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

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

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Rubi [A]  time = 0.03, antiderivative size = 56, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 56, normalized size = 1.00 \begin {gather*} \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 {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (3-x+2 x^2\right )^3 \left (2+3 x+5 x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.33, size = 44, normalized size = 0.79 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.20, size = 44, normalized size = 0.79 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.00, size = 45, normalized size = 0.80 \begin {gather*} \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 {gather*}

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 = 0.43, size = 44, normalized size = 0.79 \begin {gather*} \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 {gather*}

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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mupad [B]  time = 0.03, size = 44, normalized size = 0.79 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 - x + 3)^3*(3*x + 5*x^2 + 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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sympy [A]  time = 0.08, size = 53, normalized size = 0.95 \begin {gather*} \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 {gather*}

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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