3.135 \(\int \frac{1}{\sqrt{x} \sqrt{x (a+b x^2+c x^4)}} \, dx\)

Optimal. Leaf size=51 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x} \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x+b x^3+c x^5}}\right )}{2 \sqrt{a}} \]

[Out]

-ArcTanh[(Sqrt[x]*(2*a + b*x^2))/(2*Sqrt[a]*Sqrt[a*x + b*x^3 + c*x^5])]/(2*Sqrt[a])

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Rubi [A]  time = 0.0672684, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1997, 1913, 206} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x} \left (2 a+b x^2\right )}{2 \sqrt{a} \sqrt{a x+b x^3+c x^5}}\right )}{2 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[x*(a + b*x^2 + c*x^4)]),x]

[Out]

-ArcTanh[(Sqrt[x]*(2*a + b*x^2))/(2*Sqrt[a]*Sqrt[a*x + b*x^3 + c*x^5])]/(2*Sqrt[a])

Rule 1997

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] &&
GeneralizedTrinomialQ[u, x] &&  !GeneralizedTrinomialMatchQ[u, x]

Rule 1913

Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-2/(n - q), Sub
st[Int[1/(4*a - x^2), x], x, (x^(m + 1)*(2*a + b*x^(n - q)))/Sqrt[a*x^q + b*x^n + c*x^r]], x] /; FreeQ[{a, b,
c, m, n, q, r}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && NeQ[b^2 - 4*a*c, 0] && EqQ[m, q/2 - 1]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.0195306, size = 83, normalized size = 1.63 \[ -\frac{\sqrt{x} \sqrt{a+b x^2+c x^4} \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{2 \sqrt{a} \sqrt{x \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*Sqrt[x*(a + b*x^2 + c*x^4)]),x]

[Out]

-(Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x^2 + c*x^4])])/(2*Sqrt[a]*Sqrt[
x*(a + b*x^2 + c*x^4)])

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Maple [A]  time = 0.008, size = 72, normalized size = 1.4 \begin{align*} -{\frac{1}{2}\sqrt{x}\sqrt{c{x}^{4}+b{x}^{2}+a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) }}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(1/2)/(x*(c*x^4+b*x^2+a))^(1/2),x)

[Out]

-1/2*x^(1/2)/(x*(c*x^4+b*x^2+a))^(1/2)*(c*x^4+b*x^2+a)^(1/2)/a^(1/2)*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(
1/2))/x^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (c x^{4} + b x^{2} + a\right )} x} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(x*(c*x^4+b*x^2+a))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt((c*x^4 + b*x^2 + a)*x)*sqrt(x)), x)

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Fricas [A]  time = 1.37217, size = 327, normalized size = 6.41 \begin{align*} \left [\frac{\log \left (-\frac{{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt{c x^{5} + b x^{3} + a x}{\left (b x^{2} + 2 \, a\right )} \sqrt{a} \sqrt{x}}{x^{5}}\right )}{4 \, \sqrt{a}}, \frac{\sqrt{-a} \arctan \left (\frac{\sqrt{c x^{5} + b x^{3} + a x}{\left (b x^{2} + 2 \, a\right )} \sqrt{-a} \sqrt{x}}{2 \,{\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right )}{2 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(x*(c*x^4+b*x^2+a))^(1/2),x, algorithm="fricas")

[Out]

[1/4*log(-((b^2 + 4*a*c)*x^5 + 8*a*b*x^3 + 8*a^2*x - 4*sqrt(c*x^5 + b*x^3 + a*x)*(b*x^2 + 2*a)*sqrt(a)*sqrt(x)
)/x^5)/sqrt(a), 1/2*sqrt(-a)*arctan(1/2*sqrt(c*x^5 + b*x^3 + a*x)*(b*x^2 + 2*a)*sqrt(-a)*sqrt(x)/(a*c*x^5 + a*
b*x^3 + a^2*x))/a]

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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(1/x**(1/2)/(x*(c*x**4+b*x**2+a))**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{{\left (c x^{4} + b x^{2} + a\right )} x} \sqrt{x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(1/2)/(x*(c*x^4+b*x^2+a))^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt((c*x^4 + b*x^2 + a)*x)*sqrt(x)), x)