\(\int e^{e (c+d x)^3} (a+b x)^2 \, dx\) [392]

Optimal result

Integrand size = 19, antiderivative size = 126 \[ \int e^{e (c+d x)^3} (a+b x)^2 \, dx=\frac {b^2 e^{e (c+d x)^3}}{3 d^3 e}-\frac {(b c-a d)^2 (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^3 \sqrt [3]{-e (c+d x)^3}}+\frac {2 b (b c-a d) (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^3 \left (-e (c+d x)^3\right )^{2/3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2258, 2239, 2250, 2240} \[ \int e^{e (c+d x)^3} (a+b x)^2 \, dx=\frac {2 b (c+d x)^2 (b c-a d) \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^3 \left (-e (c+d x)^3\right )^{2/3}}-\frac {(c+d x) (b c-a d)^2 \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^3 \sqrt [3]{-e (c+d x)^3}}+\frac {b^2 e^{e (c+d x)^3}}{3 d^3 e} \]

[In]

Int[E^(e*(c + d*x)^3)*(a + b*x)^2,x]

[Out]

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

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 2240

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

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.93 \[ \int e^{e (c+d x)^3} (a+b x)^2 \, dx=\frac {\frac {b^2 e^{e (c+d x)^3}}{e}-\frac {(b c-a d)^2 (c+d x) \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{\sqrt [3]{-e (c+d x)^3}}+\frac {2 b (b c-a d) (c+d x)^2 \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{\left (-e (c+d x)^3\right )^{2/3}}}{3 d^3} \]

[In]

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

[Out]

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

Maple [F]

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

[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 = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.39 \[ \int e^{e (c+d x)^3} (a+b x)^2 \, dx=\frac {b^{2} d^{2} e^{\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + 3 \, c^{2} d e x + c^{3} e\right )} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-d^{3} e\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right ) - 2 \, {\left (b^{2} c d - a b d^{2}\right )} \left (-d^{3} e\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, d^{5} e} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)