3.5.74 \(\int \frac {x^4}{(\sqrt {10}-x^2)^{9/2}} \, dx\) [474]

Optimal. Leaf size=50 \[ \frac {x^5}{7 \sqrt {10} \left (\sqrt {10}-x^2\right )^{7/2}}+\frac {x^5}{175 \left (\sqrt {10}-x^2\right )^{5/2}} \]

[Out]

1/70*x^5*10^(1/2)/(-x^2+10^(1/2))^(7/2)+1/175*x^5/(-x^2+10^(1/2))^(5/2)

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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {277, 270} \begin {gather*} \frac {x^5}{5 \sqrt {10} \left (\sqrt {10}-x^2\right )^{7/2}}-\frac {x^7}{175 \left (\sqrt {10}-x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/(Sqrt[10] - x^2)^(9/2),x]

[Out]

x^5/(5*Sqrt[10]*(Sqrt[10] - x^2)^(7/2)) - x^7/(175*(Sqrt[10] - x^2)^(7/2))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x^4}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx &=\frac {x^5}{5 \sqrt {10} \left (\sqrt {10}-x^2\right )^{7/2}}-\frac {1}{5} \sqrt {\frac {2}{5}} \int \frac {x^6}{\left (\sqrt {10}-x^2\right )^{9/2}} \, dx\\ &=\frac {x^5}{5 \sqrt {10} \left (\sqrt {10}-x^2\right )^{7/2}}-\frac {x^7}{175 \left (\sqrt {10}-x^2\right )^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 0.22, size = 35, normalized size = 0.70 \begin {gather*} -\frac {x^5 \left (-7 \sqrt {10}+2 x^2\right )}{350 \left (\sqrt {10}-x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/(Sqrt[10] - x^2)^(9/2),x]

[Out]

-1/350*(x^5*(-7*Sqrt[10] + 2*x^2))/(Sqrt[10] - x^2)^(7/2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(120\) vs. \(2(36)=72\).
time = 0.14, size = 121, normalized size = 2.42

method	result	size
gosper	\(\frac {x^{5} \left (-2 x^{2}+7 \sqrt {10}\right )}{350 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}\)	\(28\)
meijerg	\(\frac {10^{\frac {3}{4}} x^{5} \left (-\frac {\sqrt {2}\, \sqrt {5}\, x^{2}}{5}+7\right )}{35000 \left (1-\frac {\sqrt {10}\, x^{2}}{10}\right )^{\frac {7}{2}}}\)	\(34\)
risch	\(\frac {2 x^{7}-7 \sqrt {10}\, x^{5}}{350 \left (x^{2}-\sqrt {10}\right )^{3} \sqrt {-x^{2}+\sqrt {10}}}\)	\(39\)
trager	\(-\frac {2 \sqrt {10}\, \left (\sqrt {10}\, x^{2}-35\right ) x^{5} \sqrt {-x^{2}+\sqrt {10}}}{35 \left (\sqrt {10}\, x^{2}-10\right )^{4}}\)	\(40\)
default	\(\frac {x^{3}}{4 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}-\frac {3 \sqrt {10}\, \left (\frac {x}{6 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}-\frac {\sqrt {10}\, \left (\frac {x \sqrt {10}}{70 \left (-x^{2}+\sqrt {10}\right )^{\frac {7}{2}}}+\frac {3 \sqrt {10}\, \left (\frac {x \sqrt {10}}{50 \left (-x^{2}+\sqrt {10}\right )^{\frac {5}{2}}}+\frac {2 \sqrt {10}\, \left (\frac {x \sqrt {10}}{30 \left (-x^{2}+\sqrt {10}\right )^{\frac {3}{2}}}+\frac {x}{15 \sqrt {-x^{2}+\sqrt {10}}}\right )}{25}\right )}{35}\right )}{6}\right )}{4}\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(-x^2+10^(1/2))^(9/2),x,method=_RETURNVERBOSE)

[Out]

1/4*x^3/(-x^2+10^(1/2))^(7/2)-3/4*10^(1/2)*(1/6*x/(-x^2+10^(1/2))^(7/2)-1/6*10^(1/2)*(1/70*x*10^(1/2)/(-x^2+10
^(1/2))^(7/2)+3/35*10^(1/2)*(1/50*x*10^(1/2)/(-x^2+10^(1/2))^(5/2)+2/25*10^(1/2)*(1/30*x*10^(1/2)/(-x^2+10^(1/
2))^(3/2)+1/15*x/(-x^2+10^(1/2))^(1/2)))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (36) = 72\).
time = 0.92, size = 79, normalized size = 1.58 \begin {gather*} \frac {x}{175 \, \sqrt {-x^{2} + \sqrt {10}}} + \frac {\sqrt {10} x}{350 \, {\left (-x^{2} + \sqrt {10}\right )}^{\frac {3}{2}}} + \frac {x^{3}}{4 \, {\left (-x^{2} + \sqrt {10}\right )}^{\frac {7}{2}}} + \frac {3 \, x}{140 \, {\left (-x^{2} + \sqrt {10}\right )}^{\frac {5}{2}}} - \frac {3 \, \sqrt {10} x}{28 \, {\left (-x^{2} + \sqrt {10}\right )}^{\frac {7}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^2+10^(1/2))^(9/2),x, algorithm="maxima")

[Out]

1/175*x/sqrt(-x^2 + sqrt(10)) + 1/350*sqrt(10)*x/(-x^2 + sqrt(10))^(3/2) + 1/4*x^3/(-x^2 + sqrt(10))^(7/2) + 3
/140*x/(-x^2 + sqrt(10))^(5/2) - 3/28*sqrt(10)*x/(-x^2 + sqrt(10))^(7/2)

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Fricas [A]
time = 0.64, size = 69, normalized size = 1.38 \begin {gather*} -\frac {{\left (2 \, x^{15} - 160 \, x^{11} - 2600 \, x^{7} + \sqrt {10} {\left (x^{13} - 340 \, x^{9} - 700 \, x^{5}\right )}\right )} \sqrt {-x^{2} + \sqrt {10}}}{350 \, {\left (x^{16} - 40 \, x^{12} + 600 \, x^{8} - 4000 \, x^{4} + 10000\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^2+10^(1/2))^(9/2),x, algorithm="fricas")

[Out]

-1/350*(2*x^15 - 160*x^11 - 2600*x^7 + sqrt(10)*(x^13 - 340*x^9 - 700*x^5))*sqrt(-x^2 + sqrt(10))/(x^16 - 40*x
^12 + 600*x^8 - 4000*x^4 + 10000)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(-x**2+10**(1/2))**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (36) = 72\).
time = 0.87, size = 98, normalized size = 1.96 \begin {gather*} -\frac {16 \, {\left (7 \, {\left (\frac {x}{\sqrt {-x^{2} + \sqrt {10}} - 10^{\frac {1}{4}}} - \frac {\sqrt {-x^{2} + \sqrt {10}} - 10^{\frac {1}{4}}}{x}\right )}^{2} + 20\right )}}{175 \, {\left (\frac {x}{\sqrt {-x^{2} + \sqrt {10}} - 10^{\frac {1}{4}}} - \frac {\sqrt {-x^{2} + \sqrt {10}} - 10^{\frac {1}{4}}}{x}\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(-x^2+10^(1/2))^(9/2),x, algorithm="giac")

[Out]

-16/175*(7*(x/(sqrt(-x^2 + sqrt(10)) - 10^(1/4)) - (sqrt(-x^2 + sqrt(10)) - 10^(1/4))/x)^2 + 20)/(x/(sqrt(-x^2
+ sqrt(10)) - 10^(1/4)) - (sqrt(-x^2 + sqrt(10)) - 10^(1/4))/x)^7

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^4}{{\left (\sqrt {10}-x^2\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(10^(1/2) - x^2)^(9/2),x)

[Out]

int(x^4/(10^(1/2) - x^2)^(9/2), x)

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