### 3.988 $$\int (d+e x) (c d^2+2 c d e x+c e^2 x^2)^2 \, dx$$

Optimal. Leaf size=17 $\frac{c^2 (d+e x)^6}{6 e}$

[Out]

(c^2*(d + e*x)^6)/(6*e)

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Rubi [A]  time = 0.0043832, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.107, Rules used = {27, 12, 32} $\frac{c^2 (d+e x)^6}{6 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^2*(d + e*x)^6)/(6*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 (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx &=\int c^2 (d+e x)^5 \, dx\\ &=c^2 \int (d+e x)^5 \, dx\\ &=\frac{c^2 (d+e x)^6}{6 e}\\ \end{align*}

Mathematica [A]  time = 0.0020731, size = 17, normalized size = 1. $\frac{c^2 (d+e x)^6}{6 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^2*(d + e*x)^6)/(6*e)

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Maple [B]  time = 0.038, size = 72, normalized size = 4.2 \begin{align*}{\frac{{c}^{2}{e}^{5}{x}^{6}}{6}}+d{c}^{2}{e}^{4}{x}^{5}+{\frac{5\,{d}^{2}{c}^{2}{e}^{3}{x}^{4}}{2}}+{\frac{10\,{d}^{3}{c}^{2}{e}^{2}{x}^{3}}{3}}+{\frac{5\,{c}^{2}e{d}^{4}{x}^{2}}{2}}+{c}^{2}{d}^{5}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*e^5*x^6+d*c^2*e^4*x^5+5/2*d^2*c^2*e^3*x^4+10/3*d^3*c^2*e^2*x^3+5/2*c^2*e*d^4*x^2+c^2*d^5*x

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Maxima [B]  time = 1.22369, size = 41, normalized size = 2.41 \begin{align*} \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{3}}{6 \, c e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^3/(c*e)

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Fricas [B]  time = 1.72845, size = 149, normalized size = 8.76 \begin{align*} \frac{1}{6} x^{6} e^{5} c^{2} + x^{5} e^{4} d c^{2} + \frac{5}{2} x^{4} e^{3} d^{2} c^{2} + \frac{10}{3} x^{3} e^{2} d^{3} c^{2} + \frac{5}{2} x^{2} e d^{4} c^{2} + x d^{5} c^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^6*e^5*c^2 + x^5*e^4*d*c^2 + 5/2*x^4*e^3*d^2*c^2 + 10/3*x^3*e^2*d^3*c^2 + 5/2*x^2*e*d^4*c^2 + x*d^5*c^2

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Sympy [B]  time = 0.084157, size = 80, normalized size = 4.71 \begin{align*} c^{2} d^{5} x + \frac{5 c^{2} d^{4} e x^{2}}{2} + \frac{10 c^{2} d^{3} e^{2} x^{3}}{3} + \frac{5 c^{2} d^{2} e^{3} x^{4}}{2} + c^{2} d e^{4} x^{5} + \frac{c^{2} e^{5} x^{6}}{6} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*d**5*x + 5*c**2*d**4*e*x**2/2 + 10*c**2*d**3*e**2*x**3/3 + 5*c**2*d**2*e**3*x**4/2 + c**2*d*e**4*x**5 + c
**2*e**5*x**6/6

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Giac [B]  time = 1.16845, size = 92, normalized size = 5.41 \begin{align*} \frac{1}{6} \, c^{2} x^{6} e^{5} + c^{2} d x^{5} e^{4} + \frac{5}{2} \, c^{2} d^{2} x^{4} e^{3} + \frac{10}{3} \, c^{2} d^{3} x^{3} e^{2} + \frac{5}{2} \, c^{2} d^{4} x^{2} e + c^{2} d^{5} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*c^2*x^6*e^5 + c^2*d*x^5*e^4 + 5/2*c^2*d^2*x^4*e^3 + 10/3*c^2*d^3*x^3*e^2 + 5/2*c^2*d^4*x^2*e + c^2*d^5*x