3.3.64 \(\int \frac {1}{x \sqrt {b x^2+c x^4}} \, dx\) [264]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {2039} \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 2039

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] &&
NeQ[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 = 0.03, 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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Maple [A]
time = 0.08, size = 26, normalized size = 1.13

method	result	size
trager	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}}}{b \,x^{2}}\)	\(22\)
gosper	\(-\frac {c \,x^{2}+b}{b \sqrt {c \,x^{4}+b \,x^{2}}}\)	\(26\)
default	\(-\frac {c \,x^{2}+b}{b \sqrt {c \,x^{4}+b \,x^{2}}}\)	\(26\)
risch	\(-\frac {c \,x^{2}+b}{\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, b}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, 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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Fricas [A]
time = 0.33, 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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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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Giac [A]
time = 4.60, size = 34, normalized size = 1.48 \begin {gather*} \frac {2 \, \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )} \mathrm {sgn}\left (x\right )} \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]

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

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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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