3.946 \(\int \frac{1}{x^7 (1+x^4)^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ \frac{4 x^2}{3 \sqrt{x^4+1}}+\frac{2}{3 \sqrt{x^4+1} x^2}-\frac{1}{6 \sqrt{x^4+1} x^6} \]

[Out]

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

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Rubi [A]  time = 0.010478, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {271, 264} \[ \frac{4 x^2}{3 \sqrt{x^4+1}}+\frac{2}{3 \sqrt{x^4+1} x^2}-\frac{1}{6 \sqrt{x^4+1} x^6} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(1 + x^4)^(3/2)),x]

[Out]

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

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{1}{x^7 \left (1+x^4\right )^{3/2}} \, dx &=-\frac{1}{6 x^6 \sqrt{1+x^4}}-\frac{4}{3} \int \frac{1}{x^3 \left (1+x^4\right )^{3/2}} \, dx\\ &=-\frac{1}{6 x^6 \sqrt{1+x^4}}+\frac{2}{3 x^2 \sqrt{1+x^4}}+\frac{8}{3} \int \frac{x}{\left (1+x^4\right )^{3/2}} \, dx\\ &=-\frac{1}{6 x^6 \sqrt{1+x^4}}+\frac{2}{3 x^2 \sqrt{1+x^4}}+\frac{4 x^2}{3 \sqrt{1+x^4}}\\ \end{align*}

Mathematica [A]  time = 0.0048583, size = 28, normalized size = 0.57 \[ -\frac{-8 x^8-4 x^4+1}{6 x^6 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(1 + x^4)^(3/2)),x]

[Out]

-(1 - 4*x^4 - 8*x^8)/(6*x^6*Sqrt[1 + x^4])

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Maple [A]  time = 0.028, size = 25, normalized size = 0.5 \begin{align*}{\frac{8\,{x}^{8}+4\,{x}^{4}-1}{6\,{x}^{6}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^4+1)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.970691, size = 49, normalized size = 1. \begin{align*} \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} + \frac{\sqrt{x^{4} + 1}}{x^{2}} - \frac{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^4+1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.72803, size = 96, normalized size = 1.96 \begin{align*} \frac{8 \, x^{10} + 8 \, x^{6} +{\left (8 \, x^{8} + 4 \, x^{4} - 1\right )} \sqrt{x^{4} + 1}}{6 \,{\left (x^{10} + x^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^4+1)^(3/2),x, algorithm="fricas")

[Out]

1/6*(8*x^10 + 8*x^6 + (8*x^8 + 4*x^4 - 1)*sqrt(x^4 + 1))/(x^10 + x^6)

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Sympy [A]  time = 1.66169, size = 70, normalized size = 1.43 \begin{align*} \frac{8 x^{8} \sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} + \frac{4 x^{4} \sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} - \frac{\sqrt{1 + \frac{1}{x^{4}}}}{6 x^{8} + 6 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**4+1)**(3/2),x)

[Out]

8*x**8*sqrt(1 + x**(-4))/(6*x**8 + 6*x**4) + 4*x**4*sqrt(1 + x**(-4))/(6*x**8 + 6*x**4) - sqrt(1 + x**(-4))/(6
*x**8 + 6*x**4)

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Giac [A]  time = 1.22252, size = 39, normalized size = 0.8 \begin{align*} \frac{x^{2}}{2 \, \sqrt{x^{4} + 1}} - \frac{1}{6} \,{\left (\frac{1}{x^{4}} + 1\right )}^{\frac{3}{2}} + \sqrt{\frac{1}{x^{4}} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^4+1)^(3/2),x, algorithm="giac")

[Out]

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