3.3.23 \(\int \frac {80-14552 x^2-1280 x^3-5691 x^4-160 x^5-672 x^6+e^4 (768 x^2+384 x^4+48 x^6)}{80+40 x^2+5 x^4} \, dx\)

Optimal. Leaf size=32 \[ x-16 x^3 \left (3+\frac {1}{5} \left (-1-e^4+\frac {5 (4+x)}{4+x^2}\right )\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 37, normalized size of antiderivative = 1.16, number of steps used = 4, number of rules used = 3, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {28, 1814, 1586} \begin {gather*} -\frac {16}{5} \left (14-e^4\right ) x^3-16 x^2-\frac {256 (1-x)}{x^2+4}-63 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(80 - 14552*x^2 - 1280*x^3 - 5691*x^4 - 160*x^5 - 672*x^6 + E^4*(768*x^2 + 384*x^4 + 48*x^6))/(80 + 40*x^2
+ 5*x^4),x]

[Out]

-63*x - 16*x^2 - (16*(14 - E^4)*x^3)/5 - (256*(1 - x))/(4 + x^2)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=5 \int \frac {80-14552 x^2-1280 x^3-5691 x^4-160 x^5-672 x^6+e^4 \left (768 x^2+384 x^4+48 x^6\right )}{\left (20+5 x^2\right )^2} \, dx\\ &=-\frac {256 (1-x)}{4+x^2}-\frac {1}{8} \int \frac {10080+5120 x+24 \left (1001-64 e^4\right ) x^2+1280 x^3+384 \left (14-e^4\right ) x^4}{20+5 x^2} \, dx\\ &=-\frac {256 (1-x)}{4+x^2}-\frac {1}{8} \int \left (504+256 x+\left (\frac {5376}{5}-\frac {384 e^4}{5}\right ) x^2\right ) \, dx\\ &=-63 x-16 x^2-\frac {16}{5} \left (14-e^4\right ) x^3-\frac {256 (1-x)}{4+x^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 35, normalized size = 1.09 \begin {gather*} \frac {1}{5} \left (-315 x-80 x^2+16 \left (-14+e^4\right ) x^3+\frac {1280 (-1+x)}{4+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(80 - 14552*x^2 - 1280*x^3 - 5691*x^4 - 160*x^5 - 672*x^6 + E^4*(768*x^2 + 384*x^4 + 48*x^6))/(80 +
40*x^2 + 5*x^4),x]

[Out]

(-315*x - 80*x^2 + 16*(-14 + E^4)*x^3 + (1280*(-1 + x))/(4 + x^2))/5

________________________________________________________________________________________

fricas [A]  time = 0.90, size = 47, normalized size = 1.47 \begin {gather*} -\frac {224 \, x^{5} + 80 \, x^{4} + 1211 \, x^{3} + 320 \, x^{2} - 16 \, {\left (x^{5} + 4 \, x^{3}\right )} e^{4} - 20 \, x + 1280}{5 \, {\left (x^{2} + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^6+384*x^4+768*x^2)*exp(4)-672*x^6-160*x^5-5691*x^4-1280*x^3-14552*x^2+80)/(5*x^4+40*x^2+80),x
, algorithm="fricas")

[Out]

-1/5*(224*x^5 + 80*x^4 + 1211*x^3 + 320*x^2 - 16*(x^5 + 4*x^3)*e^4 - 20*x + 1280)/(x^2 + 4)

________________________________________________________________________________________

giac [A]  time = 0.48, size = 33, normalized size = 1.03 \begin {gather*} \frac {16}{5} \, x^{3} e^{4} - \frac {224}{5} \, x^{3} - 16 \, x^{2} - 63 \, x + \frac {256 \, {\left (x - 1\right )}}{x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^6+384*x^4+768*x^2)*exp(4)-672*x^6-160*x^5-5691*x^4-1280*x^3-14552*x^2+80)/(5*x^4+40*x^2+80),x
, algorithm="giac")

[Out]

16/5*x^3*e^4 - 224/5*x^3 - 16*x^2 - 63*x + 256*(x - 1)/(x^2 + 4)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 35, normalized size = 1.09

method	result	size
risch	\(\frac {16 x^{3} {\mathrm e}^{4}}{5}-\frac {224 x^{3}}{5}-16 x^{2}-63 x +\frac {256 x -256}{x^{2}+4}\)	\(35\)
default	\(\frac {16 x^{3} {\mathrm e}^{4}}{5}-\frac {224 x^{3}}{5}-16 x^{2}-63 x -\frac {256 \left (1-x \right )}{x^{2}+4}\)	\(36\)
norman	\(\frac {\left (\frac {16 \,{\mathrm e}^{4}}{5}-\frac {224}{5}\right ) x^{5}+\left (\frac {64 \,{\mathrm e}^{4}}{5}-\frac {1211}{5}\right ) x^{3}+4 x -16 x^{4}}{x^{2}+4}\)	\(38\)
gosper	\(\frac {x \left (16 x^{4} {\mathrm e}^{4}-224 x^{4}+64 x^{2} {\mathrm e}^{4}-80 x^{3}-1211 x^{2}+20\right )}{5 x^{2}+20}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((48*x^6+384*x^4+768*x^2)*exp(4)-672*x^6-160*x^5-5691*x^4-1280*x^3-14552*x^2+80)/(5*x^4+40*x^2+80),x,metho
d=_RETURNVERBOSE)

[Out]

16/5*x^3*exp(4)-224/5*x^3-16*x^2-63*x+(256*x-256)/(x^2+4)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 30, normalized size = 0.94 \begin {gather*} \frac {16}{5} \, x^{3} {\left (e^{4} - 14\right )} - 16 \, x^{2} - 63 \, x + \frac {256 \, {\left (x - 1\right )}}{x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^6+384*x^4+768*x^2)*exp(4)-672*x^6-160*x^5-5691*x^4-1280*x^3-14552*x^2+80)/(5*x^4+40*x^2+80),x
, algorithm="maxima")

[Out]

16/5*x^3*(e^4 - 14) - 16*x^2 - 63*x + 256*(x - 1)/(x^2 + 4)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 32, normalized size = 1.00 \begin {gather*} \frac {256\,x-256}{x^2+4}-63\,x+x^3\,\left (\frac {16\,{\mathrm {e}}^4}{5}-\frac {224}{5}\right )-16\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(14552*x^2 - exp(4)*(768*x^2 + 384*x^4 + 48*x^6) + 1280*x^3 + 5691*x^4 + 160*x^5 + 672*x^6 - 80)/(40*x^2
+ 5*x^4 + 80),x)

[Out]

(256*x - 256)/(x^2 + 4) - 63*x + x^3*((16*exp(4))/5 - 224/5) - 16*x^2

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 32, normalized size = 1.00 \begin {gather*} - x^{3} \left (\frac {224}{5} - \frac {16 e^{4}}{5}\right ) - 16 x^{2} - 63 x - \frac {256 - 256 x}{x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x**6+384*x**4+768*x**2)*exp(4)-672*x**6-160*x**5-5691*x**4-1280*x**3-14552*x**2+80)/(5*x**4+40*
x**2+80),x)

[Out]

-x**3*(224/5 - 16*exp(4)/5) - 16*x**2 - 63*x - (256 - 256*x)/(x**2 + 4)

________________________________________________________________________________________