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

Optimal. Leaf size=46 \[ \frac {20 x^7}{7}-\frac {4 x^6}{3}+\frac {61 x^5}{5}+\frac {x^4}{4}+\frac {53 x^3}{3}+\frac {15 x^2}{2}+18 x \]

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Rubi [A]  time = 0.03, antiderivative size = 46, 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 {20 x^7}{7}-\frac {4 x^6}{3}+\frac {61 x^5}{5}+\frac {x^4}{4}+\frac {53 x^3}{3}+\frac {15 x^2}{2}+18 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

18*x + (15*x^2)/2 + (53*x^3)/3 + x^4/4 + (61*x^5)/5 - (4*x^6)/3 + (20*x^7)/7

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 )^2 \left (2+3 x+5 x^2\right ) \, dx &=\int \left (18+15 x+53 x^2+x^3+61 x^4-8 x^5+20 x^6\right ) \, dx\\ &=18 x+\frac {15 x^2}{2}+\frac {53 x^3}{3}+\frac {x^4}{4}+\frac {61 x^5}{5}-\frac {4 x^6}{3}+\frac {20 x^7}{7}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 46, normalized size = 1.00 \begin {gather*} \frac {20 x^7}{7}-\frac {4 x^6}{3}+\frac {61 x^5}{5}+\frac {x^4}{4}+\frac {53 x^3}{3}+\frac {15 x^2}{2}+18 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

18*x + (15*x^2)/2 + (53*x^3)/3 + x^4/4 + (61*x^5)/5 - (4*x^6)/3 + (20*x^7)/7

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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 )^2 \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)^2*(2 + 3*x + 5*x^2),x]

[Out]

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

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fricas [A]  time = 0.34, size = 34, normalized size = 0.74 \begin {gather*} \frac {20}{7} x^{7} - \frac {4}{3} x^{6} + \frac {61}{5} x^{5} + \frac {1}{4} x^{4} + \frac {53}{3} x^{3} + \frac {15}{2} x^{2} + 18 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/7*x^7 - 4/3*x^6 + 61/5*x^5 + 1/4*x^4 + 53/3*x^3 + 15/2*x^2 + 18*x

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giac [A]  time = 0.19, size = 34, normalized size = 0.74 \begin {gather*} \frac {20}{7} \, x^{7} - \frac {4}{3} \, x^{6} + \frac {61}{5} \, x^{5} + \frac {1}{4} \, x^{4} + \frac {53}{3} \, x^{3} + \frac {15}{2} \, x^{2} + 18 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/7*x^7 - 4/3*x^6 + 61/5*x^5 + 1/4*x^4 + 53/3*x^3 + 15/2*x^2 + 18*x

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maple [A]  time = 0.00, size = 35, normalized size = 0.76 \begin {gather*} \frac {20}{7} x^{7}-\frac {4}{3} x^{6}+\frac {61}{5} x^{5}+\frac {1}{4} x^{4}+\frac {53}{3} x^{3}+\frac {15}{2} x^{2}+18 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18*x+15/2*x^2+53/3*x^3+1/4*x^4+61/5*x^5-4/3*x^6+20/7*x^7

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maxima [A]  time = 0.44, size = 34, normalized size = 0.74 \begin {gather*} \frac {20}{7} \, x^{7} - \frac {4}{3} \, x^{6} + \frac {61}{5} \, x^{5} + \frac {1}{4} \, x^{4} + \frac {53}{3} \, x^{3} + \frac {15}{2} \, x^{2} + 18 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20/7*x^7 - 4/3*x^6 + 61/5*x^5 + 1/4*x^4 + 53/3*x^3 + 15/2*x^2 + 18*x

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mupad [B]  time = 0.03, size = 34, normalized size = 0.74 \begin {gather*} \frac {20\,x^7}{7}-\frac {4\,x^6}{3}+\frac {61\,x^5}{5}+\frac {x^4}{4}+\frac {53\,x^3}{3}+\frac {15\,x^2}{2}+18\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

18*x + (15*x^2)/2 + (53*x^3)/3 + x^4/4 + (61*x^5)/5 - (4*x^6)/3 + (20*x^7)/7

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sympy [A]  time = 0.07, size = 41, normalized size = 0.89 \begin {gather*} \frac {20 x^{7}}{7} - \frac {4 x^{6}}{3} + \frac {61 x^{5}}{5} + \frac {x^{4}}{4} + \frac {53 x^{3}}{3} + \frac {15 x^{2}}{2} + 18 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

20*x**7/7 - 4*x**6/3 + 61*x**5/5 + x**4/4 + 53*x**3/3 + 15*x**2/2 + 18*x

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