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

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

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Rubi [A]  time = 0.06, antiderivative size = 77, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 {gather*}

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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.02, size = 76, normalized size = 0.99 \begin {gather*} \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} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.18, size = 80, normalized size = 1.04 \begin {gather*} \frac {\sqrt {-a+b x^2+c x^4}}{2 a x^2}-\frac {b \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}-\frac {\sqrt {-a+b x^2+c x^4}}{\sqrt {a}}\right )}{2 a^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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fricas [A]  time = 1.11, size = 188, normalized size = 2.44 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.22, size = 111, normalized size = 1.44 \begin {gather*} \frac {b \arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}}{\sqrt {a}}\right )}{2 \, a^{\frac {3}{2}}} - \frac {{\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )} b - 2 \, a \sqrt {c}}{2 \, {\left ({\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} - a}\right )}^{2} + a\right )} a} \end {gather*}

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]

1/2*b*arctan(-(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 - a))/sqrt(a))/a^(3/2) - 1/2*((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^
2 - a))*b - 2*a*sqrt(c))/(((sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2 - a))^2 + a)*a)

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maple [A]  time = 0.01, size = 74, normalized size = 0.96 \begin {gather*} -\frac {b \ln \left (\frac {b \,x^{2}-2 a +2 \sqrt {-a}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}{x^{2}}\right )}{4 \sqrt {-a}\, a}+\frac {\sqrt {c \,x^{4}+b \,x^{2}-a}}{2 a \,x^{2}} \end {gather*}

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

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maxima [A]  time = 2.41, size = 62, normalized size = 0.81 \begin {gather*} -\frac {b \arcsin \left (-\frac {b}{\sqrt {b^{2} + 4 \, a c}} + \frac {2 \, a}{\sqrt {b^{2} + 4 \, a c} x^{2}}\right )}{4 \, a^{\frac {3}{2}}} + \frac {\sqrt {c x^{4} + b x^{2} - a}}{2 \, a x^{2}} \end {gather*}

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]

-1/4*b*arcsin(-b/sqrt(b^2 + 4*a*c) + 2*a/(sqrt(b^2 + 4*a*c)*x^2))/a^(3/2) + 1/2*sqrt(c*x^4 + b*x^2 - a)/(a*x^2
)

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mupad [B]  time = 4.55, size = 64, normalized size = 0.83 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2-a}}{2\,a\,x^2}-\frac {b\,\mathrm {atanh}\left (\frac {a-\frac {b\,x^2}{2}}{\sqrt {-a}\,\sqrt {c\,x^4+b\,x^2-a}}\right )}{4\,{\left (-a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \sqrt {- a + b x^{2} + c x^{4}}}\, dx \end {gather*}

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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