3.82.38 \(\int \frac {21+7 x^2-28 x (3+x^2)^2}{9+3 x^2} \, dx\)

Optimal. Leaf size=16 \[ \frac {7}{3} \left (2+x-\left (3+x^2\right )^2\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1586} \begin {gather*} -\frac {7 x^4}{3}-14 x^2+\frac {7 x}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(21 + 7*x^2 - 28*x*(3 + x^2)^2)/(9 + 3*x^2),x]

[Out]

(7*x)/3 - 14*x^2 - (7*x^4)/3

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {7}{3}-28 x-\frac {28 x^3}{3}\right ) \, dx\\ &=\frac {7 x}{3}-14 x^2-\frac {7 x^4}{3}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[(21 + 7*x^2 - 28*x*(3 + x^2)^2)/(9 + 3*x^2),x]

[Out]

(-7*(-x + 6*x^2 + x^4))/3

________________________________________________________________________________________

fricas [A]  time = 0.65, size = 14, normalized size = 0.88 \begin {gather*} -\frac {7}{3} \, x^{4} - 14 \, x^{2} + \frac {7}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-28*x*(x^2+3)^2+7*x^2+21)/(3*x^2+9),x, algorithm="fricas")

[Out]

-7/3*x^4 - 14*x^2 + 7/3*x

________________________________________________________________________________________

giac [A]  time = 0.14, size = 14, normalized size = 0.88 \begin {gather*} -\frac {7}{3} \, x^{4} - 14 \, x^{2} + \frac {7}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-28*x*(x^2+3)^2+7*x^2+21)/(3*x^2+9),x, algorithm="giac")

[Out]

-7/3*x^4 - 14*x^2 + 7/3*x

________________________________________________________________________________________

maple [A]  time = 0.32, size = 12, normalized size = 0.75

method	result	size
gosper	\(-\frac {7 x \left (x^{3}+6 x -1\right )}{3}\)	\(12\)
default	\(-\frac {7}{3} x^{4}-14 x^{2}+\frac {7}{3} x\)	\(15\)
norman	\(-\frac {7}{3} x^{4}-14 x^{2}+\frac {7}{3} x\)	\(15\)
risch	\(-\frac {7}{3} x^{4}-14 x^{2}+\frac {7}{3} x\)	\(15\)
meijerg	\(\frac {7 \sqrt {3}\, \arctan \left (\frac {x \sqrt {3}}{3}\right )}{3}+\frac {7 x^{2} \left (-x^{2}+6\right )}{3}-28 x^{2}+\frac {7 \sqrt {3}\, \left (\frac {2 x \sqrt {3}}{3}-2 \arctan \left (\frac {x \sqrt {3}}{3}\right )\right )}{6}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-28*x*(x^2+3)^2+7*x^2+21)/(3*x^2+9),x,method=_RETURNVERBOSE)

[Out]

-7/3*x*(x^3+6*x-1)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 14, normalized size = 0.88 \begin {gather*} -\frac {7}{3} \, x^{4} - 14 \, x^{2} + \frac {7}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-28*x*(x^2+3)^2+7*x^2+21)/(3*x^2+9),x, algorithm="maxima")

[Out]

-7/3*x^4 - 14*x^2 + 7/3*x

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 11, normalized size = 0.69 \begin {gather*} -\frac {7\,x\,\left (x^3+6\,x-1\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x^2 - 28*x*(x^2 + 3)^2 + 21)/(3*x^2 + 9),x)

[Out]

-(7*x*(6*x + x^3 - 1))/3

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 15, normalized size = 0.94 \begin {gather*} - \frac {7 x^{4}}{3} - 14 x^{2} + \frac {7 x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-28*x*(x**2+3)**2+7*x**2+21)/(3*x**2+9),x)

[Out]

-7*x**4/3 - 14*x**2 + 7*x/3

________________________________________________________________________________________