3.1.6 \(\int \frac {\sqrt {b x^2}}{x} \, dx\) [6]

Optimal. Leaf size=9 \[ \sqrt {b x^2} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 9, 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, 8} \begin {gather*} \sqrt {b x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[b*x^2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.56 \begin {gather*} \frac {b x^2}{\sqrt {b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 8, normalized size = 0.89

method	result	size
derivativedivides	\(\sqrt {b \,x^{2}}\)	\(8\)
default	\(\sqrt {b \,x^{2}}\)	\(8\)
risch	\(\sqrt {b \,x^{2}}\)	\(8\)
trager	\(\frac {\left (x -1\right ) \sqrt {b \,x^{2}}}{x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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,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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Fricas [A]
time = 0.35, size = 7, normalized size = 0.78 \begin {gather*} \sqrt {b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x^2)

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Sympy [A]
time = 0.05, size = 7, normalized size = 0.78 \begin {gather*} \sqrt {b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b*x**2)

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Giac [A]
time = 1.43, size = 7, normalized size = 0.78 \begin {gather*} \sqrt {b} x \mathrm {sgn}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*x*sgn(x)

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Mupad [B]
time = 0.93, size = 6, normalized size = 0.67 \begin {gather*} \sqrt {b}\,\left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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