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/(-x^4+1)^(1/2)+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 = {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 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 \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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 0.84, size = 23, normalized size = 0.74 \[ \frac {{\left (x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{2 \, {\left (x^{4} - 1\right )}} \]

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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giac [A]  time = 0.15, size = 23, normalized size = 0.74 \[ \frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2 \, \sqrt {-x^{4} + 1}} \]

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)

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maple [A]  time = 0.00, size = 28, normalized size = 0.90 \[ \frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}-2\right )}{2 \left (-x^{4}+1\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.30, size = 23, normalized size = 0.74 \[ \frac {1}{2} \, \sqrt {-x^{4} + 1} + \frac {1}{2 \, \sqrt {-x^{4} + 1}} \]

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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mupad [B]  time = 1.17, size = 16, normalized size = 0.52 \[ -\frac {x^4-2}{2\,\sqrt {1-x^4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(1 - x^4)^(3/2),x)

[Out]

-(x^4 - 2)/(2*(1 - x^4)^(1/2))

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sympy [A]  time = 1.20, size = 22, normalized size = 0.71 \[ - \frac {x^{4}}{2 \sqrt {1 - x^{4}}} + \frac {1}{\sqrt {1 - x^{4}}} \]

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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