\(\int (4-3 x)^{4/3} x^2 \, dx\) [295]

Optimal result

Integrand size = 13, antiderivative size = 40 \[ \int (4-3 x)^{4/3} x^2 \, dx=-\frac {16}{63} (4-3 x)^{7/3}+\frac {4}{45} (4-3 x)^{10/3}-\frac {1}{117} (4-3 x)^{13/3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int (4-3 x)^{4/3} x^2 \, dx=-\frac {1}{117} (4-3 x)^{13/3}+\frac {4}{45} (4-3 x)^{10/3}-\frac {16}{63} (4-3 x)^{7/3} \]

[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 45

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*} \text {integral}& = \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] (verified)

Time = 0.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int (4-3 x)^{4/3} x^2 \, dx=-\frac {1}{455} (4-3 x)^{7/3} \left (16+28 x+35 x^2\right ) \]

[In]

Integrate[(4 - 3*x)^(4/3)*x^2,x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 2.

Time = 0.32 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.45

method	result	size
meijerg	\(\frac {4 \,2^{\frac {2}{3}} x^{3} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {4}{3},3;4;\frac {3 x}{4}\right )}{3}\)	\(18\)
gosper	\(-\frac {\left (4-3 x \right )^{\frac {7}{3}} \left (35 x^{2}+28 x +16\right )}{455}\)	\(20\)
pseudoelliptic	\(-\frac {\left (35 x^{2}+28 x +16\right ) \left (-4+3 x \right )^{2} \left (4-3 x \right )^{\frac {1}{3}}}{455}\)	\(27\)
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	\(\frac {\left (315 x^{4}-588 x^{3}+32 x^{2}+64 x +256\right ) \left (-4+3 x \right )}{455 \left (4-3 x \right )^{\frac {2}{3}}}\)	\(35\)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72 \[ \int (4-3 x)^{4/3} x^2 \, dx=-\frac {1}{455} \, {\left (315 \, x^{4} - 588 \, x^{3} + 32 \, x^{2} + 64 \, x + 256\right )} {\left (-3 \, x + 4\right )}^{\frac {1}{3}} \]

[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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.88 (sec) , antiderivative size = 178, normalized size of antiderivative = 4.45 \[ \int (4-3 x)^{4/3} x^2 \, dx=\begin {cases} - \frac {9 x^{4} \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{13} + \frac {84 x^{3} \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{65} - \frac {32 x^{2} \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{455} - \frac {64 x \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{455} - \frac {256 \sqrt [3]{3 x - 4} e^{\frac {i \pi }{3}}}{455} & \text {for}\: \left |{x}\right | > \frac {4}{3} \\- \frac {9 x^{4} \sqrt [3]{4 - 3 x}}{13} + \frac {84 x^{3} \sqrt [3]{4 - 3 x}}{65} - \frac {32 x^{2} \sqrt [3]{4 - 3 x}}{455} - \frac {64 x \sqrt [3]{4 - 3 x}}{455} - \frac {256 \sqrt [3]{4 - 3 x}}{455} & \text {otherwise} \end {cases} \]

[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, Abs(
x) > 4/3), (-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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int (4-3 x)^{4/3} x^2 \, dx=-\frac {1}{117} \, {\left (-3 \, x + 4\right )}^{\frac {13}{3}} + \frac {4}{45} \, {\left (-3 \, x + 4\right )}^{\frac {10}{3}} - \frac {16}{63} \, {\left (-3 \, x + 4\right )}^{\frac {7}{3}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.22 \[ \int (4-3 x)^{4/3} x^2 \, dx=-\frac {1}{117} \, {\left (3 \, x - 4\right )}^{4} {\left (-3 \, x + 4\right )}^{\frac {1}{3}} - \frac {4}{45} \, {\left (3 \, x - 4\right )}^{3} {\left (-3 \, x + 4\right )}^{\frac {1}{3}} - \frac {16}{63} \, {\left (3 \, x - 4\right )}^{2} {\left (-3 \, x + 4\right )}^{\frac {1}{3}} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.58 \[ \int (4-3 x)^{4/3} x^2 \, dx=-\frac {{\left (4-3\,x\right )}^{7/3}\,\left (1092\,x+35\,{\left (3\,x-4\right )}^2-416\right )}{4095} \]

[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