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

Optimal. Leaf size=23 \[ -\frac {\sqrt {b x^2+c x^4}}{b x^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2014} \begin {gather*} -\frac {\sqrt {b x^2+c x^4}}{b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[b*x^2 + c*x^4]),x]

[Out]

-(Sqrt[b*x^2 + c*x^4]/(b*x^2))

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {1}{x \sqrt {b x^2+c x^4}} \, dx &=-\frac {\sqrt {b x^2+c x^4}}{b x^2}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {x^2 \left (b+c x^2\right )}}{b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[x^2*(b + c*x^2)]/(b*x^2))

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IntegrateAlgebraic [A]  time = 0.13, size = 23, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b x^2+c x^4}}{b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[b*x^2 + c*x^4]/(b*x^2))

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fricas [A]  time = 1.87, size = 21, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}}}{b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^4 + b*x^2)/(b*x^2)

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giac [A]  time = 0.18, size = 25, normalized size = 1.09 \begin {gather*} \frac {1}{\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/(sqrt(c)*x^2 - sqrt(c*x^4 + b*x^2))

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maple [A]  time = 0.00, size = 26, normalized size = 1.13 \begin {gather*} -\frac {c \,x^{2}+b}{\sqrt {c \,x^{4}+b \,x^{2}}\, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.47, size = 21, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {c x^{4} + b x^{2}}}{b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(c*x^4 + b*x^2)/(b*x^2)

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mupad [B]  time = 4.21, size = 21, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {c\,x^4+b\,x^2}}{b\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x^2 + c*x^4)^(1/2)),x)

[Out]

-(b*x^2 + c*x^4)^(1/2)/(b*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(x**2*(b + c*x**2))), x)

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