### 3.989 $$\int (c d^2+2 c d e x+c e^2 x^2)^2 \, dx$$

Optimal. Leaf size=17 $\frac{c^2 (d+e x)^5}{5 e}$

[Out]

(c^2*(d + e*x)^5)/(5*e)

________________________________________________________________________________________

Rubi [A]  time = 0.003937, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.136, Rules used = {27, 12, 32} $\frac{c^2 (d+e x)^5}{5 e}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(c^2*(d + e*x)^5)/(5*e)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx &=\int c^2 (d+e x)^4 \, dx\\ &=c^2 \int (d+e x)^4 \, dx\\ &=\frac{c^2 (d+e x)^5}{5 e}\\ \end{align*}

Mathematica [A]  time = 0.001093, size = 17, normalized size = 1. $\frac{c^2 (d+e x)^5}{5 e}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^2,x]

[Out]

(c^2*(d + e*x)^5)/(5*e)

________________________________________________________________________________________

Maple [B]  time = 0.039, size = 58, normalized size = 3.4 \begin{align*}{\frac{{c}^{2}{e}^{4}{x}^{5}}{5}}+{c}^{2}d{e}^{3}{x}^{4}+2\,{c}^{2}{d}^{2}{e}^{2}{x}^{3}+2\,{c}^{2}{d}^{3}e{x}^{2}+{c}^{2}{d}^{4}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*e^2*x^2+2*c*d*e*x+c*d^2)^2,x)

[Out]

1/5*c^2*e^4*x^5+c^2*d*e^3*x^4+2*c^2*d^2*e^2*x^3+2*c^2*d^3*e*x^2+c^2*d^4*x

________________________________________________________________________________________

Maxima [B]  time = 1.22374, size = 92, normalized size = 5.41 \begin{align*} \frac{1}{5} \, c^{2} e^{4} x^{5} + c^{2} d e^{3} x^{4} + \frac{4}{3} \, c^{2} d^{2} e^{2} x^{3} + c^{2} d^{4} x + \frac{2}{3} \,{\left (c e^{2} x^{3} + 3 \, c d e x^{2}\right )} c d^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*e^4*x^5 + c^2*d*e^3*x^4 + 4/3*c^2*d^2*e^2*x^3 + c^2*d^4*x + 2/3*(c*e^2*x^3 + 3*c*d*e*x^2)*c*d^2

________________________________________________________________________________________

Fricas [B]  time = 1.85975, size = 112, normalized size = 6.59 \begin{align*} \frac{1}{5} x^{5} e^{4} c^{2} + x^{4} e^{3} d c^{2} + 2 x^{3} e^{2} d^{2} c^{2} + 2 x^{2} e d^{3} c^{2} + x d^{4} c^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*e^4*c^2 + x^4*e^3*d*c^2 + 2*x^3*e^2*d^2*c^2 + 2*x^2*e*d^3*c^2 + x*d^4*c^2

________________________________________________________________________________________

Sympy [B]  time = 0.074471, size = 60, normalized size = 3.53 \begin{align*} c^{2} d^{4} x + 2 c^{2} d^{3} e x^{2} + 2 c^{2} d^{2} e^{2} x^{3} + c^{2} d e^{3} x^{4} + \frac{c^{2} e^{4} x^{5}}{5} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**2,x)

[Out]

c**2*d**4*x + 2*c**2*d**3*e*x**2 + 2*c**2*d**2*e**2*x**3 + c**2*d*e**3*x**4 + c**2*e**4*x**5/5

________________________________________________________________________________________

Giac [B]  time = 1.19167, size = 74, normalized size = 4.35 \begin{align*} \frac{1}{5} \, c^{2} x^{5} e^{4} + c^{2} d x^{4} e^{3} + 2 \, c^{2} d^{2} x^{3} e^{2} + 2 \, c^{2} d^{3} x^{2} e + c^{2} d^{4} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c^2*x^5*e^4 + c^2*d*x^4*e^3 + 2*c^2*d^2*x^3*e^2 + 2*c^2*d^3*x^2*e + c^2*d^4*x