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

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

[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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Rubi [A]  time = 0.0296107, antiderivative size = 46, 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} \[ \frac{20 x^7}{7}-\frac{4 x^6}{3}+\frac{61 x^5}{5}+\frac{x^4}{4}+\frac{53 x^3}{3}+\frac{15 x^2}{2}+18 x \]

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*}

Mathematica [A]  time = 0.000904, size = 46, normalized size = 1. \[ \frac{20 x^7}{7}-\frac{4 x^6}{3}+\frac{61 x^5}{5}+\frac{x^4}{4}+\frac{53 x^3}{3}+\frac{15 x^2}{2}+18 x \]

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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Maple [A]  time = 0.045, size = 35, normalized size = 0.8 \begin{align*} 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*}

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 = 0.964242, size = 46, normalized size = 1. \begin{align*} \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{align*}

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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Fricas [A]  time = 0.82888, size = 95, normalized size = 2.07 \begin{align*} \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{align*}

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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Sympy [A]  time = 0.097347, size = 41, normalized size = 0.89 \begin{align*} \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{align*}

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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Giac [A]  time = 1.17085, size = 46, normalized size = 1. \begin{align*} \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{align*}

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