^2 - (24*c^3*d*e - 36*c^2*d*e^2 + 12*c*d*e^3 - d*e^4)*b^3)*x)*e^(e/(d*x + c))/(d^5*x^2 + 2*c*d^4*x + c^2*d^
3), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 831 vs. \(2 (306) = 612\).

Time = 0.33 (sec) , antiderivative size = 831, normalized size of antiderivative = 2.60 \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\frac {{\left (\frac {24 \, b^{3} c^{3} e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {72 \, a b^{2} c^{2} d e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} + \frac {72 \, a^{2} b c d^{2} e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {24 \, a^{3} d^{3} e^{6} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {36 \, b^{3} c^{2} e^{7} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} + \frac {72 \, a b^{2} c d e^{7} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {36 \, a^{2} b d^{2} e^{7} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} + \frac {12 \, b^{3} c e^{8} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {12 \, a b^{2} d e^{8} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} - \frac {b^{3} e^{9} {\rm Ei}\left (\frac {e}{d x + c}\right )}{{\left (d x + c\right )}^{4}} + 6 \, b^{3} e^{5} e^{\left (\frac {e}{d x + c}\right )} - \frac {24 \, b^{3} c e^{5} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} + \frac {36 \, b^{3} c^{2} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {24 \, b^{3} c^{3} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {24 \, a b^{2} d e^{5} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} - \frac {72 \, a b^{2} c d e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} + \frac {72 \, a b^{2} c^{2} d e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {36 \, a^{2} b d^{2} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {72 \, a^{2} b c d^{2} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {24 \, a^{3} d^{3} e^{5} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {2 \, b^{3} e^{6} e^{\left (\frac {e}{d x + c}\right )}}{d x + c} - \frac {12 \, b^{3} c e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} + \frac {36 \, b^{3} c^{2} e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {12 \, a b^{2} d e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {72 \, a b^{2} c d e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {36 \, a^{2} b d^{2} e^{6} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {b^{3} e^{7} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{2}} - \frac {12 \, b^{3} c e^{7} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {12 \, a b^{2} d e^{7} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}} + \frac {b^{3} e^{8} e^{\left (\frac {e}{d x + c}\right )}}{{\left (d x + c\right )}^{3}}\right )} {\left (d x + c\right )}^{4}}{24 \, d^{4} e^{5}} \]

[In]

integrate(exp(e/(d*x+c))*(b*x+a)^3,x, algorithm="giac")

[Out]

1/24*(24*b^3*c^3*e^6*Ei(e/(d*x + c))/(d*x + c)^4 - 72*a*b^2*c^2*d*e^6*Ei(e/(d*x + c))/(d*x + c)^4 + 72*a^2*b*c
*d^2*e^6*Ei(e/(d*x + c))/(d*x + c)^4 - 24*a^3*d^3*e^6*Ei(e/(d*x + c))/(d*x + c)^4 - 36*b^3*c^2*e^7*Ei(e/(d*x +
c))/(d*x + c)^4 + 72*a*b^2*c*d*e^7*Ei(e/(d*x + c))/(d*x + c)^4 - 36*a^2*b*d^2*e^7*Ei(e/(d*x + c))/(d*x + c)^4
+ 12*b^3*c*e^8*Ei(e/(d*x + c))/(d*x + c)^4 - 12*a*b^2*d*e^8*Ei(e/(d*x + c))/(d*x + c)^4 - b^3*e^9*Ei(e/(d*x +
c))/(d*x + c)^4 + 6*b^3*e^5*e^(e/(d*x + c)) - 24*b^3*c*e^5*e^(e/(d*x + c))/(d*x + c) + 36*b^3*c^2*e^5*e^(e/(d
*x + c))/(d*x + c)^2 - 24*b^3*c^3*e^5*e^(e/(d*x + c))/(d*x + c)^3 + 24*a*b^2*d*e^5*e^(e/(d*x + c))/(d*x + c) -
72*a*b^2*c*d*e^5*e^(e/(d*x + c))/(d*x + c)^2 + 72*a*b^2*c^2*d*e^5*e^(e/(d*x + c))/(d*x + c)^3 + 36*a^2*b*d^2*
e^5*e^(e/(d*x + c))/(d*x + c)^2 - 72*a^2*b*c*d^2*e^5*e^(e/(d*x + c))/(d*x + c)^3 + 24*a^3*d^3*e^5*e^(e/(d*x +
c))/(d*x + c)^3 + 2*b^3*e^6*e^(e/(d*x + c))/(d*x + c) - 12*b^3*c*e^6*e^(e/(d*x + c))/(d*x + c)^2 + 36*b^3*c^2*
e^6*e^(e/(d*x + c))/(d*x + c)^3 + 12*a*b^2*d*e^6*e^(e/(d*x + c))/(d*x + c)^2 - 72*a*b^2*c*d*e^6*e^(e/(d*x + c)
)/(d*x + c)^3 + 36*a^2*b*d^2*e^6*e^(e/(d*x + c))/(d*x + c)^3 + b^3*e^7*e^(e/(d*x + c))/(d*x + c)^2 - 12*b^3*c*
e^7*e^(e/(d*x + c))/(d*x + c)^3 + 12*a*b^2*d*e^7*e^(e/(d*x + c))/(d*x + c)^3 + b^3*e^8*e^(e/(d*x + c))/(d*x +
c)^3)*(d*x + c)^4/(d^4*e^5)

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {e}{c+d x}} (a+b x)^3 \, dx=\int {\mathrm {e}}^{\frac {e}{c+d\,x}}\,{\left (a+b\,x\right )}^3 \,d x \]

[In]

int(exp(e/(c + d*x))*(a + b*x)^3,x)

[Out]

int(exp(e/(c + d*x))*(a + b*x)^3, x)