### 3.987 $$\int (d+e x)^2 (c d^2+2 c d e x+c e^2 x^2)^2 \, dx$$

Optimal. Leaf size=17 $\frac{c^2 (d+e x)^7}{7 e}$

[Out]

(c^2*(d + e*x)^7)/(7*e)

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Rubi [A]  time = 0.0048802, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.1, Rules used = {27, 12, 32} $\frac{c^2 (d+e x)^7}{7 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^2*(d + e*x)^7)/(7*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)^2 \left (c d^2+2 c d e x+c e^2 x^2\right )^2 \, dx &=\int c^2 (d+e x)^6 \, dx\\ &=c^2 \int (d+e x)^6 \, dx\\ &=\frac{c^2 (d+e x)^7}{7 e}\\ \end{align*}

Mathematica [A]  time = 0.0026172, size = 17, normalized size = 1. $\frac{c^2 (d+e x)^7}{7 e}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(c^2*(d + e*x)^7)/(7*e)

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Maple [B]  time = 0.039, size = 86, normalized size = 5.1 \begin{align*}{\frac{{e}^{6}{c}^{2}{x}^{7}}{7}}+d{e}^{5}{c}^{2}{x}^{6}+3\,{d}^{2}{c}^{2}{e}^{4}{x}^{5}+5\,{d}^{3}{c}^{2}{e}^{3}{x}^{4}+5\,{d}^{4}{c}^{2}{e}^{2}{x}^{3}+3\,{d}^{5}{c}^{2}e{x}^{2}+{d}^{6}{c}^{2}x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*e^6*c^2*x^7+d*e^5*c^2*x^6+3*d^2*c^2*e^4*x^5+5*d^3*c^2*e^3*x^4+5*d^4*c^2*e^2*x^3+3*d^5*c^2*e*x^2+d^6*c^2*x

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Maxima [B]  time = 1.14724, size = 115, normalized size = 6.76 \begin{align*} \frac{1}{7} \, c^{2} e^{6} x^{7} + c^{2} d e^{5} x^{6} + 3 \, c^{2} d^{2} e^{4} x^{5} + 5 \, c^{2} d^{3} e^{3} x^{4} + 5 \, c^{2} d^{4} e^{2} x^{3} + 3 \, c^{2} d^{5} e x^{2} + c^{2} d^{6} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^2*e^6*x^7 + c^2*d*e^5*x^6 + 3*c^2*d^2*e^4*x^5 + 5*c^2*d^3*e^3*x^4 + 5*c^2*d^4*e^2*x^3 + 3*c^2*d^5*e*x^2
+ c^2*d^6*x

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Fricas [B]  time = 1.7823, size = 166, normalized size = 9.76 \begin{align*} \frac{1}{7} x^{7} e^{6} c^{2} + x^{6} e^{5} d c^{2} + 3 x^{5} e^{4} d^{2} c^{2} + 5 x^{4} e^{3} d^{3} c^{2} + 5 x^{3} e^{2} d^{4} c^{2} + 3 x^{2} e d^{5} c^{2} + x d^{6} c^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e^6*c^2 + x^6*e^5*d*c^2 + 3*x^5*e^4*d^2*c^2 + 5*x^4*e^3*d^3*c^2 + 5*x^3*e^2*d^4*c^2 + 3*x^2*e*d^5*c^2
+ x*d^6*c^2

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Sympy [B]  time = 0.089141, size = 90, normalized size = 5.29 \begin{align*} c^{2} d^{6} x + 3 c^{2} d^{5} e x^{2} + 5 c^{2} d^{4} e^{2} x^{3} + 5 c^{2} d^{3} e^{3} x^{4} + 3 c^{2} d^{2} e^{4} x^{5} + c^{2} d e^{5} x^{6} + \frac{c^{2} e^{6} x^{7}}{7} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*d**6*x + 3*c**2*d**5*e*x**2 + 5*c**2*d**4*e**2*x**3 + 5*c**2*d**3*e**3*x**4 + 3*c**2*d**2*e**4*x**5 + c**
2*d*e**5*x**6 + c**2*e**6*x**7/7

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Giac [B]  time = 1.21732, size = 109, normalized size = 6.41 \begin{align*} \frac{1}{7} \, c^{2} x^{7} e^{6} + c^{2} d x^{6} e^{5} + 3 \, c^{2} d^{2} x^{5} e^{4} + 5 \, c^{2} d^{3} x^{4} e^{3} + 5 \, c^{2} d^{4} x^{3} e^{2} + 3 \, c^{2} d^{5} x^{2} e + c^{2} d^{6} x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c^2*x^7*e^6 + c^2*d*x^6*e^5 + 3*c^2*d^2*x^5*e^4 + 5*c^2*d^3*x^4*e^3 + 5*c^2*d^4*x^3*e^2 + 3*c^2*d^5*x^2*e
+ c^2*d^6*x