3.895 \(\int \frac{x^7}{(1-x^4)^{3/2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\sqrt{1-x^4}}{2}+\frac{1}{2 \sqrt{1-x^4}} \]

[Out]

1/(2*Sqrt[1 - x^4]) + Sqrt[1 - x^4]/2

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Rubi [A]  time = 0.0138459, 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} \[ \frac{\sqrt{1-x^4}}{2}+\frac{1}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(1 - x^4)^(3/2),x]

[Out]

1/(2*Sqrt[1 - x^4]) + Sqrt[1 - x^4]/2

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

Mathematica [A]  time = 0.0061457, size = 22, normalized size = 0.71 \[ \frac{2-x^4}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 - x^4)^(3/2),x]

[Out]

(2 - x^4)/(2*Sqrt[1 - x^4])

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Maple [A]  time = 0.005, 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 ) }{2} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.01243, size = 31, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{-x^{4} + 1} + \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)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.46528, size = 54, normalized size = 1.74 \begin{align*} \frac{{\left (x^{4} - 2\right )} \sqrt{-x^{4} + 1}}{2 \,{\left (x^{4} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.915426, size = 22, normalized size = 0.71 \begin{align*} - \frac{x^{4}}{2 \sqrt{1 - x^{4}}} + \frac{1}{\sqrt{1 - x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**4/(2*sqrt(1 - x**4)) + 1/sqrt(1 - x**4)

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Giac [A]  time = 1.20726, size = 31, normalized size = 1. \begin{align*} \frac{1}{2} \, \sqrt{-x^{4} + 1} + \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)^(3/2),x, algorithm="giac")

[Out]

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