3.1.8 \(\int \frac {\sqrt {b x^2}}{x^3} \, dx\)

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

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \begin {gather*} -\frac {\sqrt {b x^2}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^2]/x^3,x]

[Out]

-(Sqrt[b*x^2]/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 {\sqrt {b x^2}}{x^3} \, dx &=\frac {\sqrt {b x^2} \int \frac {1}{x^2} \, dx}{x}\\ &=-\frac {\sqrt {b x^2}}{x^2}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b x^2}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^2]/x^3,x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b x^2}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b*x^2]/x^3,x]

[Out]

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

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fricas [A]  time = 0.87, size = 12, normalized size = 0.86 \begin {gather*} -\frac {\sqrt {b x^{2}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 10, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {b} \mathrm {sgn}\relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(1/2)/x^3,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(1/2)/x^3,x)

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(1/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 0.90, size = 10, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {b}}{\sqrt {x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(1/2)/x^3,x)

[Out]

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

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sympy [A]  time = 0.39, size = 15, normalized size = 1.07 \begin {gather*} - \frac {\sqrt {b} \sqrt {x^{2}}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(1/2)/x**3,x)

[Out]

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

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