3.10.46 \(\int \frac {1}{x^7 (1+x^4)^{3/2}} \, dx\) [946]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {277, 270} \begin {gather*} -\frac {1}{6 \sqrt {x^4+1} x^6}+\frac {4 x^2}{3 \sqrt {x^4+1}}+\frac {2}{3 \sqrt {x^4+1} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(1 + x^4)^(3/2)),x]

[Out]

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

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 {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*}

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Mathematica [A]
time = 0.13, size = 28, normalized size = 0.57 \begin {gather*} \frac {-1+4 x^4+8 x^8}{6 x^6 \sqrt {1+x^4}} \end {gather*}

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 = 0.15, size = 25, normalized size = 0.51

method	result	size
gosper	\(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\)	\(25\)
default	\(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\)	\(25\)
trager	\(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\)	\(25\)
meijerg	\(-\frac {-8 x^{8}-4 x^{4}+1}{6 x^{6} \sqrt {x^{4}+1}}\)	\(25\)
risch	\(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\)	\(25\)
elliptic	\(\frac {8 x^{8}+4 x^{4}-1}{6 x^{6} \sqrt {x^{4}+1}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 36, normalized size = 0.73 \begin {gather*} \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 {gather*}

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 = 0.36, size = 42, normalized size = 0.86 \begin {gather*} \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 {gather*}

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 = 0.58, size = 70, normalized size = 1.43 \begin {gather*} \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 {gather*}

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 = 1.96, size = 70, normalized size = 1.43 \begin {gather*} \frac {x^{2}}{2 \, \sqrt {x^{4} + 1}} - \frac {3 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{4} - 12 \, {\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} + 5}{3 \, {\left ({\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1\right )}^{3}} \end {gather*}

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) - 1/3*(3*(x^2 - sqrt(x^4 + 1))^4 - 12*(x^2 - sqrt(x^4 + 1))^2 + 5)/((x^2 - sqrt(x^4 + 1)
)^2 - 1)^3

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Mupad [B]
time = 1.23, size = 28, normalized size = 0.57 \begin {gather*} -\frac {12\,x^4-8\,{\left (x^4+1\right )}^2+9}{6\,x^6\,\sqrt {x^4+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^7*(x^4 + 1)^(3/2)),x)

[Out]

-(12*x^4 - 8*(x^4 + 1)^2 + 9)/(6*x^6*(x^4 + 1)^(1/2))

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