3.3.59 \(\int x^7 (1+x^4)^{2/3} \, dx\)

Optimal. Leaf size=25 \[ \frac {3}{160} \left (x^4+1\right )^{2/3} \left (5 x^8+2 x^4-3\right ) \]

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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 {3}{32} \left (x^4+1\right )^{8/3}-\frac {3}{20} \left (x^4+1\right )^{5/3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(1 + x^4)^(2/3),x]

[Out]

(-3*(1 + x^4)^(5/3))/20 + (3*(1 + x^4)^(8/3))/32

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^7 \left (1+x^4\right )^{2/3} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x (1+x)^{2/3} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-(1+x)^{2/3}+(1+x)^{5/3}\right ) \, dx,x,x^4\right )\\ &=-\frac {3}{20} \left (1+x^4\right )^{5/3}+\frac {3}{32} \left (1+x^4\right )^{8/3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {3}{160} \left (x^4+1\right )^{5/3} \left (5 x^4-3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(1 + x^4)^(2/3),x]

[Out]

(3*(1 + x^4)^(5/3)*(-3 + 5*x^4))/160

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IntegrateAlgebraic [A]  time = 0.02, size = 20, normalized size = 0.80 \begin {gather*} \frac {3}{160} \left (1+x^4\right )^{5/3} \left (-3+5 x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^7*(1 + x^4)^(2/3),x]

[Out]

(3*(1 + x^4)^(5/3)*(-3 + 5*x^4))/160

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fricas [A]  time = 0.44, size = 21, normalized size = 0.84 \begin {gather*} \frac {3}{160} \, {\left (5 \, x^{8} + 2 \, x^{4} - 3\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(x^4+1)^(2/3),x, algorithm="fricas")

[Out]

3/160*(5*x^8 + 2*x^4 - 3)*(x^4 + 1)^(2/3)

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giac [A]  time = 0.45, size = 19, normalized size = 0.76 \begin {gather*} \frac {3}{32} \, {\left (x^{4} + 1\right )}^{\frac {8}{3}} - \frac {3}{20} \, {\left (x^{4} + 1\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(x^4+1)^(2/3),x, algorithm="giac")

[Out]

3/32*(x^4 + 1)^(8/3) - 3/20*(x^4 + 1)^(5/3)

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maple [A]  time = 0.07, size = 17, normalized size = 0.68

method	result	size
gosper	\(\frac {3 \left (x^{4}+1\right )^{\frac {5}{3}} \left (5 x^{4}-3\right )}{160}\)	\(17\)
meijerg	\(\frac {\hypergeom \left (\left [-\frac {2}{3}, 2\right ], \relax [3], -x^{4}\right ) x^{8}}{8}\)	\(17\)
trager	\(\left (\frac {3}{32} x^{8}+\frac {3}{80} x^{4}-\frac {9}{160}\right ) \left (x^{4}+1\right )^{\frac {2}{3}}\)	\(21\)
risch	\(\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}} \left (5 x^{8}+2 x^{4}-3\right )}{160}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(x^4+1)^(2/3),x,method=_RETURNVERBOSE)

[Out]

3/160*(x^4+1)^(5/3)*(5*x^4-3)

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maxima [A]  time = 0.44, size = 19, normalized size = 0.76 \begin {gather*} \frac {3}{32} \, {\left (x^{4} + 1\right )}^{\frac {8}{3}} - \frac {3}{20} \, {\left (x^{4} + 1\right )}^{\frac {5}{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(x^4+1)^(2/3),x, algorithm="maxima")

[Out]

3/32*(x^4 + 1)^(8/3) - 3/20*(x^4 + 1)^(5/3)

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mupad [B]  time = 0.22, size = 20, normalized size = 0.80 \begin {gather*} {\left (x^4+1\right )}^{2/3}\,\left (\frac {3\,x^8}{32}+\frac {3\,x^4}{80}-\frac {9}{160}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(x^4 + 1)^(2/3),x)

[Out]

(x^4 + 1)^(2/3)*((3*x^4)/80 + (3*x^8)/32 - 9/160)

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sympy [A]  time = 1.00, size = 41, normalized size = 1.64 \begin {gather*} \frac {3 x^{8} \left (x^{4} + 1\right )^{\frac {2}{3}}}{32} + \frac {3 x^{4} \left (x^{4} + 1\right )^{\frac {2}{3}}}{80} - \frac {9 \left (x^{4} + 1\right )^{\frac {2}{3}}}{160} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(x**4+1)**(2/3),x)

[Out]

3*x**8*(x**4 + 1)**(2/3)/32 + 3*x**4*(x**4 + 1)**(2/3)/80 - 9*(x**4 + 1)**(2/3)/160

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