3.1.44 \(\int \frac {x}{(b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=14 \[ -\frac {1}{b \sqrt {b x^2}} \]

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Rubi [A]  time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {15, 30} \begin {gather*} -\frac {1}{b \sqrt {b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(b*x^2)^(3/2),x]

[Out]

-(1/(b*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 {x}{\left (b x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {1}{x^2} \, dx}{b \sqrt {b x^2}}\\ &=-\frac {1}{b \sqrt {b x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {x^2}{\left (b x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(b*x^2)^(3/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.01, size = 17, normalized size = 1.21 \begin {gather*} -\frac {\sqrt {b x^2}}{b^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x/(b*x^2)^(3/2),x]

[Out]

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

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fricas [A]  time = 1.04, size = 15, normalized size = 1.07 \begin {gather*} -\frac {\sqrt {b x^{2}}}{b^{2} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.15, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{\sqrt {b x^{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \begin {gather*} -\frac {x^{2}}{\left (b \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2)^(3/2),x)

[Out]

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

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maxima [A]  time = 1.33, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{\sqrt {b x^{2}} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.93, size = 10, normalized size = 0.71 \begin {gather*} -\frac {1}{b^{3/2}\,\sqrt {x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2)^(3/2),x)

[Out]

-1/(b^(3/2)*(x^2)^(1/2))

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sympy [A]  time = 0.52, size = 15, normalized size = 1.07 \begin {gather*} - \frac {x^{2}}{b^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2)**(3/2),x)

[Out]

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

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