\(\int \sqrt [3]{3-2 x} (7+x) \, dx\) [1456]

Optimal result

Integrand size = 13, antiderivative size = 27 \[ \int \sqrt [3]{3-2 x} (7+x) \, dx=-\frac {51}{16} (3-2 x)^{4/3}+\frac {3}{28} (3-2 x)^{7/3} \]

[Out]

-51/16*(3-2*x)^(4/3)+3/28*(3-2*x)^(7/3)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, 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} \[ \int \sqrt [3]{3-2 x} (7+x) \, dx=\frac {3}{28} (3-2 x)^{7/3}-\frac {51}{16} (3-2 x)^{4/3} \]

[In]

Int[(3 - 2*x)^(1/3)*(7 + x),x]

[Out]

(-51*(3 - 2*x)^(4/3))/16 + (3*(3 - 2*x)^(7/3))/28

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 {17}{2} \sqrt [3]{3-2 x}-\frac {1}{2} (3-2 x)^{4/3}\right ) \, dx \\ & = -\frac {51}{16} (3-2 x)^{4/3}+\frac {3}{28} (3-2 x)^{7/3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \sqrt [3]{3-2 x} (7+x) \, dx=-\frac {3}{112} (3-2 x)^{4/3} (107+8 x) \]

[In]

Integrate[(3 - 2*x)^(1/3)*(7 + x),x]

[Out]

(-3*(3 - 2*x)^(4/3)*(107 + 8*x))/112

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.56

method	result	size
gosper	\(-\frac {3 \left (8 x +107\right ) \left (3-2 x \right )^{\frac {4}{3}}}{112}\)	\(15\)
trager	\(\left (\frac {3}{7} x^{2}+\frac {285}{56} x -\frac {963}{112}\right ) \left (3-2 x \right )^{\frac {1}{3}}\)	\(19\)
derivativedivides	\(-\frac {51 \left (3-2 x \right )^{\frac {4}{3}}}{16}+\frac {3 \left (3-2 x \right )^{\frac {7}{3}}}{28}\)	\(20\)
default	\(-\frac {51 \left (3-2 x \right )^{\frac {4}{3}}}{16}+\frac {3 \left (3-2 x \right )^{\frac {7}{3}}}{28}\)	\(20\)
pseudoelliptic	\(\frac {3 \left (3-2 x \right )^{\frac {1}{3}} \left (16 x^{2}+190 x -321\right )}{112}\)	\(20\)
risch	\(-\frac {3 \left (16 x^{2}+190 x -321\right ) \left (-3+2 x \right )}{112 \left (3-2 x \right )^{\frac {2}{3}}}\)	\(25\)
meijerg	\(7 \,3^{\frac {1}{3}} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},1;2;\frac {2 x}{3}\right )+\frac {3^{\frac {1}{3}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {1}{3},2;3;\frac {2 x}{3}\right )}{2}\)	\(34\)

[In]

int((3-2*x)^(1/3)*(7+x),x,method=_RETURNVERBOSE)

[Out]

-3/112*(8*x+107)*(3-2*x)^(4/3)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \sqrt [3]{3-2 x} (7+x) \, dx=\frac {3}{112} \, {\left (16 \, x^{2} + 190 \, x - 321\right )} {\left (-2 \, x + 3\right )}^{\frac {1}{3}} \]

[In]

integrate((3-2*x)^(1/3)*(7+x),x, algorithm="fricas")

[Out]

3/112*(16*x^2 + 190*x - 321)*(-2*x + 3)^(1/3)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.15 \[ \int \sqrt [3]{3-2 x} (7+x) \, dx=\begin {cases} \frac {3 \left (x + 7\right )^{2} \sqrt [3]{2 x - 3} e^{\frac {i \pi }{3}}}{7} - \frac {51 \left (x + 7\right ) \sqrt [3]{2 x - 3} e^{\frac {i \pi }{3}}}{56} - \frac {2601 \sqrt [3]{2 x - 3} e^{\frac {i \pi }{3}}}{112} & \text {for}\: \left |{x + 7}\right | > \frac {17}{2} \\\frac {3 \sqrt [3]{3 - 2 x} \left (x + 7\right )^{2}}{7} - \frac {51 \sqrt [3]{3 - 2 x} \left (x + 7\right )}{56} - \frac {2601 \sqrt [3]{3 - 2 x}}{112} & \text {otherwise} \end {cases} \]

[In]

integrate((3-2*x)**(1/3)*(7+x),x)

[Out]

Piecewise((3*(x + 7)**2*(2*x - 3)**(1/3)*exp(I*pi/3)/7 - 51*(x + 7)*(2*x - 3)**(1/3)*exp(I*pi/3)/56 - 2601*(2*
x - 3)**(1/3)*exp(I*pi/3)/112, Abs(x + 7) > 17/2), (3*(3 - 2*x)**(1/3)*(x + 7)**2/7 - 51*(3 - 2*x)**(1/3)*(x +
7)/56 - 2601*(3 - 2*x)**(1/3)/112, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \sqrt [3]{3-2 x} (7+x) \, dx=\frac {3}{28} \, {\left (-2 \, x + 3\right )}^{\frac {7}{3}} - \frac {51}{16} \, {\left (-2 \, x + 3\right )}^{\frac {4}{3}} \]

[In]

integrate((3-2*x)^(1/3)*(7+x),x, algorithm="maxima")

[Out]

3/28*(-2*x + 3)^(7/3) - 51/16*(-2*x + 3)^(4/3)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \sqrt [3]{3-2 x} (7+x) \, dx=\frac {3}{28} \, {\left (2 \, x - 3\right )}^{2} {\left (-2 \, x + 3\right )}^{\frac {1}{3}} - \frac {51}{16} \, {\left (-2 \, x + 3\right )}^{\frac {4}{3}} \]

[In]

integrate((3-2*x)^(1/3)*(7+x),x, algorithm="giac")

[Out]

3/28*(2*x - 3)^2*(-2*x + 3)^(1/3) - 51/16*(-2*x + 3)^(4/3)

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \sqrt [3]{3-2 x} (7+x) \, dx=-\frac {3\,{\left (3-2\,x\right )}^{4/3}\,\left (8\,x+107\right )}{112} \]

[In]

int((3 - 2*x)^(1/3)*(x + 7),x)

[Out]

-(3*(3 - 2*x)^(4/3)*(8*x + 107))/112