3.3.73 \(\int \frac {-3 (-3 x+x^3)+(-3 x+x^3)^3}{\sqrt {-4+x^2}} \, dx\) [273]

Optimal. Leaf size=56 \[ \sqrt {-4+x^2}+\frac {10}{3} \left (-4+x^2\right )^{3/2}+3 \left (-4+x^2\right )^{5/2}+\left (-4+x^2\right )^{7/2}+\frac {1}{9} \left (-4+x^2\right )^{9/2} \]

[Out]

(x^2-4)^(1/2)+10/3*(x^2-4)^(3/2)+3*(x^2-4)^(5/2)+(x^2-4)^(7/2)+1/9*(x^2-4)^(9/2)

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Rubi [A]
time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1825, 1813, 1864} \begin {gather*} \frac {1}{9} \left (x^2-4\right )^{9/2}+\left (x^2-4\right )^{7/2}+3 \left (x^2-4\right )^{5/2}+\frac {10}{3} \left (x^2-4\right )^{3/2}+\sqrt {x^2-4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3*(-3*x + x^3) + (-3*x + x^3)^3)/Sqrt[-4 + x^2],x]

[Out]

Sqrt[-4 + x^2] + (10*(-4 + x^2)^(3/2))/3 + 3*(-4 + x^2)^(5/2) + (-4 + x^2)^(7/2) + (-4 + x^2)^(9/2)/9

Rule 1813

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1825

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[x*PolynomialQuotient[Pq, x, x]*(a + b*x^2)^p, x] /; Fre
eQ[{a, b, p}, x] && PolyQ[Pq, x] && EqQ[Coeff[Pq, x, 0], 0] &&  !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \frac {x \left (9-30 x^2+27 x^4-9 x^6+x^8\right )}{\sqrt {-4+x^2}} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {9-30 x+27 x^2-9 x^3+x^4}{\sqrt {-4+x}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{\sqrt {-4+x}}+10 \sqrt {-4+x}+15 (-4+x)^{3/2}+7 (-4+x)^{5/2}+(-4+x)^{7/2}\right ) \, dx,x,x^2\right )\\ &=\sqrt {-4+x^2}+\frac {10}{3} \left (-4+x^2\right )^{3/2}+3 \left (-4+x^2\right )^{5/2}+\left (-4+x^2\right )^{7/2}+\frac {1}{9} \left (-4+x^2\right )^{9/2}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 33, normalized size = 0.59 \begin {gather*} \frac {1}{9} \sqrt {-4+x^2} \left (1-10 x^2+15 x^4-7 x^6+x^8\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*(-3*x + x^3) + (-3*x + x^3)^3)/Sqrt[-4 + x^2],x]

[Out]

(Sqrt[-4 + x^2]*(1 - 10*x^2 + 15*x^4 - 7*x^6 + x^8))/9

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Maple [A]
time = 0.12, size = 59, normalized size = 1.05

method	result	size
risch	\(\frac {\left (x^{8}-7 x^{6}+15 x^{4}-10 x^{2}+1\right ) \sqrt {x^{2}-4}}{9}\)	\(30\)
pseudoelliptic	\(\frac {\left (x^{8}-7 x^{6}+15 x^{4}-10 x^{2}+1\right ) \sqrt {x^{2}-4}}{9}\)	\(30\)
trager	\(\left (\frac {1}{9} x^{8}-\frac {7}{9} x^{6}+\frac {5}{3} x^{4}-\frac {10}{9} x^{2}+\frac {1}{9}\right ) \sqrt {x^{2}-4}\)	\(31\)
gosper	\(\frac {\left (x^{8}-7 x^{6}+15 x^{4}-10 x^{2}+1\right ) \left (x -2\right ) \left (2+x \right )}{9 \sqrt {x^{2}-4}}\)	\(36\)
default	\(\frac {x^{8} \sqrt {x^{2}-4}}{9}-\frac {7 x^{6} \sqrt {x^{2}-4}}{9}+\frac {5 x^{4} \sqrt {x^{2}-4}}{3}-\frac {10 x^{2} \sqrt {x^{2}-4}}{9}+\frac {\sqrt {x^{2}-4}}{9}\)	\(59\)
meijerg	\(-\frac {256 \sqrt {-\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (\frac {35}{128} x^{8}+\frac {5}{4} x^{6}+6 x^{4}+32 x^{2}+256\right ) \sqrt {-\frac {x^{2}}{4}+1}}{315}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}}-\frac {576 \sqrt {-\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (\frac {5}{8} x^{6}+3 x^{4}+16 x^{2}+128\right ) \sqrt {-\frac {x^{2}}{4}+1}}{140}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}}-\frac {432 \sqrt {-\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {3}{8} x^{4}+2 x^{2}+16\right ) \sqrt {-\frac {x^{2}}{4}+1}}{15}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}}-\frac {120 \sqrt {-\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (x^{2}+8\right ) \sqrt {-\frac {x^{2}}{4}+1}}{6}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}}-\frac {9 \sqrt {-\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-\frac {x^{2}}{4}+1}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+\frac {x^{2}}{4}\right )}}\)	\(293\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3-3*x)^3-3*x^3+9*x)/(x^2-4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*x^8*(x^2-4)^(1/2)-7/9*x^6*(x^2-4)^(1/2)+5/3*x^4*(x^2-4)^(1/2)-10/9*x^2*(x^2-4)^(1/2)+1/9*(x^2-4)^(1/2)

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Maxima [A]
time = 0.35, size = 58, normalized size = 1.04 \begin {gather*} \frac {1}{9} \, \sqrt {x^{2} - 4} x^{8} - \frac {7}{9} \, \sqrt {x^{2} - 4} x^{6} + \frac {5}{3} \, \sqrt {x^{2} - 4} x^{4} - \frac {10}{9} \, \sqrt {x^{2} - 4} x^{2} + \frac {1}{9} \, \sqrt {x^{2} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-3*x)^3-3*x^3+9*x)/(x^2-4)^(1/2),x, algorithm="maxima")

[Out]

1/9*sqrt(x^2 - 4)*x^8 - 7/9*sqrt(x^2 - 4)*x^6 + 5/3*sqrt(x^2 - 4)*x^4 - 10/9*sqrt(x^2 - 4)*x^2 + 1/9*sqrt(x^2
- 4)

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Fricas [A]
time = 0.57, size = 29, normalized size = 0.52 \begin {gather*} \frac {1}{9} \, {\left (x^{8} - 7 \, x^{6} + 15 \, x^{4} - 10 \, x^{2} + 1\right )} \sqrt {x^{2} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-3*x)^3-3*x^3+9*x)/(x^2-4)^(1/2),x, algorithm="fricas")

[Out]

1/9*(x^8 - 7*x^6 + 15*x^4 - 10*x^2 + 1)*sqrt(x^2 - 4)

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Sympy [A]
time = 0.97, size = 68, normalized size = 1.21 \begin {gather*} \frac {x^{8} \sqrt {x^{2} - 4}}{9} - \frac {7 x^{6} \sqrt {x^{2} - 4}}{9} + \frac {5 x^{4} \sqrt {x^{2} - 4}}{3} - \frac {10 x^{2} \sqrt {x^{2} - 4}}{9} + \frac {\sqrt {x^{2} - 4}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3-3*x)**3-3*x**3+9*x)/(x**2-4)**(1/2),x)

[Out]

x**8*sqrt(x**2 - 4)/9 - 7*x**6*sqrt(x**2 - 4)/9 + 5*x**4*sqrt(x**2 - 4)/3 - 10*x**2*sqrt(x**2 - 4)/9 + sqrt(x*
*2 - 4)/9

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Giac [A]
time = 0.41, size = 42, normalized size = 0.75 \begin {gather*} \frac {1}{9} \, {\left (x^{2} - 4\right )}^{\frac {9}{2}} + {\left (x^{2} - 4\right )}^{\frac {7}{2}} + 3 \, {\left (x^{2} - 4\right )}^{\frac {5}{2}} + \frac {10}{3} \, {\left (x^{2} - 4\right )}^{\frac {3}{2}} + \sqrt {x^{2} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3-3*x)^3-3*x^3+9*x)/(x^2-4)^(1/2),x, algorithm="giac")

[Out]

1/9*(x^2 - 4)^(9/2) + (x^2 - 4)^(7/2) + 3*(x^2 - 4)^(5/2) + 10/3*(x^2 - 4)^(3/2) + sqrt(x^2 - 4)

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Mupad [B]
time = 0.13, size = 30, normalized size = 0.54 \begin {gather*} \sqrt {x^2-4}\,\left (\frac {x^8}{9}-\frac {7\,x^6}{9}+\frac {5\,x^4}{3}-\frac {10\,x^2}{9}+\frac {1}{9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((3*x - x^3)^3 - 9*x + 3*x^3)/(x^2 - 4)^(1/2),x)

[Out]

(x^2 - 4)^(1/2)*((5*x^4)/3 - (10*x^2)/9 - (7*x^6)/9 + x^8/9 + 1/9)

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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(((x^3-3*x)^3-3*x^3+9*x)/(x^2-4)^(1/2),x)

[Out]

not solved

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