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

Optimal. Leaf size=320 \[ \frac{b^3 e^4 \text{Gamma}\left (-4,-\frac{e}{c+d x}\right )}{d^4}+\frac{b^2 e^3 (b c-a d) \text{Ei}\left (\frac{e}{c+d x}\right )}{2 d^4}-\frac{b^2 e^2 (c+d x) (b c-a d) e^{\frac{e}{c+d x}}}{2 d^4}-\frac{b^2 e (c+d x)^2 (b c-a d) e^{\frac{e}{c+d x}}}{2 d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{c+d x}}}{d^4}-\frac{3 b e^2 (b c-a d)^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{2 d^4}+\frac{e (b c-a d)^3 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^4}+\frac{3 b e (c+d x) (b c-a d)^2 e^{\frac{e}{c+d x}}}{2 d^4}+\frac{3 b (c+d x)^2 (b c-a d)^2 e^{\frac{e}{c+d x}}}{2 d^4}-\frac{(c+d x) (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^4} \]

[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

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Rubi [A]  time = 0.31705, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {2226, 2206, 2210, 2214, 2218} \[ \frac{b^3 e^4 \text{Gamma}\left (-4,-\frac{e}{c+d x}\right )}{d^4}+\frac{b^2 e^3 (b c-a d) \text{Ei}\left (\frac{e}{c+d x}\right )}{2 d^4}-\frac{b^2 e^2 (c+d x) (b c-a d) e^{\frac{e}{c+d x}}}{2 d^4}-\frac{b^2 e (c+d x)^2 (b c-a d) e^{\frac{e}{c+d x}}}{2 d^4}-\frac{b^2 (c+d x)^3 (b c-a d) e^{\frac{e}{c+d x}}}{d^4}-\frac{3 b e^2 (b c-a d)^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{2 d^4}+\frac{e (b c-a d)^3 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^4}+\frac{3 b e (c+d x) (b c-a d)^2 e^{\frac{e}{c+d x}}}{2 d^4}+\frac{3 b (c+d x)^2 (b c-a d)^2 e^{\frac{e}{c+d x}}}{2 d^4}-\frac{(c+d x) (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^4} \]

Antiderivative was successfully verified.

[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 2226

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]

Rule 2206

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 2210

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 2214

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 2218

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

Rubi steps

\begin{align*} \int e^{\frac{e}{c+d x}} (a+b x)^3 \, dx &=\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]  time = 0.316403, size = 292, normalized size = 0.91 \[ \frac{d x e^{\frac{e}{c+d x}} \left (36 a^2 b d^2 (d x+e)+24 a^3 d^3+12 a b^2 d \left (-4 c e+2 d^2 x^2+d e x+e^2\right )+b^3 \left (18 c^2 e-2 c e (3 d x+5 e)+2 d^2 e x^2+6 d^3 x^3+d e^2 x+e^3\right )\right )-e \left (36 a^2 b d^2 (e-2 c)+24 a^3 d^3+12 a b^2 d \left (6 c^2-6 c e+e^2\right )+b^3 \left (36 c^2 e-24 c^3-12 c e^2+e^3\right )\right ) \text{Ei}\left (\frac{e}{c+d x}\right )}{24 d^4}-\frac{c e^{\frac{e}{c+d x}} \left (36 a^2 b d^2 (c-e)-24 a^3 d^3-12 a b^2 d \left (2 c^2-5 c e+e^2\right )+b^3 \left (-26 c^2 e+6 c^3+11 c e^2-e^3\right )\right )}{24 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)))/(24*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)

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Maple [B]  time = 0.01, size = 682, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (6 \, b^{3} d^{3} x^{4} + 2 \,{\left (12 \, a b^{2} d^{3} + b^{3} d^{2} e\right )} x^{3} +{\left (36 \, a^{2} b d^{3} + 12 \, a b^{2} d^{2} e -{\left (6 \, c d e - d e^{2}\right )} b^{3}\right )} x^{2} +{\left (24 \, a^{3} d^{3} + 36 \, a^{2} b d^{2} e - 12 \,{\left (4 \, c d e - d e^{2}\right )} a b^{2} +{\left (18 \, c^{2} e - 10 \, c e^{2} + e^{3}\right )} b^{3}\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{24 \, d^{3}} + \int -\frac{{\left (36 \, a^{2} b c^{2} d^{2} e - 12 \,{\left (4 \, c^{3} d e - c^{2} d e^{2}\right )} a b^{2} +{\left (18 \, c^{4} e - 10 \, c^{3} e^{2} + c^{2} e^{3}\right )} b^{3} -{\left (24 \, a^{3} d^{4} e - 36 \,{\left (2 \, c d^{3} e - d^{3} e^{2}\right )} a^{2} b + 12 \,{\left (6 \, c^{2} d^{2} e - 6 \, c d^{2} e^{2} + d^{2} e^{3}\right )} a b^{2} -{\left (24 \, c^{3} d e - 36 \, c^{2} d e^{2} + 12 \, c d e^{3} - d e^{4}\right )} b^{3}\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{24 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b x + a\right )}^{3} e^{\left (\frac{e}{d x + c}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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