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

Optimal. Leaf size=63 \[ -\frac{4096 x^{19}}{19}+\frac{24576 x^{17}}{17}-\frac{21248 x^{15}}{5}+\frac{93440 x^{13}}{13}-\frac{84912 x^{11}}{11}+\frac{16448 x^9}{3}-2605 x^7+\frac{4113 x^5}{5}-171 x^3+27 x \]

[Out]

27*x - 171*x^3 + (4113*x^5)/5 - 2605*x^7 + (16448*x^9)/3 - (84912*x^11)/11 + (93440*x^13)/13 - (21248*x^15)/5
+ (24576*x^17)/17 - (4096*x^19)/19

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Rubi [A]  time = 0.0757429, antiderivative size = 63, 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} \[ -\frac{4096 x^{19}}{19}+\frac{24576 x^{17}}{17}-\frac{21248 x^{15}}{5}+\frac{93440 x^{13}}{13}-\frac{84912 x^{11}}{11}+\frac{16448 x^9}{3}-2605 x^7+\frac{4113 x^5}{5}-171 x^3+27 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

27*x - 171*x^3 + (4113*x^5)/5 - 2605*x^7 + (16448*x^9)/3 - (84912*x^11)/11 + (93440*x^13)/13 - (21248*x^15)/5
+ (24576*x^17)/17 - (4096*x^19)/19

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 )^3 \, dx &=-\int (-1+x)^3 (1+x)^3 (-1+2 x)^3 (1+2 x)^3 \left (-3+4 x^2\right )^3 \, dx\\ &=-\int (-1+2 x)^3 (1+2 x)^3 \left (-1+x^2\right )^3 \left (-3+4 x^2\right )^3 \, dx\\ &=-\int \left (-1+x^2\right )^3 \left (-3+4 x^2\right )^3 \left (-1+4 x^2\right )^3 \, dx\\ &=-\int \left (-27+513 x^2-4113 x^4+18235 x^6-49344 x^8+84912 x^{10}-93440 x^{12}+63744 x^{14}-24576 x^{16}+4096 x^{18}\right ) \, dx\\ &=27 x-171 x^3+\frac{4113 x^5}{5}-2605 x^7+\frac{16448 x^9}{3}-\frac{84912 x^{11}}{11}+\frac{93440 x^{13}}{13}-\frac{21248 x^{15}}{5}+\frac{24576 x^{17}}{17}-\frac{4096 x^{19}}{19}\\ \end{align*}

Mathematica [A]  time = 0.0020171, size = 63, normalized size = 1. \[ -\frac{4096 x^{19}}{19}+\frac{24576 x^{17}}{17}-\frac{21248 x^{15}}{5}+\frac{93440 x^{13}}{13}-\frac{84912 x^{11}}{11}+\frac{16448 x^9}{3}-2605 x^7+\frac{4113 x^5}{5}-171 x^3+27 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

27*x - 171*x^3 + (4113*x^5)/5 - 2605*x^7 + (16448*x^9)/3 - (84912*x^11)/11 + (93440*x^13)/13 - (21248*x^15)/5
+ (24576*x^17)/17 - (4096*x^19)/19

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Maple [A]  time = 0.001, size = 50, normalized size = 0.8 \begin{align*} 27\,x-171\,{x}^{3}+{\frac{4113\,{x}^{5}}{5}}-2605\,{x}^{7}+{\frac{16448\,{x}^{9}}{3}}-{\frac{84912\,{x}^{11}}{11}}+{\frac{93440\,{x}^{13}}{13}}-{\frac{21248\,{x}^{15}}{5}}+{\frac{24576\,{x}^{17}}{17}}-{\frac{4096\,{x}^{19}}{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27*x-171*x^3+4113/5*x^5-2605*x^7+16448/3*x^9-84912/11*x^11+93440/13*x^13-21248/5*x^15+24576/17*x^17-4096/19*x^
19

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Maxima [A]  time = 1.20483, size = 66, normalized size = 1.05 \begin{align*} -\frac{4096}{19} \, x^{19} + \frac{24576}{17} \, x^{17} - \frac{21248}{5} \, x^{15} + \frac{93440}{13} \, x^{13} - \frac{84912}{11} \, x^{11} + \frac{16448}{3} \, x^{9} - 2605 \, x^{7} + \frac{4113}{5} \, x^{5} - 171 \, x^{3} + 27 \, 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)^3,x, algorithm="maxima")

[Out]

-4096/19*x^19 + 24576/17*x^17 - 21248/5*x^15 + 93440/13*x^13 - 84912/11*x^11 + 16448/3*x^9 - 2605*x^7 + 4113/5
*x^5 - 171*x^3 + 27*x

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Fricas [A]  time = 1.5149, size = 180, normalized size = 2.86 \begin{align*} -\frac{4096}{19} x^{19} + \frac{24576}{17} x^{17} - \frac{21248}{5} x^{15} + \frac{93440}{13} x^{13} - \frac{84912}{11} x^{11} + \frac{16448}{3} x^{9} - 2605 x^{7} + \frac{4113}{5} x^{5} - 171 x^{3} + 27 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)^3,x, algorithm="fricas")

[Out]

-4096/19*x^19 + 24576/17*x^17 - 21248/5*x^15 + 93440/13*x^13 - 84912/11*x^11 + 16448/3*x^9 - 2605*x^7 + 4113/5
*x^5 - 171*x^3 + 27*x

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Sympy [A]  time = 0.068592, size = 60, normalized size = 0.95 \begin{align*} - \frac{4096 x^{19}}{19} + \frac{24576 x^{17}}{17} - \frac{21248 x^{15}}{5} + \frac{93440 x^{13}}{13} - \frac{84912 x^{11}}{11} + \frac{16448 x^{9}}{3} - 2605 x^{7} + \frac{4113 x^{5}}{5} - 171 x^{3} + 27 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)**3,x)

[Out]

-4096*x**19/19 + 24576*x**17/17 - 21248*x**15/5 + 93440*x**13/13 - 84912*x**11/11 + 16448*x**9/3 - 2605*x**7 +
4113*x**5/5 - 171*x**3 + 27*x

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Giac [A]  time = 1.11731, size = 66, normalized size = 1.05 \begin{align*} -\frac{4096}{19} \, x^{19} + \frac{24576}{17} \, x^{17} - \frac{21248}{5} \, x^{15} + \frac{93440}{13} \, x^{13} - \frac{84912}{11} \, x^{11} + \frac{16448}{3} \, x^{9} - 2605 \, x^{7} + \frac{4113}{5} \, x^{5} - 171 \, x^{3} + 27 \, 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)^3,x, algorithm="giac")

[Out]

-4096/19*x^19 + 24576/17*x^17 - 21248/5*x^15 + 93440/13*x^13 - 84912/11*x^11 + 16448/3*x^9 - 2605*x^7 + 4113/5
*x^5 - 171*x^3 + 27*x