\(\int x^5 (-1+x^3)^{3/4} \, dx\) [255]

Optimal result

Integrand size = 13, antiderivative size = 25 \[ \int x^5 \left (-1+x^3\right )^{3/4} \, dx=\frac {4}{231} \left (-1+x^3\right )^{3/4} \left (-4-3 x^3+7 x^6\right ) \]

[Out]

4/231*(x^3-1)^(3/4)*(7*x^6-3*x^3-4)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int x^5 \left (-1+x^3\right )^{3/4} \, dx=\frac {4}{33} \left (x^3-1\right )^{11/4}+\frac {4}{21} \left (x^3-1\right )^{7/4} \]

[In]

Int[x^5*(-1 + x^3)^(3/4),x]

[Out]

(4*(-1 + x^3)^(7/4))/21 + (4*(-1 + x^3)^(11/4))/33

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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int (-1+x)^{3/4} x \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left ((-1+x)^{3/4}+(-1+x)^{7/4}\right ) \, dx,x,x^3\right ) \\ & = \frac {4}{21} \left (-1+x^3\right )^{7/4}+\frac {4}{33} \left (-1+x^3\right )^{11/4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int x^5 \left (-1+x^3\right )^{3/4} \, dx=\frac {4}{231} \left (-1+x^3\right )^{7/4} \left (4+7 x^3\right ) \]

[In]

Integrate[x^5*(-1 + x^3)^(3/4),x]

[Out]

(4*(-1 + x^3)^(7/4)*(4 + 7*x^3))/231

Maple [A] (verified)

Time = 0.85 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68

method	result	size
pseudoelliptic	\(\frac {4 \left (x^{3}-1\right )^{\frac {7}{4}} \left (7 x^{3}+4\right )}{231}\)	\(17\)
trager	\(\left (\frac {4}{33} x^{6}-\frac {4}{77} x^{3}-\frac {16}{231}\right ) \left (x^{3}-1\right )^{\frac {3}{4}}\)	\(21\)
risch	\(\frac {4 \left (x^{3}-1\right )^{\frac {3}{4}} \left (7 x^{6}-3 x^{3}-4\right )}{231}\)	\(22\)
gosper	\(\frac {4 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (7 x^{3}+4\right ) \left (x^{3}-1\right )^{\frac {3}{4}}}{231}\)	\(26\)
meijerg	\(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {3}{4}} x^{6} \operatorname {hypergeom}\left (\left [-\frac {3}{4}, 2\right ], \left [3\right ], x^{3}\right )}{6 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {3}{4}}}\)	\(33\)

[In]

int(x^5*(x^3-1)^(3/4),x,method=_RETURNVERBOSE)

[Out]

4/231*(x^3-1)^(7/4)*(7*x^3+4)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int x^5 \left (-1+x^3\right )^{3/4} \, dx=\frac {4}{231} \, {\left (7 \, x^{6} - 3 \, x^{3} - 4\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}} \]

[In]

integrate(x^5*(x^3-1)^(3/4),x, algorithm="fricas")

[Out]

4/231*(7*x^6 - 3*x^3 - 4)*(x^3 - 1)^(3/4)

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int x^5 \left (-1+x^3\right )^{3/4} \, dx=\frac {4 x^{6} \left (x^{3} - 1\right )^{\frac {3}{4}}}{33} - \frac {4 x^{3} \left (x^{3} - 1\right )^{\frac {3}{4}}}{77} - \frac {16 \left (x^{3} - 1\right )^{\frac {3}{4}}}{231} \]

[In]

integrate(x**5*(x**3-1)**(3/4),x)

[Out]

4*x**6*(x**3 - 1)**(3/4)/33 - 4*x**3*(x**3 - 1)**(3/4)/77 - 16*(x**3 - 1)**(3/4)/231

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^5 \left (-1+x^3\right )^{3/4} \, dx=\frac {4}{33} \, {\left (x^{3} - 1\right )}^{\frac {11}{4}} + \frac {4}{21} \, {\left (x^{3} - 1\right )}^{\frac {7}{4}} \]

[In]

integrate(x^5*(x^3-1)^(3/4),x, algorithm="maxima")

[Out]

4/33*(x^3 - 1)^(11/4) + 4/21*(x^3 - 1)^(7/4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int x^5 \left (-1+x^3\right )^{3/4} \, dx=\frac {4}{33} \, {\left (x^{3} - 1\right )}^{\frac {11}{4}} + \frac {4}{21} \, {\left (x^{3} - 1\right )}^{\frac {7}{4}} \]

[In]

integrate(x^5*(x^3-1)^(3/4),x, algorithm="giac")

[Out]

4/33*(x^3 - 1)^(11/4) + 4/21*(x^3 - 1)^(7/4)

Mupad [B] (verification not implemented)

Time = 5.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int x^5 \left (-1+x^3\right )^{3/4} \, dx=-{\left (x^3-1\right )}^{3/4}\,\left (-\frac {4\,x^6}{33}+\frac {4\,x^3}{77}+\frac {16}{231}\right ) \]

[In]

int(x^5*(x^3 - 1)^(3/4),x)

[Out]

-(x^3 - 1)^(3/4)*((4*x^3)/77 - (4*x^6)/33 + 16/231)