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

Optimal. Leaf size=84 \[ \frac{65536 x^{25}}{25}-\frac{524288 x^{23}}{23}+\frac{1884160 x^{21}}{21}-\frac{4014080 x^{19}}{19}+\frac{5633536 x^{17}}{17}-\frac{1094656 x^{15}}{3}+\frac{3764416 x^{13}}{13}-\frac{1841600 x^{11}}{11}+\frac{634321 x^9}{9}-\frac{149700 x^7}{7}+4590 x^5-684 x^3+81 x \]

[Out]

81*x - 684*x^3 + 4590*x^5 - (149700*x^7)/7 + (634321*x^9)/9 - (1841600*x^11)/11 + (3764416*x^13)/13 - (1094656
*x^15)/3 + (5633536*x^17)/17 - (4014080*x^19)/19 + (1884160*x^21)/21 - (524288*x^23)/23 + (65536*x^25)/25

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Rubi [A]  time = 0.0855259, antiderivative size = 84, 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} \[ \frac{65536 x^{25}}{25}-\frac{524288 x^{23}}{23}+\frac{1884160 x^{21}}{21}-\frac{4014080 x^{19}}{19}+\frac{5633536 x^{17}}{17}-\frac{1094656 x^{15}}{3}+\frac{3764416 x^{13}}{13}-\frac{1841600 x^{11}}{11}+\frac{634321 x^9}{9}-\frac{149700 x^7}{7}+4590 x^5-684 x^3+81 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

81*x - 684*x^3 + 4590*x^5 - (149700*x^7)/7 + (634321*x^9)/9 - (1841600*x^11)/11 + (3764416*x^13)/13 - (1094656
*x^15)/3 + (5633536*x^17)/17 - (4014080*x^19)/19 + (1884160*x^21)/21 - (524288*x^23)/23 + (65536*x^25)/25

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 )^4 \, dx &=\int (-1+x)^4 (1+x)^4 (-1+2 x)^4 (1+2 x)^4 \left (-3+4 x^2\right )^4 \, dx\\ &=\int (-1+2 x)^4 (1+2 x)^4 \left (-1+x^2\right )^4 \left (-3+4 x^2\right )^4 \, dx\\ &=\int \left (-1+x^2\right )^4 \left (-3+4 x^2\right )^4 \left (-1+4 x^2\right )^4 \, dx\\ &=\int \left (81-2052 x^2+22950 x^4-149700 x^6+634321 x^8-1841600 x^{10}+3764416 x^{12}-5473280 x^{14}+5633536 x^{16}-4014080 x^{18}+1884160 x^{20}-524288 x^{22}+65536 x^{24}\right ) \, dx\\ &=81 x-684 x^3+4590 x^5-\frac{149700 x^7}{7}+\frac{634321 x^9}{9}-\frac{1841600 x^{11}}{11}+\frac{3764416 x^{13}}{13}-\frac{1094656 x^{15}}{3}+\frac{5633536 x^{17}}{17}-\frac{4014080 x^{19}}{19}+\frac{1884160 x^{21}}{21}-\frac{524288 x^{23}}{23}+\frac{65536 x^{25}}{25}\\ \end{align*}

Mathematica [A]  time = 0.001617, size = 84, normalized size = 1. \[ \frac{65536 x^{25}}{25}-\frac{524288 x^{23}}{23}+\frac{1884160 x^{21}}{21}-\frac{4014080 x^{19}}{19}+\frac{5633536 x^{17}}{17}-\frac{1094656 x^{15}}{3}+\frac{3764416 x^{13}}{13}-\frac{1841600 x^{11}}{11}+\frac{634321 x^9}{9}-\frac{149700 x^7}{7}+4590 x^5-684 x^3+81 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

81*x - 684*x^3 + 4590*x^5 - (149700*x^7)/7 + (634321*x^9)/9 - (1841600*x^11)/11 + (3764416*x^13)/13 - (1094656
*x^15)/3 + (5633536*x^17)/17 - (4014080*x^19)/19 + (1884160*x^21)/21 - (524288*x^23)/23 + (65536*x^25)/25

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Maple [A]  time = 0.002, size = 65, normalized size = 0.8 \begin{align*} 81\,x-684\,{x}^{3}+4590\,{x}^{5}-{\frac{149700\,{x}^{7}}{7}}+{\frac{634321\,{x}^{9}}{9}}-{\frac{1841600\,{x}^{11}}{11}}+{\frac{3764416\,{x}^{13}}{13}}-{\frac{1094656\,{x}^{15}}{3}}+{\frac{5633536\,{x}^{17}}{17}}-{\frac{4014080\,{x}^{19}}{19}}+{\frac{1884160\,{x}^{21}}{21}}-{\frac{524288\,{x}^{23}}{23}}+{\frac{65536\,{x}^{25}}{25}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81*x-684*x^3+4590*x^5-149700/7*x^7+634321/9*x^9-1841600/11*x^11+3764416/13*x^13-1094656/3*x^15+5633536/17*x^17
-4014080/19*x^19+1884160/21*x^21-524288/23*x^23+65536/25*x^25

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Maxima [A]  time = 1.75358, size = 86, normalized size = 1.02 \begin{align*} \frac{65536}{25} \, x^{25} - \frac{524288}{23} \, x^{23} + \frac{1884160}{21} \, x^{21} - \frac{4014080}{19} \, x^{19} + \frac{5633536}{17} \, x^{17} - \frac{1094656}{3} \, x^{15} + \frac{3764416}{13} \, x^{13} - \frac{1841600}{11} \, x^{11} + \frac{634321}{9} \, x^{9} - \frac{149700}{7} \, x^{7} + 4590 \, x^{5} - 684 \, x^{3} + 81 \, 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)^4,x, algorithm="maxima")

[Out]

65536/25*x^25 - 524288/23*x^23 + 1884160/21*x^21 - 4014080/19*x^19 + 5633536/17*x^17 - 1094656/3*x^15 + 376441
6/13*x^13 - 1841600/11*x^11 + 634321/9*x^9 - 149700/7*x^7 + 4590*x^5 - 684*x^3 + 81*x

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Fricas [A]  time = 1.54675, size = 266, normalized size = 3.17 \begin{align*} \frac{65536}{25} x^{25} - \frac{524288}{23} x^{23} + \frac{1884160}{21} x^{21} - \frac{4014080}{19} x^{19} + \frac{5633536}{17} x^{17} - \frac{1094656}{3} x^{15} + \frac{3764416}{13} x^{13} - \frac{1841600}{11} x^{11} + \frac{634321}{9} x^{9} - \frac{149700}{7} x^{7} + 4590 x^{5} - 684 x^{3} + 81 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)^4,x, algorithm="fricas")

[Out]

65536/25*x^25 - 524288/23*x^23 + 1884160/21*x^21 - 4014080/19*x^19 + 5633536/17*x^17 - 1094656/3*x^15 + 376441
6/13*x^13 - 1841600/11*x^11 + 634321/9*x^9 - 149700/7*x^7 + 4590*x^5 - 684*x^3 + 81*x

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Sympy [A]  time = 0.0718, size = 80, normalized size = 0.95 \begin{align*} \frac{65536 x^{25}}{25} - \frac{524288 x^{23}}{23} + \frac{1884160 x^{21}}{21} - \frac{4014080 x^{19}}{19} + \frac{5633536 x^{17}}{17} - \frac{1094656 x^{15}}{3} + \frac{3764416 x^{13}}{13} - \frac{1841600 x^{11}}{11} + \frac{634321 x^{9}}{9} - \frac{149700 x^{7}}{7} + 4590 x^{5} - 684 x^{3} + 81 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)**4,x)

[Out]

65536*x**25/25 - 524288*x**23/23 + 1884160*x**21/21 - 4014080*x**19/19 + 5633536*x**17/17 - 1094656*x**15/3 +
3764416*x**13/13 - 1841600*x**11/11 + 634321*x**9/9 - 149700*x**7/7 + 4590*x**5 - 684*x**3 + 81*x

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Giac [A]  time = 1.1395, size = 86, normalized size = 1.02 \begin{align*} \frac{65536}{25} \, x^{25} - \frac{524288}{23} \, x^{23} + \frac{1884160}{21} \, x^{21} - \frac{4014080}{19} \, x^{19} + \frac{5633536}{17} \, x^{17} - \frac{1094656}{3} \, x^{15} + \frac{3764416}{13} \, x^{13} - \frac{1841600}{11} \, x^{11} + \frac{634321}{9} \, x^{9} - \frac{149700}{7} \, x^{7} + 4590 \, x^{5} - 684 \, x^{3} + 81 \, 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)^4,x, algorithm="giac")

[Out]

65536/25*x^25 - 524288/23*x^23 + 1884160/21*x^21 - 4014080/19*x^19 + 5633536/17*x^17 - 1094656/3*x^15 + 376441
6/13*x^13 - 1841600/11*x^11 + 634321/9*x^9 - 149700/7*x^7 + 4590*x^5 - 684*x^3 + 81*x