3.33 \(\int (3-x+2 x^2)^3 (2+3 x+5 x^2) \, dx\)

Optimal. Leaf size=56 \[ \frac{40 x^9}{9}-\frac{9 x^8}{2}+\frac{190 x^7}{7}-\frac{83 x^6}{6}+\frac{288 x^5}{5}-5 x^4+60 x^3+\frac{27 x^2}{2}+54 x \]

[Out]

54*x + (27*x^2)/2 + 60*x^3 - 5*x^4 + (288*x^5)/5 - (83*x^6)/6 + (190*x^7)/7 - (9*x^8)/2 + (40*x^9)/9

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Rubi [A]  time = 0.0318655, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1657} \[ \frac{40 x^9}{9}-\frac{9 x^8}{2}+\frac{190 x^7}{7}-\frac{83 x^6}{6}+\frac{288 x^5}{5}-5 x^4+60 x^3+\frac{27 x^2}{2}+54 x \]

Antiderivative was successfully verified.

[In]

Int[(3 - x + 2*x^2)^3*(2 + 3*x + 5*x^2),x]

[Out]

54*x + (27*x^2)/2 + 60*x^3 - 5*x^4 + (288*x^5)/5 - (83*x^6)/6 + (190*x^7)/7 - (9*x^8)/2 + (40*x^9)/9

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

\begin{align*} \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right ) \, dx &=\int \left (54+27 x+180 x^2-20 x^3+288 x^4-83 x^5+190 x^6-36 x^7+40 x^8\right ) \, dx\\ &=54 x+\frac{27 x^2}{2}+60 x^3-5 x^4+\frac{288 x^5}{5}-\frac{83 x^6}{6}+\frac{190 x^7}{7}-\frac{9 x^8}{2}+\frac{40 x^9}{9}\\ \end{align*}

Mathematica [A]  time = 0.0012394, size = 56, normalized size = 1. \[ \frac{40 x^9}{9}-\frac{9 x^8}{2}+\frac{190 x^7}{7}-\frac{83 x^6}{6}+\frac{288 x^5}{5}-5 x^4+60 x^3+\frac{27 x^2}{2}+54 x \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - x + 2*x^2)^3*(2 + 3*x + 5*x^2),x]

[Out]

54*x + (27*x^2)/2 + 60*x^3 - 5*x^4 + (288*x^5)/5 - (83*x^6)/6 + (190*x^7)/7 - (9*x^8)/2 + (40*x^9)/9

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Maple [A]  time = 0.043, size = 45, normalized size = 0.8 \begin{align*} 54\,x+{\frac{27\,{x}^{2}}{2}}+60\,{x}^{3}-5\,{x}^{4}+{\frac{288\,{x}^{5}}{5}}-{\frac{83\,{x}^{6}}{6}}+{\frac{190\,{x}^{7}}{7}}-{\frac{9\,{x}^{8}}{2}}+{\frac{40\,{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-x+3)^3*(5*x^2+3*x+2),x)

[Out]

54*x+27/2*x^2+60*x^3-5*x^4+288/5*x^5-83/6*x^6+190/7*x^7-9/2*x^8+40/9*x^9

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Maxima [A]  time = 0.967082, size = 59, normalized size = 1.05 \begin{align*} \frac{40}{9} \, x^{9} - \frac{9}{2} \, x^{8} + \frac{190}{7} \, x^{7} - \frac{83}{6} \, x^{6} + \frac{288}{5} \, x^{5} - 5 \, x^{4} + 60 \, x^{3} + \frac{27}{2} \, x^{2} + 54 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^3*(5*x^2+3*x+2),x, algorithm="maxima")

[Out]

40/9*x^9 - 9/2*x^8 + 190/7*x^7 - 83/6*x^6 + 288/5*x^5 - 5*x^4 + 60*x^3 + 27/2*x^2 + 54*x

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Fricas [A]  time = 0.882358, size = 122, normalized size = 2.18 \begin{align*} \frac{40}{9} x^{9} - \frac{9}{2} x^{8} + \frac{190}{7} x^{7} - \frac{83}{6} x^{6} + \frac{288}{5} x^{5} - 5 x^{4} + 60 x^{3} + \frac{27}{2} x^{2} + 54 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^3*(5*x^2+3*x+2),x, algorithm="fricas")

[Out]

40/9*x^9 - 9/2*x^8 + 190/7*x^7 - 83/6*x^6 + 288/5*x^5 - 5*x^4 + 60*x^3 + 27/2*x^2 + 54*x

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Sympy [A]  time = 0.107796, size = 53, normalized size = 0.95 \begin{align*} \frac{40 x^{9}}{9} - \frac{9 x^{8}}{2} + \frac{190 x^{7}}{7} - \frac{83 x^{6}}{6} + \frac{288 x^{5}}{5} - 5 x^{4} + 60 x^{3} + \frac{27 x^{2}}{2} + 54 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-x+3)**3*(5*x**2+3*x+2),x)

[Out]

40*x**9/9 - 9*x**8/2 + 190*x**7/7 - 83*x**6/6 + 288*x**5/5 - 5*x**4 + 60*x**3 + 27*x**2/2 + 54*x

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Giac [A]  time = 1.20353, size = 59, normalized size = 1.05 \begin{align*} \frac{40}{9} \, x^{9} - \frac{9}{2} \, x^{8} + \frac{190}{7} \, x^{7} - \frac{83}{6} \, x^{6} + \frac{288}{5} \, x^{5} - 5 \, x^{4} + 60 \, x^{3} + \frac{27}{2} \, x^{2} + 54 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-x+3)^3*(5*x^2+3*x+2),x, algorithm="giac")

[Out]

40/9*x^9 - 9/2*x^8 + 190/7*x^7 - 83/6*x^6 + 288/5*x^5 - 5*x^4 + 60*x^3 + 27/2*x^2 + 54*x