3.72 \(\int (3-19 x^2+32 x^4-16 x^6)^2 \, dx\)

Optimal. Leaf size=44 \[ \frac{256 x^{13}}{13}-\frac{1024 x^{11}}{11}+\frac{544 x^9}{3}-\frac{1312 x^7}{7}+\frac{553 x^5}{5}-38 x^3+9 x \]

[Out]

9*x - 38*x^3 + (553*x^5)/5 - (1312*x^7)/7 + (544*x^9)/3 - (1024*x^11)/11 + (256*x^13)/13

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Rubi [A]  time = 0.0693192, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2059, 517, 521} \[ \frac{256 x^{13}}{13}-\frac{1024 x^{11}}{11}+\frac{544 x^9}{3}-\frac{1312 x^7}{7}+\frac{553 x^5}{5}-38 x^3+9 x \]

Antiderivative was successfully verified.

[In]

Int[(3 - 19*x^2 + 32*x^4 - 16*x^6)^2,x]

[Out]

9*x - 38*x^3 + (553*x^5)/5 - (1312*x^7)/7 + (544*x^9)/3 - (1024*x^11)/11 + (256*x^13)/13

Rule 2059

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[u^p, x] /;  !SumQ[NonfreeFactors[u, x]]] /; PolyQ[P, x]
&& IntegerQ[p]

Rule 517

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^
(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x]
&& EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 521

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :>
Int[ExpandIntegrand[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && I
GtQ[p, 0] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin{align*} \int \left (3-19 x^2+32 x^4-16 x^6\right )^2 \, dx &=\int (-1+x)^2 (1+x)^2 (-1+2 x)^2 (1+2 x)^2 \left (-3+4 x^2\right )^2 \, dx\\ &=\int (-1+2 x)^2 (1+2 x)^2 \left (-1+x^2\right )^2 \left (-3+4 x^2\right )^2 \, dx\\ &=\int \left (-1+x^2\right )^2 \left (-3+4 x^2\right )^2 \left (-1+4 x^2\right )^2 \, dx\\ &=\int \left (9-114 x^2+553 x^4-1312 x^6+1632 x^8-1024 x^{10}+256 x^{12}\right ) \, dx\\ &=9 x-38 x^3+\frac{553 x^5}{5}-\frac{1312 x^7}{7}+\frac{544 x^9}{3}-\frac{1024 x^{11}}{11}+\frac{256 x^{13}}{13}\\ \end{align*}

Mathematica [A]  time = 0.0007037, size = 44, normalized size = 1. \[ \frac{256 x^{13}}{13}-\frac{1024 x^{11}}{11}+\frac{544 x^9}{3}-\frac{1312 x^7}{7}+\frac{553 x^5}{5}-38 x^3+9 x \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 19*x^2 + 32*x^4 - 16*x^6)^2,x]

[Out]

9*x - 38*x^3 + (553*x^5)/5 - (1312*x^7)/7 + (544*x^9)/3 - (1024*x^11)/11 + (256*x^13)/13

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Maple [A]  time = 0., size = 35, normalized size = 0.8 \begin{align*} 9\,x-38\,{x}^{3}+{\frac{553\,{x}^{5}}{5}}-{\frac{1312\,{x}^{7}}{7}}+{\frac{544\,{x}^{9}}{3}}-{\frac{1024\,{x}^{11}}{11}}+{\frac{256\,{x}^{13}}{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x^6+32*x^4-19*x^2+3)^2,x)

[Out]

9*x-38*x^3+553/5*x^5-1312/7*x^7+544/3*x^9-1024/11*x^11+256/13*x^13

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Maxima [A]  time = 1.14573, size = 46, normalized size = 1.05 \begin{align*} \frac{256}{13} \, x^{13} - \frac{1024}{11} \, x^{11} + \frac{544}{3} \, x^{9} - \frac{1312}{7} \, x^{7} + \frac{553}{5} \, x^{5} - 38 \, x^{3} + 9 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^6+32*x^4-19*x^2+3)^2,x, algorithm="maxima")

[Out]

256/13*x^13 - 1024/11*x^11 + 544/3*x^9 - 1312/7*x^7 + 553/5*x^5 - 38*x^3 + 9*x

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Fricas [A]  time = 1.5264, size = 108, normalized size = 2.45 \begin{align*} \frac{256}{13} x^{13} - \frac{1024}{11} x^{11} + \frac{544}{3} x^{9} - \frac{1312}{7} x^{7} + \frac{553}{5} x^{5} - 38 x^{3} + 9 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^6+32*x^4-19*x^2+3)^2,x, algorithm="fricas")

[Out]

256/13*x^13 - 1024/11*x^11 + 544/3*x^9 - 1312/7*x^7 + 553/5*x^5 - 38*x^3 + 9*x

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Sympy [A]  time = 0.061542, size = 41, normalized size = 0.93 \begin{align*} \frac{256 x^{13}}{13} - \frac{1024 x^{11}}{11} + \frac{544 x^{9}}{3} - \frac{1312 x^{7}}{7} + \frac{553 x^{5}}{5} - 38 x^{3} + 9 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x**6+32*x**4-19*x**2+3)**2,x)

[Out]

256*x**13/13 - 1024*x**11/11 + 544*x**9/3 - 1312*x**7/7 + 553*x**5/5 - 38*x**3 + 9*x

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Giac [A]  time = 1.12267, size = 46, normalized size = 1.05 \begin{align*} \frac{256}{13} \, x^{13} - \frac{1024}{11} \, x^{11} + \frac{544}{3} \, x^{9} - \frac{1312}{7} \, x^{7} + \frac{553}{5} \, x^{5} - 38 \, x^{3} + 9 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^6+32*x^4-19*x^2+3)^2,x, algorithm="giac")

[Out]

256/13*x^13 - 1024/11*x^11 + 544/3*x^9 - 1312/7*x^7 + 553/5*x^5 - 38*x^3 + 9*x