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

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

[Out]

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

________________________________________________________________________________________

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} \[ \sqrt {b x^2} \]

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*}

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.72, size = 7, normalized size = 0.78 \[ \sqrt {b x^{2}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 7, normalized size = 0.78 \[ \sqrt {b} x \mathrm {sgn}\relax (x) \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 8, normalized size = 0.89 \[ \sqrt {b \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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.

________________________________________________________________________________________

mupad [B]  time = 0.93, size = 6, normalized size = 0.67 \[ \sqrt {b}\,\relax |x| \]

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)

________________________________________________________________________________________

sympy [A]  time = 0.17, size = 10, normalized size = 1.11 \[ \sqrt {b} \sqrt {x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*sqrt(x**2)

________________________________________________________________________________________