3.973 \(\int \frac{1}{x^3 \sqrt{-a+b x^2+c x^4}} \, dx\)

Optimal. Leaf size=77 \[ \frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2}-\frac{b \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0619014, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {1114, 730, 724, 204} \[ \frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2}-\frac{b \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

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

Rule 1114

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

Rule 730

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e
^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]

Rule 724

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

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])

Rubi steps

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

Mathematica [A]  time = 0.0229716, size = 76, normalized size = 0.99 \[ \frac{b \tan ^{-1}\left (\frac{b x^2-2 a}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{4 a^{3/2}}+\frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[-a + b*x^2 + c*x^4]),x]

[Out]

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

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Maple [A]  time = 0.169, size = 74, normalized size = 1. \begin{align*}{\frac{1}{2\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}-a}}-{\frac{b}{4\,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{-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(c*x^4+b*x^2-a)^(1/2),x)

[Out]

1/2*(c*x^4+b*x^2-a)^(1/2)/a/x^2-1/4*b/a/(-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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4+b*x^2-a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.64411, size = 421, normalized size = 5.47 \begin{align*} \left [-\frac{\sqrt{-a} b x^{2} \log \left (\frac{{\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} - 4 \, \sqrt{c x^{4} + b x^{2} - a}{\left (b x^{2} - 2 \, a\right )} \sqrt{-a} + 8 \, a^{2}}{x^{4}}\right ) - 4 \, \sqrt{c x^{4} + b x^{2} - a} a}{8 \, a^{2} x^{2}}, \frac{\sqrt{a} b x^{2} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2} - a}{\left (b x^{2} - 2 \, a\right )} \sqrt{a}}{2 \,{\left (a c x^{4} + a b x^{2} - a^{2}\right )}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} - a} a}{4 \, a^{2} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4+b*x^2-a)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \sqrt{- a + b x^{2} + c x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(c*x**4+b*x**2-a)**(1/2),x)

[Out]

Integral(1/(x**3*sqrt(-a + b*x**2 + c*x**4)), x)

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Giac [A]  time = 1.33424, size = 93, normalized size = 1.21 \begin{align*} \frac{b \log \left ({\left | -2 \, \sqrt{-a}{\left (\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}} - \frac{\sqrt{-a}}{x^{2}}\right )} + b \right |}\right )}{4 \, \sqrt{-a} a} + \frac{\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(c*x^4+b*x^2-a)^(1/2),x, algorithm="giac")

[Out]

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