3.942 \(\int \frac{x^9}{(1+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=41 \[ -\frac{x^6}{2 \sqrt{x^4+1}}+\frac{3}{4} \sqrt{x^4+1} x^2-\frac{3}{4} \sinh ^{-1}\left (x^2\right ) \]

[Out]

-x^6/(2*Sqrt[1 + x^4]) + (3*x^2*Sqrt[1 + x^4])/4 - (3*ArcSinh[x^2])/4

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Rubi [A]  time = 0.01531, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {275, 288, 321, 215} \[ -\frac{x^6}{2 \sqrt{x^4+1}}+\frac{3}{4} \sqrt{x^4+1} x^2-\frac{3}{4} \sinh ^{-1}\left (x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^9/(1 + x^4)^(3/2),x]

[Out]

-x^6/(2*Sqrt[1 + x^4]) + (3*x^2*Sqrt[1 + x^4])/4 - (3*ArcSinh[x^2])/4

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 288

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

Rule 321

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

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{x^9}{\left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{x^6}{2 \sqrt{1+x^4}}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^2}} \, dx,x,x^2\right )\\ &=-\frac{x^6}{2 \sqrt{1+x^4}}+\frac{3}{4} x^2 \sqrt{1+x^4}-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,x^2\right )\\ &=-\frac{x^6}{2 \sqrt{1+x^4}}+\frac{3}{4} x^2 \sqrt{1+x^4}-\frac{3}{4} \sinh ^{-1}\left (x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0093248, size = 37, normalized size = 0.9 \[ \frac{x^6+3 x^2-3 \sqrt{x^4+1} \sinh ^{-1}\left (x^2\right )}{4 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(1 + x^4)^(3/2),x]

[Out]

(3*x^2 + x^6 - 3*Sqrt[1 + x^4]*ArcSinh[x^2])/(4*Sqrt[1 + x^4])

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Maple [A]  time = 0.091, size = 32, normalized size = 0.8 \begin{align*}{\frac{{x}^{6}}{4}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{3\,{x}^{2}}{4}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{3\,{\it Arcsinh} \left ({x}^{2} \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.993062, size = 99, normalized size = 2.41 \begin{align*} -\frac{\frac{3 \,{\left (x^{4} + 1\right )}}{x^{4}} - 2}{4 \,{\left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{x^{6}}\right )}} - \frac{3}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} + 1\right ) + \frac{3}{8} \, \log \left (\frac{\sqrt{x^{4} + 1}}{x^{2}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(3*(x^4 + 1)/x^4 - 2)/(sqrt(x^4 + 1)/x^2 - (x^4 + 1)^(3/2)/x^6) - 3/8*log(sqrt(x^4 + 1)/x^2 + 1) + 3/8*lo
g(sqrt(x^4 + 1)/x^2 - 1)

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Fricas [A]  time = 1.40538, size = 131, normalized size = 3.2 \begin{align*} \frac{2 \, x^{4} + 3 \,{\left (x^{4} + 1\right )} \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) +{\left (x^{6} + 3 \, x^{2}\right )} \sqrt{x^{4} + 1} + 2}{4 \,{\left (x^{4} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.96966, size = 36, normalized size = 0.88 \begin{align*} \frac{x^{6}}{4 \sqrt{x^{4} + 1}} + \frac{3 x^{2}}{4 \sqrt{x^{4} + 1}} - \frac{3 \operatorname{asinh}{\left (x^{2} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**6/(4*sqrt(x**4 + 1)) + 3*x**2/(4*sqrt(x**4 + 1)) - 3*asinh(x**2)/4

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Giac [A]  time = 1.17706, size = 46, normalized size = 1.12 \begin{align*} \frac{{\left (x^{4} + 3\right )} x^{2}}{4 \, \sqrt{x^{4} + 1}} + \frac{3}{4} \, \log \left (-x^{2} + \sqrt{x^{4} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x^4 + 3)*x^2/sqrt(x^4 + 1) + 3/4*log(-x^2 + sqrt(x^4 + 1))