3.474 \(\int \frac{x^4}{(\sqrt{10}-x^2)^{9/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0151225, antiderivative size = 50, normalized size of antiderivative = 1., 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 = {271, 264} \[ \frac{x^5}{5 \sqrt{10} \left (\sqrt{10}-x^2\right )^{7/2}}-\frac{x^7}{175 \left (\sqrt{10}-x^2\right )^{7/2}} \]

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 271

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]

Rule 264

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]

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*}

Mathematica [A]  time = 0.0232837, size = 35, normalized size = 0.7 \[ \frac{7 \sqrt{10} x^5-2 x^7}{350 \left (\sqrt{10}-x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.032, size = 28, normalized size = 0.6 \begin{align*}{\frac{{x}^{5} \left ( -2\,{x}^{2}+7\,\sqrt{10} \right ) }{350} \left ( -{x}^{2}+\sqrt{10} \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.41785, size = 107, normalized size = 2.14 \begin{align*} \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{align*}

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 = 2.2025, size = 196, normalized size = 3.92 \begin{align*} -\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{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.16634, size = 132, normalized size = 2.64 \begin{align*} -\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{align*}

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