3.5.39 \(\int \frac {x^8}{\sqrt {1+x^3}} \, dx\) [439]

Optimal. Leaf size=40 \[ \frac {2 \sqrt {1+x^3}}{3}-\frac {4}{9} \left (1+x^3\right )^{3/2}+\frac {2}{15} \left (1+x^3\right )^{5/2} \]

[Out]

-4/9*(x^3+1)^(3/2)+2/15*(x^3+1)^(5/2)+2/3*(x^3+1)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {272, 45} \begin {gather*} \frac {2}{15} \left (x^3+1\right )^{5/2}-\frac {4}{9} \left (x^3+1\right )^{3/2}+\frac {2 \sqrt {x^3+1}}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^8/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3])/3 - (4*(1 + x^3)^(3/2))/9 + (2*(1 + x^3)^(5/2))/15

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*} \int \frac {x^8}{\sqrt {1+x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{\sqrt {1+x}}-2 \sqrt {1+x}+(1+x)^{3/2}\right ) \, dx,x,x^3\right )\\ &=\frac {2 \sqrt {1+x^3}}{3}-\frac {4}{9} \left (1+x^3\right )^{3/2}+\frac {2}{15} \left (1+x^3\right )^{5/2}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 25, normalized size = 0.62 \begin {gather*} \frac {2}{45} \sqrt {1+x^3} \left (8-4 x^3+3 x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^8/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3]*(8 - 4*x^3 + 3*x^6))/45

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Maple [A]
time = 0.12, size = 35, normalized size = 0.88

method	result	size
trager	\(\left (\frac {2}{15} x^{6}-\frac {8}{45} x^{3}+\frac {16}{45}\right ) \sqrt {x^{3}+1}\)	\(21\)
risch	\(\frac {2 \left (3 x^{6}-4 x^{3}+8\right ) \sqrt {x^{3}+1}}{45}\)	\(22\)
gosper	\(\frac {2 \left (x +1\right ) \left (x^{2}-x +1\right ) \left (3 x^{6}-4 x^{3}+8\right )}{45 \sqrt {x^{3}+1}}\)	\(33\)
default	\(\frac {2 x^{6} \sqrt {x^{3}+1}}{15}-\frac {8 x^{3} \sqrt {x^{3}+1}}{45}+\frac {16 \sqrt {x^{3}+1}}{45}\)	\(35\)
elliptic	\(\frac {2 x^{6} \sqrt {x^{3}+1}}{15}-\frac {8 x^{3} \sqrt {x^{3}+1}}{45}+\frac {16 \sqrt {x^{3}+1}}{45}\)	\(35\)
meijerg	\(\frac {-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 x^{6}-8 x^{3}+16\right ) \sqrt {x^{3}+1}}{15}}{3 \sqrt {\pi }}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/15*x^6*(x^3+1)^(1/2)-8/45*x^3*(x^3+1)^(1/2)+16/45*(x^3+1)^(1/2)

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Maxima [A]
time = 0.29, size = 28, normalized size = 0.70 \begin {gather*} \frac {2}{15} \, {\left (x^{3} + 1\right )}^{\frac {5}{2}} - \frac {4}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {x^{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^3+1)^(1/2),x, algorithm="maxima")

[Out]

2/15*(x^3 + 1)^(5/2) - 4/9*(x^3 + 1)^(3/2) + 2/3*sqrt(x^3 + 1)

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Fricas [A]
time = 0.34, size = 21, normalized size = 0.52 \begin {gather*} \frac {2}{45} \, {\left (3 \, x^{6} - 4 \, x^{3} + 8\right )} \sqrt {x^{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^3+1)^(1/2),x, algorithm="fricas")

[Out]

2/45*(3*x^6 - 4*x^3 + 8)*sqrt(x^3 + 1)

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Sympy [A]
time = 0.14, size = 41, normalized size = 1.02 \begin {gather*} \frac {2 x^{6} \sqrt {x^{3} + 1}}{15} - \frac {8 x^{3} \sqrt {x^{3} + 1}}{45} + \frac {16 \sqrt {x^{3} + 1}}{45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(x**3+1)**(1/2),x)

[Out]

2*x**6*sqrt(x**3 + 1)/15 - 8*x**3*sqrt(x**3 + 1)/45 + 16*sqrt(x**3 + 1)/45

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Giac [A]
time = 0.95, size = 28, normalized size = 0.70 \begin {gather*} \frac {2}{15} \, {\left (x^{3} + 1\right )}^{\frac {5}{2}} - \frac {4}{9} \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} + \frac {2}{3} \, \sqrt {x^{3} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(x^3+1)^(1/2),x, algorithm="giac")

[Out]

2/15*(x^3 + 1)^(5/2) - 4/9*(x^3 + 1)^(3/2) + 2/3*sqrt(x^3 + 1)

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Mupad [B]
time = 0.03, size = 21, normalized size = 0.52 \begin {gather*} \frac {2\,\sqrt {x^3+1}\,\left (3\,x^6-4\,x^3+8\right )}{45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(x^3 + 1)^(1/2),x)

[Out]

(2*(x^3 + 1)^(1/2)*(3*x^6 - 4*x^3 + 8))/45

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