\(\int \sqrt {b x^2} \, dx\) [5]

Optimal result

Integrand size = 9, antiderivative size = 14 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} x \sqrt {b x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} x \sqrt {b x^2} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} x \sqrt {b x^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.79

method	result	size
gosper	\(\frac {x \sqrt {b \,x^{2}}}{2}\)	\(11\)
default	\(\frac {x \sqrt {b \,x^{2}}}{2}\)	\(11\)
risch	\(\frac {x \sqrt {b \,x^{2}}}{2}\)	\(11\)
trager	\(\frac {\left (-1+x \right ) \left (1+x \right ) \sqrt {b \,x^{2}}}{2 x}\)	\(19\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} \, \sqrt {b x^{2}} x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^2} \, dx=\frac {x \sqrt {b x^{2}}}{2} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} \, \sqrt {b x^{2}} x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^2} \, dx=\frac {1}{2} \, \sqrt {b} x^{2} \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.77 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \sqrt {b x^2} \, dx=\frac {\sqrt {b}\,x\,\left |x\right |}{2} \]

[In]

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

[Out]

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