3.295 \(\int (4-3 x)^{4/3} x^2 \, dx\)

Optimal. Leaf size=40 \[ -\frac{1}{117} (4-3 x)^{13/3}+\frac{4}{45} (4-3 x)^{10/3}-\frac{16}{63} (4-3 x)^{7/3} \]

[Out]

(-16*(4 - 3*x)^(7/3))/63 + (4*(4 - 3*x)^(10/3))/45 - (4 - 3*x)^(13/3)/117

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Rubi [A]  time = 0.0072651, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ -\frac{1}{117} (4-3 x)^{13/3}+\frac{4}{45} (4-3 x)^{10/3}-\frac{16}{63} (4-3 x)^{7/3} \]

Antiderivative was successfully verified.

[In]

Int[(4 - 3*x)^(4/3)*x^2,x]

[Out]

(-16*(4 - 3*x)^(7/3))/63 + (4*(4 - 3*x)^(10/3))/45 - (4 - 3*x)^(13/3)/117

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (4-3 x)^{4/3} x^2 \, dx &=\int \left (\frac{16}{9} (4-3 x)^{4/3}-\frac{8}{9} (4-3 x)^{7/3}+\frac{1}{9} (4-3 x)^{10/3}\right ) \, dx\\ &=-\frac{16}{63} (4-3 x)^{7/3}+\frac{4}{45} (4-3 x)^{10/3}-\frac{1}{117} (4-3 x)^{13/3}\\ \end{align*}

Mathematica [A]  time = 0.012353, size = 23, normalized size = 0.57 \[ -\frac{1}{455} (4-3 x)^{7/3} \left (35 x^2+28 x+16\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(4 - 3*x)^(4/3)*x^2,x]

[Out]

-((4 - 3*x)^(7/3)*(16 + 28*x + 35*x^2))/455

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Maple [A]  time = 0.003, size = 20, normalized size = 0.5 \begin{align*} -{\frac{35\,{x}^{2}+28\,x+16}{455} \left ( 4-3\,x \right ) ^{{\frac{7}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4-3*x)^(4/3)*x^2,x)

[Out]

-1/455*(35*x^2+28*x+16)*(4-3*x)^(7/3)

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Maxima [A]  time = 0.959427, size = 38, normalized size = 0.95 \begin{align*} -\frac{1}{117} \,{\left (-3 \, x + 4\right )}^{\frac{13}{3}} + \frac{4}{45} \,{\left (-3 \, x + 4\right )}^{\frac{10}{3}} - \frac{16}{63} \,{\left (-3 \, x + 4\right )}^{\frac{7}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4-3*x)^(4/3)*x^2,x, algorithm="maxima")

[Out]

-1/117*(-3*x + 4)^(13/3) + 4/45*(-3*x + 4)^(10/3) - 16/63*(-3*x + 4)^(7/3)

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Fricas [A]  time = 1.76462, size = 90, normalized size = 2.25 \begin{align*} -\frac{1}{455} \,{\left (315 \, x^{4} - 588 \, x^{3} + 32 \, x^{2} + 64 \, x + 256\right )}{\left (-3 \, x + 4\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4-3*x)^(4/3)*x^2,x, algorithm="fricas")

[Out]

-1/455*(315*x^4 - 588*x^3 + 32*x^2 + 64*x + 256)*(-3*x + 4)^(1/3)

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Sympy [B]  time = 1.71752, size = 180, normalized size = 4.5 \begin{align*} \begin{cases} - \frac{9 x^{4} \sqrt [3]{3 x - 4} e^{\frac{i \pi }{3}}}{13} + \frac{84 x^{3} \sqrt [3]{3 x - 4} e^{\frac{i \pi }{3}}}{65} - \frac{32 x^{2} \sqrt [3]{3 x - 4} e^{\frac{i \pi }{3}}}{455} - \frac{64 x \sqrt [3]{3 x - 4} e^{\frac{i \pi }{3}}}{455} - \frac{256 \sqrt [3]{3 x - 4} e^{\frac{i \pi }{3}}}{455} & \text{for}\: \frac{3 \left |{x}\right |}{4} > 1 \\- \frac{9 x^{4} \sqrt [3]{4 - 3 x}}{13} + \frac{84 x^{3} \sqrt [3]{4 - 3 x}}{65} - \frac{32 x^{2} \sqrt [3]{4 - 3 x}}{455} - \frac{64 x \sqrt [3]{4 - 3 x}}{455} - \frac{256 \sqrt [3]{4 - 3 x}}{455} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4-3*x)**(4/3)*x**2,x)

[Out]

Piecewise((-9*x**4*(3*x - 4)**(1/3)*exp(I*pi/3)/13 + 84*x**3*(3*x - 4)**(1/3)*exp(I*pi/3)/65 - 32*x**2*(3*x -
4)**(1/3)*exp(I*pi/3)/455 - 64*x*(3*x - 4)**(1/3)*exp(I*pi/3)/455 - 256*(3*x - 4)**(1/3)*exp(I*pi/3)/455, 3*Ab
s(x)/4 > 1), (-9*x**4*(4 - 3*x)**(1/3)/13 + 84*x**3*(4 - 3*x)**(1/3)/65 - 32*x**2*(4 - 3*x)**(1/3)/455 - 64*x*
(4 - 3*x)**(1/3)/455 - 256*(4 - 3*x)**(1/3)/455, True))

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Giac [A]  time = 1.07843, size = 66, normalized size = 1.65 \begin{align*} -\frac{1}{117} \,{\left (3 \, x - 4\right )}^{4}{\left (-3 \, x + 4\right )}^{\frac{1}{3}} - \frac{4}{45} \,{\left (3 \, x - 4\right )}^{3}{\left (-3 \, x + 4\right )}^{\frac{1}{3}} - \frac{16}{63} \,{\left (3 \, x - 4\right )}^{2}{\left (-3 \, x + 4\right )}^{\frac{1}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4-3*x)^(4/3)*x^2,x, algorithm="giac")

[Out]

-1/117*(3*x - 4)^4*(-3*x + 4)^(1/3) - 4/45*(3*x - 4)^3*(-3*x + 4)^(1/3) - 16/63*(3*x - 4)^2*(-3*x + 4)^(1/3)