3.10.60 \(\int \frac {x^7}{\sqrt {16-x^4}} \, dx\) [960]

Optimal. Leaf size=29 \[ -8 \sqrt {16-x^4}+\frac {1}{6} \left (16-x^4\right )^{3/2} \]

[Out]

1/6*(-x^4+16)^(3/2)-8*(-x^4+16)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 29, 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 (16-x^4\right )^{3/2}-8 \sqrt {16-x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7/Sqrt[16 - x^4],x]

[Out]

-8*Sqrt[16 - x^4] + (16 - 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 {16-x^4}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{\sqrt {16-x}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {16}{\sqrt {16-x}}-\sqrt {16-x}\right ) \, dx,x,x^4\right )\\ &=-8 \sqrt {16-x^4}+\frac {1}{6} \left (16-x^4\right )^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{6} \left (-32-x^4\right ) \sqrt {16-x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7/Sqrt[16 - x^4],x]

[Out]

((-32 - x^4)*Sqrt[16 - x^4])/6

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Maple [A]
time = 0.16, size = 27, normalized size = 0.93

method	result	size
trager	\(\left (-\frac {x^{4}}{6}-\frac {16}{3}\right ) \sqrt {-x^{4}+16}\)	\(18\)
risch	\(\frac {\left (x^{4}+32\right ) \left (x^{4}-16\right )}{6 \sqrt {-x^{4}+16}}\)	\(22\)
default	\(\frac {\left (x^{2}-4\right ) \left (x^{2}+4\right ) \left (x^{4}+32\right )}{6 \sqrt {-x^{4}+16}}\)	\(27\)
elliptic	\(\frac {\left (x^{2}-4\right ) \left (x^{2}+4\right ) \left (x^{4}+32\right )}{6 \sqrt {-x^{4}+16}}\)	\(27\)
gosper	\(\frac {\left (x -2\right ) \left (2+x \right ) \left (x^{2}+4\right ) \left (x^{4}+32\right )}{6 \sqrt {-x^{4}+16}}\)	\(28\)
meijerg	\(\frac {\frac {64 \sqrt {\pi }}{3}-\frac {8 \sqrt {\pi }\, \left (8+\frac {x^{4}}{4}\right ) \sqrt {1-\frac {x^{4}}{16}}}{3}}{\sqrt {\pi }}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(-x^4+16)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(x^2-4)*(x^2+4)*(x^4+32)/(-x^4+16)^(1/2)

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Maxima [A]
time = 0.29, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, {\left (-x^{4} + 16\right )}^{\frac {3}{2}} - 8 \, \sqrt {-x^{4} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^4+16)^(1/2),x, algorithm="maxima")

[Out]

1/6*(-x^4 + 16)^(3/2) - 8*sqrt(-x^4 + 16)

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Fricas [A]
time = 0.34, size = 16, normalized size = 0.55 \begin {gather*} -\frac {1}{6} \, {\left (x^{4} + 32\right )} \sqrt {-x^{4} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^4+16)^(1/2),x, algorithm="fricas")

[Out]

-1/6*(x^4 + 32)*sqrt(-x^4 + 16)

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Sympy [A]
time = 0.12, size = 26, normalized size = 0.90 \begin {gather*} - \frac {x^{4} \sqrt {16 - x^{4}}}{6} - \frac {16 \sqrt {16 - x^{4}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(-x**4+16)**(1/2),x)

[Out]

-x**4*sqrt(16 - x**4)/6 - 16*sqrt(16 - x**4)/3

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Giac [A]
time = 1.52, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{6} \, {\left (-x^{4} + 16\right )}^{\frac {3}{2}} - 8 \, \sqrt {-x^{4} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(-x^4+16)^(1/2),x, algorithm="giac")

[Out]

1/6*(-x^4 + 16)^(3/2) - 8*sqrt(-x^4 + 16)

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Mupad [B]
time = 1.19, size = 16, normalized size = 0.55 \begin {gather*} -\frac {\sqrt {16-x^4}\,\left (x^4+32\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(16 - x^4)^(1/2),x)

[Out]

-((16 - x^4)^(1/2)*(x^4 + 32))/6

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