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

Optimal. Leaf size=255 \[ \frac{b e^2 (b c-a d) \text{Ei}\left (\frac{e}{c+d x}\right )}{d^3}-\frac{e (b c-a d)^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^3}-\frac{b e (c+d x) (b c-a d) e^{\frac{e}{c+d x}}}{d^3}-\frac{b (c+d x)^2 (b c-a d) e^{\frac{e}{c+d x}}}{d^3}+\frac{(c+d x) (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^3}-\frac{b^2 e^3 \text{Ei}\left (\frac{e}{c+d x}\right )}{6 d^3}+\frac{b^2 e^2 (c+d x) e^{\frac{e}{c+d x}}}{6 d^3}+\frac{b^2 e (c+d x)^2 e^{\frac{e}{c+d x}}}{6 d^3}+\frac{b^2 (c+d x)^3 e^{\frac{e}{c+d x}}}{3 d^3} \]

[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)

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Rubi [A]  time = 0.256185, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2226, 2206, 2210, 2214} \[ \frac{b e^2 (b c-a d) \text{Ei}\left (\frac{e}{c+d x}\right )}{d^3}-\frac{e (b c-a d)^2 \text{Ei}\left (\frac{e}{c+d x}\right )}{d^3}-\frac{b e (c+d x) (b c-a d) e^{\frac{e}{c+d x}}}{d^3}-\frac{b (c+d x)^2 (b c-a d) e^{\frac{e}{c+d x}}}{d^3}+\frac{(c+d x) (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^3}-\frac{b^2 e^3 \text{Ei}\left (\frac{e}{c+d x}\right )}{6 d^3}+\frac{b^2 e^2 (c+d x) e^{\frac{e}{c+d x}}}{6 d^3}+\frac{b^2 e (c+d x)^2 e^{\frac{e}{c+d x}}}{6 d^3}+\frac{b^2 (c+d x)^3 e^{\frac{e}{c+d x}}}{3 d^3} \]

Antiderivative was successfully verified.

[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 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]))

Rubi steps

\begin{align*} \int e^{\frac{e}{c+d x}} (a+b x)^2 \, dx &=\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]  time = 0.199511, size = 170, normalized size = 0.67 \[ \frac{d x e^{\frac{e}{c+d x}} \left (6 a^2 d^2+6 a b d (d x+e)+b^2 \left (-4 c e+2 d^2 x^2+d e x+e^2\right )\right )-e \left (6 a^2 d^2+6 a b d (e-2 c)+b^2 \left (6 c^2-6 c e+e^2\right )\right ) \text{Ei}\left (\frac{e}{c+d x}\right )}{6 d^3}+\frac{c e^{\frac{e}{c+d x}} \left (6 a^2 d^2+6 a b d (e-c)+b^2 \left (2 c^2-5 c e+e^2\right )\right )}{6 d^3} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.007, size = 356, normalized size = 1.4 \begin{align*} -{\frac{e}{d} \left ({a}^{2} \left ( -{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) +{\frac{{b}^{2}{e}^{2}}{{d}^{2}} \left ( -{\frac{ \left ( dx+c \right ) ^{3}}{3\,{e}^{3}}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\frac{ \left ( dx+c \right ) ^{2}}{6\,{e}^{2}}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\frac{dx+c}{6\,e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\frac{1}{6}{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) } \right ) }+{\frac{{c}^{2}{b}^{2}}{{d}^{2}} \left ( -{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) }+2\,{\frac{abe}{d} \left ( -1/2\,{\frac{ \left ( dx+c \right ) ^{2}}{{e}^{2}}{{\rm e}^{{\frac{e}{dx+c}}}}}-1/2\,{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-1/2\,{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) }-2\,{\frac{{b}^{2}ec}{{d}^{2}} \left ( -1/2\,{\frac{ \left ( dx+c \right ) ^{2}}{{e}^{2}}{{\rm e}^{{\frac{e}{dx+c}}}}}-1/2\,{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-1/2\,{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) }-2\,{\frac{abc}{d} \left ( -{\frac{dx+c}{e}{{\rm e}^{{\frac{e}{dx+c}}}}}-{\it Ei} \left ( 1,-{\frac{e}{dx+c}} \right ) \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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))))

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

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Fricas [A]  time = 1.54076, size = 408, normalized size = 1.6 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b x\right )^{2} 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)**2,x)

[Out]

Integral((a + b*x)**2*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 )}^{2} 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)^2,x, algorithm="giac")

[Out]

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