3.11 \(\int (-4 x+3 x^3)^6 \, dx\)

Optimal. Leaf size=46 \[ \frac{729 x^{19}}{19}-\frac{5832 x^{17}}{17}+1296 x^{15}-\frac{34560 x^{13}}{13}+\frac{34560 x^{11}}{11}-2048 x^9+\frac{4096 x^7}{7} \]

[Out]

(4096*x^7)/7 - 2048*x^9 + (34560*x^11)/11 - (34560*x^13)/13 + 1296*x^15 - (5832*x^17)/17 + (729*x^19)/19

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Rubi [A]  time = 0.0157418, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1593, 270} \[ \frac{729 x^{19}}{19}-\frac{5832 x^{17}}{17}+1296 x^{15}-\frac{34560 x^{13}}{13}+\frac{34560 x^{11}}{11}-2048 x^9+\frac{4096 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(-4*x + 3*x^3)^6,x]

[Out]

(4096*x^7)/7 - 2048*x^9 + (34560*x^11)/11 - (34560*x^13)/13 + 1296*x^15 - (5832*x^17)/17 + (729*x^19)/19

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (-4 x+3 x^3\right )^6 \, dx &=\int x^6 \left (-4+3 x^2\right )^6 \, dx\\ &=\int \left (4096 x^6-18432 x^8+34560 x^{10}-34560 x^{12}+19440 x^{14}-5832 x^{16}+729 x^{18}\right ) \, dx\\ &=\frac{4096 x^7}{7}-2048 x^9+\frac{34560 x^{11}}{11}-\frac{34560 x^{13}}{13}+1296 x^{15}-\frac{5832 x^{17}}{17}+\frac{729 x^{19}}{19}\\ \end{align*}

Mathematica [A]  time = 0.0018519, size = 46, normalized size = 1. \[ \frac{729 x^{19}}{19}-\frac{5832 x^{17}}{17}+1296 x^{15}-\frac{34560 x^{13}}{13}+\frac{34560 x^{11}}{11}-2048 x^9+\frac{4096 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(-4*x + 3*x^3)^6,x]

[Out]

(4096*x^7)/7 - 2048*x^9 + (34560*x^11)/11 - (34560*x^13)/13 + 1296*x^15 - (5832*x^17)/17 + (729*x^19)/19

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Maple [A]  time = 0.002, size = 37, normalized size = 0.8 \begin{align*}{\frac{4096\,{x}^{7}}{7}}-2048\,{x}^{9}+{\frac{34560\,{x}^{11}}{11}}-{\frac{34560\,{x}^{13}}{13}}+1296\,{x}^{15}-{\frac{5832\,{x}^{17}}{17}}+{\frac{729\,{x}^{19}}{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^3-4*x)^6,x)

[Out]

4096/7*x^7-2048*x^9+34560/11*x^11-34560/13*x^13+1296*x^15-5832/17*x^17+729/19*x^19

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Maxima [A]  time = 1.04227, size = 49, normalized size = 1.07 \begin{align*} \frac{729}{19} \, x^{19} - \frac{5832}{17} \, x^{17} + 1296 \, x^{15} - \frac{34560}{13} \, x^{13} + \frac{34560}{11} \, x^{11} - 2048 \, x^{9} + \frac{4096}{7} \, x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3-4*x)^6,x, algorithm="maxima")

[Out]

729/19*x^19 - 5832/17*x^17 + 1296*x^15 - 34560/13*x^13 + 34560/11*x^11 - 2048*x^9 + 4096/7*x^7

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Fricas [A]  time = 1.20751, size = 130, normalized size = 2.83 \begin{align*} \frac{729}{19} x^{19} - \frac{5832}{17} x^{17} + 1296 x^{15} - \frac{34560}{13} x^{13} + \frac{34560}{11} x^{11} - 2048 x^{9} + \frac{4096}{7} x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3-4*x)^6,x, algorithm="fricas")

[Out]

729/19*x^19 - 5832/17*x^17 + 1296*x^15 - 34560/13*x^13 + 34560/11*x^11 - 2048*x^9 + 4096/7*x^7

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Sympy [A]  time = 0.062123, size = 42, normalized size = 0.91 \begin{align*} \frac{729 x^{19}}{19} - \frac{5832 x^{17}}{17} + 1296 x^{15} - \frac{34560 x^{13}}{13} + \frac{34560 x^{11}}{11} - 2048 x^{9} + \frac{4096 x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**3-4*x)**6,x)

[Out]

729*x**19/19 - 5832*x**17/17 + 1296*x**15 - 34560*x**13/13 + 34560*x**11/11 - 2048*x**9 + 4096*x**7/7

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Giac [A]  time = 1.25294, size = 49, normalized size = 1.07 \begin{align*} \frac{729}{19} \, x^{19} - \frac{5832}{17} \, x^{17} + 1296 \, x^{15} - \frac{34560}{13} \, x^{13} + \frac{34560}{11} \, x^{11} - 2048 \, x^{9} + \frac{4096}{7} \, x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^3-4*x)^6,x, algorithm="giac")

[Out]

729/19*x^19 - 5832/17*x^17 + 1296*x^15 - 34560/13*x^13 + 34560/11*x^11 - 2048*x^9 + 4096/7*x^7