3.1.37 \(\int \frac {2+3 x^2}{x^3 \sqrt {5+x^4}} \, dx\) [37]

Optimal. Leaf size=42 \[ -\frac {\sqrt {5+x^4}}{5 x^2}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )}{2 \sqrt {5}} \]

[Out]

-3/10*arctanh(1/5*(x^4+5)^(1/2)*5^(1/2))*5^(1/2)-1/5*(x^4+5)^(1/2)/x^2

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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1266, 821, 272, 65, 213} \begin {gather*} -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {x^4+5}}{\sqrt {5}}\right )}{2 \sqrt {5}}-\frac {\sqrt {x^4+5}}{5 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x^2)/(x^3*Sqrt[5 + x^4]),x]

[Out]

-1/5*Sqrt[5 + x^4]/x^2 - (3*ArcTanh[Sqrt[5 + x^4]/Sqrt[5]])/(2*Sqrt[5])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 272

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

Rule 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0
] && EqQ[Simplify[m + 2*p + 3], 0]

Rule 1266

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m
- 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]

Rubi steps

\begin {align*} \int \frac {2+3 x^2}{x^3 \sqrt {5+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {2+3 x}{x^2 \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {5+x^4}}{5 x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {5+x^4}}{5 x^2}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {5+x}} \, dx,x,x^4\right )\\ &=-\frac {\sqrt {5+x^4}}{5 x^2}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {5+x^4}\right )\\ &=-\frac {\sqrt {5+x^4}}{5 x^2}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {5+x^4}}{\sqrt {5}}\right )}{2 \sqrt {5}}\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 46, normalized size = 1.10 \begin {gather*} -\frac {\sqrt {5+x^4}}{5 x^2}+\frac {3 \tanh ^{-1}\left (\frac {x^2-\sqrt {5+x^4}}{\sqrt {5}}\right )}{\sqrt {5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x^2)/(x^3*Sqrt[5 + x^4]),x]

[Out]

-1/5*Sqrt[5 + x^4]/x^2 + (3*ArcTanh[(x^2 - Sqrt[5 + x^4])/Sqrt[5]])/Sqrt[5]

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Maple [A]
time = 0.18, size = 31, normalized size = 0.74

method	result	size
default	\(-\frac {\sqrt {x^{4}+5}}{5 x^{2}}-\frac {3 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{10}\)	\(31\)
risch	\(-\frac {\sqrt {x^{4}+5}}{5 x^{2}}-\frac {3 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{10}\)	\(31\)
elliptic	\(-\frac {\sqrt {x^{4}+5}}{5 x^{2}}-\frac {3 \sqrt {5}\, \arctanh \left (\frac {\sqrt {5}}{\sqrt {x^{4}+5}}\right )}{10}\)	\(31\)
trager	\(-\frac {\sqrt {x^{4}+5}}{5 x^{2}}+\frac {3 \RootOf \left (\textit {\_Z}^{2}-5\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-5\right )-\sqrt {x^{4}+5}}{x^{2}}\right )}{10}\)	\(44\)
meijerg	\(-\frac {\sqrt {5}\, \sqrt {1+\frac {x^{4}}{5}}}{5 x^{2}}+\frac {3 \sqrt {5}\, \left (-2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1+\frac {x^{4}}{5}}}{2}\right )+\left (-2 \ln \left (2\right )+4 \ln \left (x \right )-\ln \left (5\right )\right ) \sqrt {\pi }\right )}{20 \sqrt {\pi }}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2+2)/x^3/(x^4+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/5*(x^4+5)^(1/2)/x^2-3/10*5^(1/2)*arctanh(5^(1/2)/(x^4+5)^(1/2))

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Maxima [A]
time = 0.49, size = 47, normalized size = 1.12 \begin {gather*} \frac {3}{20} \, \sqrt {5} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{\sqrt {5} + \sqrt {x^{4} + 5}}\right ) - \frac {\sqrt {x^{4} + 5}}{5 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^3/(x^4+5)^(1/2),x, algorithm="maxima")

[Out]

3/20*sqrt(5)*log(-(sqrt(5) - sqrt(x^4 + 5))/(sqrt(5) + sqrt(x^4 + 5))) - 1/5*sqrt(x^4 + 5)/x^2

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Fricas [A]
time = 0.37, size = 47, normalized size = 1.12 \begin {gather*} \frac {3 \, \sqrt {5} x^{2} \log \left (-\frac {\sqrt {5} - \sqrt {x^{4} + 5}}{x^{2}}\right ) - 2 \, x^{2} - 2 \, \sqrt {x^{4} + 5}}{10 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^3/(x^4+5)^(1/2),x, algorithm="fricas")

[Out]

1/10*(3*sqrt(5)*x^2*log(-(sqrt(5) - sqrt(x^4 + 5))/x^2) - 2*x^2 - 2*sqrt(x^4 + 5))/x^2

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Sympy [A]
time = 1.76, size = 31, normalized size = 0.74 \begin {gather*} - \frac {\sqrt {1 + \frac {5}{x^{4}}}}{5} - \frac {3 \sqrt {5} \operatorname {asinh}{\left (\frac {\sqrt {5}}{x^{2}} \right )}}{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x**2+2)/x**3/(x**4+5)**(1/2),x)

[Out]

-sqrt(1 + 5/x**4)/5 - 3*sqrt(5)*asinh(sqrt(5)/x**2)/10

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (31) = 62\).
time = 4.78, size = 66, normalized size = 1.57 \begin {gather*} \frac {3}{10} \, \sqrt {5} \log \left (-\frac {x^{2} + \sqrt {5} - \sqrt {x^{4} + 5}}{x^{2} - \sqrt {5} - \sqrt {x^{4} + 5}}\right ) + \frac {2}{{\left (x^{2} - \sqrt {x^{4} + 5}\right )}^{2} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*x^2+2)/x^3/(x^4+5)^(1/2),x, algorithm="giac")

[Out]

3/10*sqrt(5)*log(-(x^2 + sqrt(5) - sqrt(x^4 + 5))/(x^2 - sqrt(5) - sqrt(x^4 + 5))) + 2/((x^2 - sqrt(x^4 + 5))^
2 - 5)

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Mupad [B]
time = 0.33, size = 31, normalized size = 0.74 \begin {gather*} -\frac {3\,\sqrt {5}\,\mathrm {atanh}\left (\frac {\sqrt {5}\,\sqrt {x^4+5}}{5}\right )}{10}-\frac {\sqrt {x^4+5}}{5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 + 2)/(x^3*(x^4 + 5)^(1/2)),x)

[Out]

- (3*5^(1/2)*atanh((5^(1/2)*(x^4 + 5)^(1/2))/5))/10 - (x^4 + 5)^(1/2)/(5*x^2)

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