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

Optimal. Leaf size=126 \[ \frac {2 b (c+d x)^2 (b c-a d) \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^3 \left (-e (c+d x)^3\right )^{2/3}}-\frac {(c+d x) (b c-a d)^2 \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^3 \sqrt [3]{-e (c+d x)^3}}+\frac {b^2 e^{e (c+d x)^3}}{3 d^3 e} \]

[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)

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Rubi [A]  time = 0.11, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2226, 2208, 2218, 2209} \[ \frac {2 b (c+d x)^2 (b c-a d) \text {Gamma}\left (\frac {2}{3},-e (c+d x)^3\right )}{3 d^3 \left (-e (c+d x)^3\right )^{2/3}}-\frac {(c+d x) (b c-a d)^2 \text {Gamma}\left (\frac {1}{3},-e (c+d x)^3\right )}{3 d^3 \sqrt [3]{-e (c+d x)^3}}+\frac {b^2 e^{e (c+d x)^3}}{3 d^3 e} \]

Antiderivative was successfully verified.

[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 2208

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 2209

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 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]

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]

Rubi steps

\begin {align*} \int e^{e (c+d x)^3} (a+b x)^2 \, dx &=\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*}

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Mathematica [A]  time = 0.08, size = 117, normalized size = 0.93 \[ \frac {\frac {2 b (c+d x)^2 (b c-a d) \Gamma \left (\frac {2}{3},-e (c+d x)^3\right )}{\left (-e (c+d x)^3\right )^{2/3}}-\frac {(c+d x) (b c-a d)^2 \Gamma \left (\frac {1}{3},-e (c+d x)^3\right )}{\sqrt [3]{-e (c+d x)^3}}+\frac {b^2 e^{e (c+d x)^3}}{e}}{3 d^3} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.44, size = 175, normalized size = 1.39 \[ \frac {b^{2} d^{2} e^{\left (d^{3} e x^{3} + 3 \, c d^{2} e x^{2} + 3 \, c^{2} d e x + c^{3} e\right )} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-d^{3} e\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right ) - 2 \, {\left (b^{2} c d - a b d^{2}\right )} \left (-d^{3} e\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -d^{3} e x^{3} - 3 \, c d^{2} e x^{2} - 3 \, c^{2} d e x - c^{3} e\right )}{3 \, d^{5} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{2} e^{\left ({\left (d x + c\right )}^{3} e\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{2} {\mathrm e}^{\left (d x +c \right )^{3} e}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{2} e^{\left ({\left (d x + c\right )}^{3} e\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{e\,{\left (c+d\,x\right )}^3}\,{\left (a+b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \left (\int a^{2} e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx + \int b^{2} x^{2} e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx + \int 2 a b x e^{d^{3} e x^{3}} e^{3 c d^{2} e x^{2}} e^{3 c^{2} d e x}\, dx\right ) e^{c^{3} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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