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

Optimal. Leaf size=49 \[ -\frac {\tanh ^{-1}\left (\frac {x \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^2+b x^4+c x^6}}\right )}{2 \sqrt {a}} \]

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Rubi [A]  time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1996, 1904, 206} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {x \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^2+b x^4+c x^6}}\right )}{2 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[(x*(2*a + b*x^2))/(2*Sqrt[a]*Sqrt[a*x^2 + b*x^4 + c*x^6])]/(2*Sqrt[a])

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])

Rule 1904

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

Rule 1996

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedTrinomialQ[u, x] &&  !Gen
eralizedTrinomialMatchQ[u, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x^2 \left (a+b x^2+c x^4\right )}} \, dx &=\int \frac {1}{\sqrt {a x^2+b x^4+c x^6}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x \left (2 a+b x^2\right )}{\sqrt {a x^2+b x^4+c x^6}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {x \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^2+b x^4+c x^6}}\right )}{2 \sqrt {a}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 81, normalized size = 1.65 \begin {gather*} -\frac {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^2 \left (a+b x^2+c x^4\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.09, size = 48, normalized size = 0.98 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {c} x^3-\sqrt {a x^2+b x^4+c x^6}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[(Sqrt[a]*x)/(Sqrt[c]*x^3 - Sqrt[a*x^2 + b*x^4 + c*x^6])]/Sqrt[a]

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fricas [A]  time = 1.32, size = 135, normalized size = 2.76 \begin {gather*} \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^{6} + b x^{4} + a x^{2}} {\left (b x^{2} + 2 \, a\right )} \sqrt {a}}{x^{5}}\right )}{4 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{6} + b x^{4} + a x^{2}} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right )}{2 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2*(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^6 + b*x^4 + a*x^2)*(b*x^2 + 2*a)*sqrt(a))/x^5)
/sqrt(a), 1/2*sqrt(-a)*arctan(1/2*sqrt(c*x^6 + b*x^4 + a*x^2)*(b*x^2 + 2*a)*sqrt(-a)/(a*c*x^5 + a*b*x^3 + a^2*
x))/a]

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giac [A]  time = 0.44, size = 62, normalized size = 1.27 \begin {gather*} -\frac {\arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-a}} + \frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} \mathrm {sgn}\relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctan(sqrt(a)/sqrt(-a))*sgn(x)/sqrt(-a) + arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 + a))/sqrt(-a))/(sqrt(-a
)*sgn(x))

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maple [A]  time = 0.01, size = 72, normalized size = 1.47 \begin {gather*} -\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{2 \sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x^{2}}\, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {{\left (c x^{4} + b x^{2} + a\right )} x^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x^2\,\left (c\,x^4+b\,x^2+a\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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