\(\int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx\) [417]

Optimal result

Integrand size = 19, antiderivative size = 151 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{3 d^3}-\frac {b^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )}{3 d^3}-\frac {2 b (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {(b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2258, 2239, 2250, 2245, 2241} \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=-\frac {2 b (c+d x)^2 (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {(c+d x) (b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}-\frac {b^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {b^2 (c+d x)^3 e^{\frac {e}{(c+d x)^3}}}{3 d^3} \]

[In]

Int[E^(e/(c + d*x)^3)*(a + b*x)^2,x]

[Out]

(b^2*E^(e/(c + d*x)^3)*(c + d*x)^3)/(3*d^3) - (b^2*e*ExpIntegralEi[e/(c + d*x)^3])/(3*d^3) - (2*b*(b*c - a*d)*
(-(e/(c + d*x)^3))^(2/3)*(c + d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)])/(3*d^3) + ((b*c - a*d)^2*(-(e/(c + d*x)^3)
)^(1/3)*(c + d*x)*Gamma[-1/3, -(e/(c + d*x)^3)])/(3*d^3)

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

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)^2 e^{\frac {e}{(c+d x)^3}}}{d^2}-\frac {2 b (b c-a d) e^{\frac {e}{(c+d x)^3}} (c+d x)}{d^2}+\frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^2}{d^2}\right ) \, dx \\ & = \frac {b^2 \int e^{\frac {e}{(c+d x)^3}} (c+d x)^2 \, dx}{d^2}-\frac {(2 b (b c-a d)) \int e^{\frac {e}{(c+d x)^3}} (c+d x) \, dx}{d^2}+\frac {(b c-a d)^2 \int e^{\frac {e}{(c+d x)^3}} \, dx}{d^2} \\ & = \frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{3 d^3}-\frac {2 b (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {(b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {\left (b^2 e\right ) \int \frac {e^{\frac {e}{(c+d x)^3}}}{c+d x} \, dx}{d^2} \\ & = \frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^3}{3 d^3}-\frac {b^2 e \text {Ei}\left (\frac {e}{(c+d x)^3}\right )}{3 d^3}-\frac {2 b (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3}+\frac {(b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.90 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\frac {b^2 e^{\frac {e}{(c+d x)^3}} (c+d x)^3-b^2 e \operatorname {ExpIntegralEi}\left (\frac {e}{(c+d x)^3}\right )-2 b (b c-a d) \left (-\frac {e}{(c+d x)^3}\right )^{2/3} (c+d x)^2 \Gamma \left (-\frac {2}{3},-\frac {e}{(c+d x)^3}\right )+(b c-a d)^2 \sqrt [3]{-\frac {e}{(c+d x)^3}} (c+d x) \Gamma \left (-\frac {1}{3},-\frac {e}{(c+d x)^3}\right )}{3 d^3} \]

[In]

Integrate[E^(e/(c + d*x)^3)*(a + b*x)^2,x]

[Out]

(b^2*E^(e/(c + d*x)^3)*(c + d*x)^3 - b^2*e*ExpIntegralEi[e/(c + d*x)^3] - 2*b*(b*c - a*d)*(-(e/(c + d*x)^3))^(
2/3)*(c + d*x)^2*Gamma[-2/3, -(e/(c + d*x)^3)] + (b*c - a*d)^2*(-(e/(c + d*x)^3))^(1/3)*(c + d*x)*Gamma[-1/3,
-(e/(c + d*x)^3)])/(3*d^3)

Maple [F]

\[\int {\mathrm e}^{\frac {e}{\left (d x +c \right )^{3}}} \left (b x +a \right )^{2}d x\]

[In]

int(exp(e/(d*x+c)^3)*(b*x+a)^2,x)

[Out]

int(exp(e/(d*x+c)^3)*(b*x+a)^2,x)

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.72 \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=-\frac {b^{2} e {\rm Ei}\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - 3 \, {\left (b^{2} c d^{2} - a b d^{3}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) + 3 \, {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \left (-\frac {e}{d^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) - {\left (b^{2} d^{3} x^{3} + 3 \, a b d^{3} x^{2} + 3 \, a^{2} d^{3} x + b^{2} c^{3} - 3 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} e^{\left (\frac {e}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right )}}{3 \, d^{3}} \]

[In]

integrate(exp(e/(d*x+c)^3)*(b*x+a)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\int \left (a + b x\right )^{2} e^{\frac {e}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}}\, dx \]

[In]

integrate(exp(e/(d*x+c)**3)*(b*x+a)**2,x)

[Out]

Integral((a + b*x)**2*exp(e/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3)), x)

Maxima [F]

\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\int { {\left (b x + a\right )}^{2} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \]

[In]

integrate(exp(e/(d*x+c)^3)*(b*x+a)^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\int { {\left (b x + a\right )}^{2} e^{\left (\frac {e}{{\left (d x + c\right )}^{3}}\right )} \,d x } \]

[In]

integrate(exp(e/(d*x+c)^3)*(b*x+a)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int e^{\frac {e}{(c+d x)^3}} (a+b x)^2 \, dx=\int {\mathrm {e}}^{\frac {e}{{\left (c+d\,x\right )}^3}}\,{\left (a+b\,x\right )}^2 \,d x \]

[In]

int(exp(e/(c + d*x)^3)*(a + b*x)^2,x)

[Out]

int(exp(e/(c + d*x)^3)*(a + b*x)^2, x)