3.875 \(\int \frac{x^7}{\sqrt{1-x^4}} \, dx\)

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

[Out]

-Sqrt[1 - x^4]/2 + (1 - x^4)^(3/2)/6

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Rubi [A]  time = 0.0146928, antiderivative size = 31, normalized size of antiderivative = 1., 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 = {266, 43} \[ \frac{1}{6} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]

Antiderivative was successfully verified.

[In]

Int[x^7/Sqrt[1 - x^4],x]

[Out]

-Sqrt[1 - x^4]/2 + (1 - x^4)^(3/2)/6

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]]

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])

Rubi steps

\begin{align*} \int \frac{x^7}{\sqrt{1-x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{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*}

Mathematica [A]  time = 0.0058819, size = 20, normalized size = 0.65 \[ -\frac{1}{6} \sqrt{1-x^4} \left (x^4+2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/Sqrt[1 - x^4],x]

[Out]

-(Sqrt[1 - x^4]*(2 + x^4))/6

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Maple [A]  time = 0.004, size = 28, normalized size = 0.9 \begin{align*}{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ({x}^{4}+2 \right ) }{6}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.975793, size = 31, normalized size = 1. \begin{align*} \frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}

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 = 1.48706, size = 42, normalized size = 1.35 \begin{align*} -\frac{1}{6} \,{\left (x^{4} + 2\right )} \sqrt{-x^{4} + 1} \end{align*}

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 = 0.609201, size = 24, normalized size = 0.77 \begin{align*} - \frac{x^{4} \sqrt{1 - x^{4}}}{6} - \frac{\sqrt{1 - x^{4}}}{3} \end{align*}

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 = 1.12525, size = 31, normalized size = 1. \begin{align*} \frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{4} + 1} \end{align*}

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)