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

Optimal. Leaf size=44 \[ -\frac {\tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {c}} \]

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Rubi [A]  time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1107, 621, 204} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b*x^2 - c*x^4],x]

[Out]

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

Rule 204

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

Rule 621

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

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 44, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + b*x^2 - c*x^4],x]

[Out]

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

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IntegrateAlgebraic [B]  time = 0.20, size = 127, normalized size = 2.89 \begin {gather*} \frac {\sqrt {-c} \log \left (-8 \sqrt {-c} c x^2 \sqrt {a+b x^2-c x^4}+4 a c+b^2+4 b c x^2-8 c^2 x^4\right )}{4 c}-\frac {\tan ^{-1}\left (\frac {2 \sqrt {-c} \sqrt {c} x^2}{b}-\frac {2 \sqrt {c} \sqrt {a+b x^2-c x^4}}{b}\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/Sqrt[a + b*x^2 - c*x^4],x]

[Out]

-1/2*ArcTan[(2*Sqrt[-c]*Sqrt[c]*x^2)/b - (2*Sqrt[c]*Sqrt[a + b*x^2 - c*x^4])/b]/Sqrt[c] + (Sqrt[-c]*Log[b^2 +
4*a*c + 4*b*c*x^2 - 8*c^2*x^4 - 8*Sqrt[-c]*c*x^2*Sqrt[a + b*x^2 - c*x^4]])/(4*c)

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fricas [A]  time = 0.85, size = 124, normalized size = 2.82 \begin {gather*} \left [-\frac {\sqrt {-c} \log \left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, \sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} - b\right )} \sqrt {-c} - 4 \, a c\right )}{4 \, c}, -\frac {\arctan \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} - b\right )} \sqrt {c}}{2 \, {\left (c^{2} x^{4} - b c x^{2} - a c\right )}}\right )}{2 \, \sqrt {c}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.21, size = 45, normalized size = 1.02 \begin {gather*} -\frac {\log \left ({\left | 2 \, {\left (\sqrt {-c} x^{2} - \sqrt {-c x^{4} + b x^{2} + a}\right )} \sqrt {-c} + b \right |}\right )}{2 \, \sqrt {-c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(abs(2*(sqrt(-c)*x^2 - sqrt(-c*x^4 + b*x^2 + a))*sqrt(-c) + b))/sqrt(-c)

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maple [A]  time = 0.01, size = 36, normalized size = 0.82 \begin {gather*} \frac {\arctan \left (\frac {\left (x^{2}-\frac {b}{2 c}\right ) \sqrt {c}}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{2 \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.39, size = 28, normalized size = 0.64 \begin {gather*} -\frac {\arcsin \left (-\frac {2 \, c x^{2} - b}{\sqrt {b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.79, size = 40, normalized size = 0.91 \begin {gather*} \frac {\ln \left (\frac {\frac {b}{2}-c\,x^2}{\sqrt {-c}}+\sqrt {-c\,x^4+b\,x^2+a}\right )}{2\,\sqrt {-c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2 - c*x^4)^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a + b x^{2} - c x^{4}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/sqrt(a + b*x**2 - c*x**4), x)

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