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

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

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Rubi [A]  time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 0.58 \[ -\frac {1}{455} (4-3 x)^{7/3} \left (35 x^2+28 x+16\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(4 - 3*x)^(4/3)*x^2,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.01, size = 33, normalized size = 0.82 \[ \frac {1}{455} \sqrt [3]{4-3 x} \left (-315 x^4+588 x^3-32 x^2-64 x-256\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(4 - 3*x)^(4/3)*x^2,x]

[Out]

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

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fricas [A]  time = 0.62, size = 29, normalized size = 0.72 \[ -\frac {1}{455} \, {\left (315 \, x^{4} - 588 \, x^{3} + 32 \, x^{2} + 64 \, x + 256\right )} {\left (-3 \, x + 4\right )}^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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giac [A]  time = 0.63, size = 49, normalized size = 1.22 \[ -\frac {1}{117} \, {\left (3 \, x - 4\right )}^{4} {\left (-3 \, x + 4\right )}^{\frac {1}{3}} - \frac {4}{45} \, {\left (3 \, x - 4\right )}^{3} {\left (-3 \, x + 4\right )}^{\frac {1}{3}} - \frac {16}{63} \, {\left (3 \, x - 4\right )}^{2} {\left (-3 \, x + 4\right )}^{\frac {1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)

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maple [C]  time = 0.30, size = 18, normalized size = 0.45

method	result	size
meijerg	\(\frac {4 \,2^{\frac {2}{3}} x^{3} \hypergeom \left (\left [-\frac {4}{3}, 3\right ], \relax [4], \frac {3 x}{4}\right )}{3}\)	\(18\)
gosper	\(-\frac {\left (35 x^{2}+28 x +16\right ) \left (4-3 x \right )^{\frac {7}{3}}}{455}\)	\(20\)
derivativedivides	\(-\frac {16 \left (4-3 x \right )^{\frac {7}{3}}}{63}+\frac {4 \left (4-3 x \right )^{\frac {10}{3}}}{45}-\frac {\left (4-3 x \right )^{\frac {13}{3}}}{117}\)	\(29\)
default	\(-\frac {16 \left (4-3 x \right )^{\frac {7}{3}}}{63}+\frac {4 \left (4-3 x \right )^{\frac {10}{3}}}{45}-\frac {\left (4-3 x \right )^{\frac {13}{3}}}{117}\)	\(29\)
trager	\(\left (-\frac {9}{13} x^{4}+\frac {84}{65} x^{3}-\frac {32}{455} x^{2}-\frac {64}{455} x -\frac {256}{455}\right ) \left (4-3 x \right )^{\frac {1}{3}}\)	\(29\)
risch	\(\frac {\left (315 x^{4}-588 x^{3}+32 x^{2}+64 x +256\right ) \left (-4+3 x \right )}{455 \left (4-3 x \right )^{\frac {2}{3}}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*2^(2/3)*x^3*hypergeom([-4/3,3],[4],3/4*x)

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maxima [A]  time = 0.61, size = 28, normalized size = 0.70 \[ -\frac {1}{117} \, {\left (-3 \, x + 4\right )}^{\frac {13}{3}} + \frac {4}{45} \, {\left (-3 \, x + 4\right )}^{\frac {10}{3}} - \frac {16}{63} \, {\left (-3 \, x + 4\right )}^{\frac {7}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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mupad [B]  time = 0.20, size = 23, normalized size = 0.58 \[ -\frac {{\left (4-3\,x\right )}^{7/3}\,\left (1092\,x+35\,{\left (3\,x-4\right )}^2-416\right )}{4095} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(4 - 3*x)^(4/3),x)

[Out]

-((4 - 3*x)^(7/3)*(1092*x + 35*(3*x - 4)^2 - 416))/4095

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sympy [B]  time = 1.75, size = 180, normalized size = 4.50 \[ \begin {cases} - \frac {9 x^{4} \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{13} + \frac {84 x^{3} \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{65} - \frac {32 x^{2} \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{455} - \frac {64 x \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{455} - \frac {256 \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{455} & \text {for}\: \frac {3 \left |{x}\right |}{4} > 1 \\- \frac {9 x^{4} \sqrt [3]{4 - 3 x}}{13} + \frac {84 x^{3} \sqrt [3]{4 - 3 x}}{65} - \frac {32 x^{2} \sqrt [3]{4 - 3 x}}{455} - \frac {64 x \sqrt [3]{4 - 3 x}}{455} - \frac {256 \sqrt [3]{4 - 3 x}}{455} & \text {otherwise} \end {cases} \]

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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