3.9.75 \(\int \frac {x^7}{\sqrt {1-x^4}} \, dx\) [875]

Optimal. Leaf size=31 \[ -\frac {1}{2} \sqrt {1-x^4}+\frac {1}{6} \left (1-x^4\right )^{3/2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \begin {gather*} \frac {1}{6} \left (1-x^4\right )^{3/2}-\frac {\sqrt {1-x^4}}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/Sqrt[1 - x^4],x]

[Out]

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

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^7}{\sqrt {1-x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{\sqrt {1-x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-\sqrt {1-x}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{2} \sqrt {1-x^4}+\frac {1}{6} \left (1-x^4\right )^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.71 \begin {gather*} \frac {1}{6} \left (-2-x^4\right ) \sqrt {1-x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/Sqrt[1 - x^4],x]

[Out]

((-2 - x^4)*Sqrt[1 - x^4])/6

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Maple [A]
time = 0.16, size = 27, normalized size = 0.87

method	result	size
trager	\(\left (-\frac {x^{4}}{6}-\frac {1}{3}\right ) \sqrt {-x^{4}+1}\)	\(18\)
risch	\(\frac {\left (x^{4}+2\right ) \left (x^{4}-1\right )}{6 \sqrt {-x^{4}+1}}\)	\(22\)
default	\(\frac {\left (x^{2}-1\right ) \left (x^{2}+1\right ) \left (x^{4}+2\right )}{6 \sqrt {-x^{4}+1}}\)	\(27\)
elliptic	\(\frac {\left (x^{2}-1\right ) \left (x^{2}+1\right ) \left (x^{4}+2\right )}{6 \sqrt {-x^{4}+1}}\)	\(27\)
gosper	\(\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}+2\right )}{6 \sqrt {-x^{4}+1}}\)	\(28\)
meijerg	\(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 x^{4}+8\right ) \sqrt {-x^{4}+1}}{6}}{4 \sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{6} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 16, normalized size = 0.52 \begin {gather*} -\frac {1}{6} \, {\left (x^{4} + 2\right )} \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(x^4 + 2)*sqrt(-x^4 + 1)

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Sympy [A]
time = 0.11, size = 24, normalized size = 0.77 \begin {gather*} - \frac {x^{4} \sqrt {1 - x^{4}}}{6} - \frac {\sqrt {1 - x^{4}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**4*sqrt(1 - x**4)/6 - sqrt(1 - x**4)/3

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Giac [A]
time = 2.08, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{6} \, {\left (-x^{4} + 1\right )}^{\frac {3}{2}} - \frac {1}{2} \, \sqrt {-x^{4} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.13, size = 16, normalized size = 0.52 \begin {gather*} -\frac {\sqrt {1-x^4}\,\left (x^4+2\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(1 - x^4)^(1/2),x)

[Out]

-((1 - x^4)^(1/2)*(x^4 + 2))/6

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