3.1.37 \(\int \frac {1}{x \sqrt {b x^2}} \, dx\) [37]

Optimal. Leaf size=11 \[ -\frac {1}{\sqrt {b x^2}} \]

[Out]

-1/(b*x^2)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \begin {gather*} -\frac {1}{\sqrt {b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[b*x^2]),x]

[Out]

-(1/Sqrt[b*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 15, normalized size = 1.36 \begin {gather*} -\frac {b x^2}{\left (b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*x^2)/(b*x^2)^(3/2))

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 10, normalized size = 0.91

method	result	size
gosper	\(-\frac {1}{\sqrt {b \,x^{2}}}\)	\(10\)
derivativedivides	\(-\frac {1}{\sqrt {b \,x^{2}}}\)	\(10\)
default	\(-\frac {1}{\sqrt {b \,x^{2}}}\)	\(10\)
risch	\(-\frac {1}{\sqrt {b \,x^{2}}}\)	\(10\)
trager	\(\frac {\left (x -1\right ) \sqrt {b \,x^{2}}}{b \,x^{2}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b*x^2)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 8, normalized size = 0.73 \begin {gather*} -\frac {1}{\sqrt {b} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(b)*x)

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 15, normalized size = 1.36 \begin {gather*} -\frac {\sqrt {b x^{2}}}{b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b*x^2)/(b*x^2)

________________________________________________________________________________________

Sympy [A]
time = 0.18, size = 10, normalized size = 0.91 \begin {gather*} - \frac {1}{\sqrt {b x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x**2)**(1/2),x)

[Out]

-1/sqrt(b*x**2)

________________________________________________________________________________________

Giac [A]
time = 1.80, size = 12, normalized size = 1.09 \begin {gather*} -\frac {1}{\sqrt {b} x \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(b)*x*sgn(x))

________________________________________________________________________________________

Mupad [B]
time = 0.96, size = 10, normalized size = 0.91 \begin {gather*} -\frac {1}{\sqrt {b}\,\sqrt {x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x^2)^(1/2)),x)

[Out]

-1/(b^(1/2)*(x^2)^(1/2))

________________________________________________________________________________________