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

Optimal. Leaf size=25 \[ \frac {4}{231} \left (x^3-1\right )^{3/4} \left (7 x^6-3 x^3-4\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {4}{33} \left (x^3-1\right )^{11/4}+\frac {4}{21} \left (x^3-1\right )^{7/4} \end {gather*}

Antiderivative was successfully verified.

[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 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])

Rule 266

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*} \int x^5 \left (-1+x^3\right )^{3/4} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int (-1+x)^{3/4} x \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {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]  time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {4}{231} \left (x^3-1\right )^{7/4} \left (7 x^3+4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^5*(-1 + x^3)^(3/4),x]

[Out]

(4*(-1 + x^3)^(7/4)*(4 + 7*x^3))/231

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 20, normalized size = 0.80 \begin {gather*} \frac {4}{231} \left (-1+x^3\right )^{7/4} \left (4+7 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5*(-1 + x^3)^(3/4),x]

[Out]

(4*(-1 + x^3)^(7/4)*(4 + 7*x^3))/231

________________________________________________________________________________________

fricas [A]  time = 0.45, size = 21, normalized size = 0.84 \begin {gather*} \frac {4}{231} \, {\left (7 \, x^{6} - 3 \, x^{3} - 4\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.61, size = 19, normalized size = 0.76 \begin {gather*} \frac {4}{33} \, {\left (x^{3} - 1\right )}^{\frac {11}{4}} + \frac {4}{21} \, {\left (x^{3} - 1\right )}^{\frac {7}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 21, normalized size = 0.84

method	result	size
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	\(\frac {4 \left (-1+x \right ) \left (x^{2}+x +1\right ) \left (7 x^{3}+4\right ) \left (x^{3}-1\right )^{\frac {3}{4}}}{231}\)	\(26\)
meijerg	\(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {3}{4}, 2\right ], \relax [3], x^{3}\right ) x^{6}}{6 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {3}{4}}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 19, normalized size = 0.76 \begin {gather*} \frac {4}{33} \, {\left (x^{3} - 1\right )}^{\frac {11}{4}} + \frac {4}{21} \, {\left (x^{3} - 1\right )}^{\frac {7}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 21, normalized size = 0.84 \begin {gather*} -{\left (x^3-1\right )}^{3/4}\,\left (-\frac {4\,x^6}{33}+\frac {4\,x^3}{77}+\frac {16}{231}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 1.25, size = 41, normalized size = 1.64 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________