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

Optimal. Leaf size=30 \[ 2 x^5+\frac {x^4}{4}+\frac {16 x^3}{3}+\frac {7 x^2}{2}+6 x \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1657} \begin {gather*} 2 x^5+\frac {x^4}{4}+\frac {16 x^3}{3}+\frac {7 x^2}{2}+6 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

6*x + (7*x^2)/2 + (16*x^3)/3 + x^4/4 + 2*x^5

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 ) \left (2+3 x+5 x^2\right ) \, dx &=\int \left (6+7 x+16 x^2+x^3+10 x^4\right ) \, dx\\ &=6 x+\frac {7 x^2}{2}+\frac {16 x^3}{3}+\frac {x^4}{4}+2 x^5\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} 2 x^5+\frac {x^4}{4}+\frac {16 x^3}{3}+\frac {7 x^2}{2}+6 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

6*x + (7*x^2)/2 + (16*x^3)/3 + x^4/4 + 2*x^5

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.34, size = 24, normalized size = 0.80 \begin {gather*} 2 x^{5} + \frac {1}{4} x^{4} + \frac {16}{3} x^{3} + \frac {7}{2} x^{2} + 6 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^5 + 1/4*x^4 + 16/3*x^3 + 7/2*x^2 + 6*x

________________________________________________________________________________________

giac [A]  time = 0.18, size = 24, normalized size = 0.80 \begin {gather*} 2 \, x^{5} + \frac {1}{4} \, x^{4} + \frac {16}{3} \, x^{3} + \frac {7}{2} \, x^{2} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^5 + 1/4*x^4 + 16/3*x^3 + 7/2*x^2 + 6*x

________________________________________________________________________________________

maple [A]  time = 0.00, size = 25, normalized size = 0.83 \begin {gather*} 2 x^{5}+\frac {1}{4} x^{4}+\frac {16}{3} x^{3}+\frac {7}{2} x^{2}+6 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*x+7/2*x^2+16/3*x^3+1/4*x^4+2*x^5

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 24, normalized size = 0.80 \begin {gather*} 2 \, x^{5} + \frac {1}{4} \, x^{4} + \frac {16}{3} \, x^{3} + \frac {7}{2} \, x^{2} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^5 + 1/4*x^4 + 16/3*x^3 + 7/2*x^2 + 6*x

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 24, normalized size = 0.80 \begin {gather*} 2\,x^5+\frac {x^4}{4}+\frac {16\,x^3}{3}+\frac {7\,x^2}{2}+6\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*x + (7*x^2)/2 + (16*x^3)/3 + x^4/4 + 2*x^5

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 26, normalized size = 0.87 \begin {gather*} 2 x^{5} + \frac {x^{4}}{4} + \frac {16 x^{3}}{3} + \frac {7 x^{2}}{2} + 6 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**5 + x**4/4 + 16*x**3/3 + 7*x**2/2 + 6*x

________________________________________________________________________________________