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 = 0.0234212, 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 \[ -\frac{1}{117} (4-3 x)^{13/3}+\frac{4}{45} (4-3 x)^{10/3}-\frac{16}{63} (4-3 x)^{7/3} \]

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

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Rubi in Sympy [A]  time = 2.12378, size = 32, normalized size = 0.8 \[ - \frac{\left (- 3 x + 4\right )^{\frac{13}{3}}}{117} + \frac{4 \left (- 3 x + 4\right )^{\frac{10}{3}}}{45} - \frac{16 \left (- 3 x + 4\right )^{\frac{7}{3}}}{63} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4-3*x)**(4/3)*x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0177789, 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 = 0.004, size = 20, normalized size = 0.5 \[ -{\frac{35\,{x}^{2}+28\,x+16}{455} \left ( 4-3\,x \right ) ^{{\frac{7}{3}}}} \]

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 = 1.53388, size = 38, normalized size = 0.95 \[ -\frac{1}{117} \,{\left (-3 \, x + 4\right )}^{\frac{13}{3}} + \frac{4}{45} \,{\left (-3 \, x + 4\right )}^{\frac{10}{3}} - \frac{16}{63} \,{\left (-3 \, x + 4\right )}^{\frac{7}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*(-3*x + 4)^(4/3),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 = 0.201517, size = 39, normalized size = 0.98 \[ -\frac{1}{455} \,{\left (315 \, x^{4} - 588 \, x^{3} + 32 \, x^{2} + 64 \, x + 256\right )}{\left (-3 \, x + 4\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*(-3*x + 4)^(4/3),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 [A]  time = 3.27038, size = 189, normalized size = 4.72 \[ \begin{cases} - \frac{9 x^{4} \sqrt [3]{3 x - 4} e^{\frac{13 i \pi }{3}}}{13} + \frac{84 x^{3} \sqrt [3]{3 x - 4} e^{\frac{13 i \pi }{3}}}{65} - \frac{32 x^{2} \sqrt [3]{3 x - 4} e^{\frac{13 i \pi }{3}}}{455} - \frac{64 x \sqrt [3]{3 x - 4} e^{\frac{13 i \pi }{3}}}{455} - \frac{256 \sqrt [3]{3 x - 4} e^{\frac{13 i \pi }{3}}}{455} & \text{for}\: \frac{3 \left |{x}\right |}{4} > 1 \\- \frac{9 x^{4} \sqrt [3]{- 3 x + 4}}{13} + \frac{84 x^{3} \sqrt [3]{- 3 x + 4}}{65} - \frac{32 x^{2} \sqrt [3]{- 3 x + 4}}{455} - \frac{64 x \sqrt [3]{- 3 x + 4}}{455} - \frac{256 \sqrt [3]{- 3 x + 4}}{455} & \text{otherwise} \end{cases} \]

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(13*I*pi/3)/13 + 84*x**3*(3*x - 4)**(1/3)
*exp(13*I*pi/3)/65 - 32*x**2*(3*x - 4)**(1/3)*exp(13*I*pi/3)/455 - 64*x*(3*x - 4
)**(1/3)*exp(13*I*pi/3)/455 - 256*(3*x - 4)**(1/3)*exp(13*I*pi/3)/455, 3*Abs(x)/
4 > 1), (-9*x**4*(-3*x + 4)**(1/3)/13 + 84*x**3*(-3*x + 4)**(1/3)/65 - 32*x**2*(
-3*x + 4)**(1/3)/455 - 64*x*(-3*x + 4)**(1/3)/455 - 256*(-3*x + 4)**(1/3)/455, T
rue))

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GIAC/XCAS [A]  time = 0.199665, size = 66, normalized size = 1.65 \[ -\frac{1}{117} \,{\left (3 \, x - 4\right )}^{4}{\left (-3 \, x + 4\right )}^{\frac{1}{3}} - \frac{4}{45} \,{\left (3 \, x - 4\right )}^{3}{\left (-3 \, x + 4\right )}^{\frac{1}{3}} - \frac{16}{63} \,{\left (3 \, x - 4\right )}^{2}{\left (-3 \, x + 4\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2*(-3*x + 4)^(4/3),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)