3.9.95 \(\int \frac {x^7}{(1-x^4)^{3/2}} \, dx\) [895]

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

[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 = {272, 45} \begin {gather*} \frac {\sqrt {1-x^4}}{2}+\frac {1}{2 \sqrt {1-x^4}} \end {gather*}

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 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}{\left (1-x^4\right )^{3/2}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{(1-x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {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 = 0.02, size = 20, normalized size = 0.65 \begin {gather*} -\frac {-2+x^4}{2 \sqrt {1-x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/(1 - x^4)^(3/2),x]

[Out]

-1/2*(-2 + x^4)/Sqrt[1 - x^4]

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Maple [A]
time = 0.15, size = 17, normalized size = 0.55

method	result	size
default	\(-\frac {x^{4}-2}{2 \sqrt {-x^{4}+1}}\)	\(17\)
risch	\(-\frac {x^{4}-2}{2 \sqrt {-x^{4}+1}}\)	\(17\)
elliptic	\(-\frac {x^{4}-2}{2 \sqrt {-x^{4}+1}}\)	\(17\)
trager	\(\frac {\left (x^{4}-2\right ) \sqrt {-x^{4}+1}}{2 x^{4}-2}\)	\(24\)
gosper	\(\frac {\left (x -1\right ) \left (x +1\right ) \left (x^{2}+1\right ) \left (x^{4}-2\right )}{2 \left (-x^{4}+1\right )^{\frac {3}{2}}}\)	\(28\)
meijerg	\(\frac {-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 x^{4}+8\right )}{4 \sqrt {-x^{4}+1}}}{2 \sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{4} + 1} + \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)^(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 = 0.36, size = 23, normalized size = 0.74 \begin {gather*} \frac {{\left (x^{4} - 2\right )} \sqrt {-x^{4} + 1}}{2 \, {\left (x^{4} - 1\right )}} \end {gather*}

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 = 0.21, size = 34, normalized size = 1.10 \begin {gather*} \frac {x^{4} \sqrt {1 - x^{4}}}{2 x^{4} - 2} - \frac {2 \sqrt {1 - x^{4}}}{2 x^{4} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 1.47, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{2} \, \sqrt {-x^{4} + 1} + \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)^(3/2),x, algorithm="giac")

[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 \begin {gather*} -\frac {x^4-2}{2\,\sqrt {1-x^4}} \end {gather*}

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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