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

Optimal result

Integrand size = 13, antiderivative size = 9 \[ \int \frac {\sqrt {b x^2}}{x} \, dx=\sqrt {b x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 8} \[ \int \frac {\sqrt {b x^2}}{x} \, dx=\sqrt {b x^2} \]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b x^2}}{x} \, dx=\sqrt {b x^2} \]

[In]

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

[Out]

Sqrt[b*x^2]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.89

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {b x^2}}{x} \, dx=\sqrt {b x^{2}} \]

[In]

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

[Out]

sqrt(b*x^2)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {b x^2}}{x} \, dx=\sqrt {b x^{2}} \]

[In]

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

[Out]

sqrt(b*x**2)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {b x^2}}{x} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {b x^2}}{x} \, dx=\sqrt {b} x \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

sqrt(b)*x*sgn(x)

Mupad [B] (verification not implemented)

Time = 5.74 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {b x^2}}{x} \, dx=\sqrt {b}\,\left |x\right | \]

[In]

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

[Out]

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