\(\int \frac {\sqrt {b x^2}}{x^2} \, dx\) [7]

Optimal result

Integrand size = 13, antiderivative size = 15 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {\sqrt {b x^2} \log (x)}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, 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, 29} \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {\sqrt {b x^2} \log (x)}{x} \]

[In]

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

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x^2} \int \frac {1}{x} \, dx}{x} \\ & = \frac {\sqrt {b x^2} \log (x)}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {b x \log (x)}{\sqrt {b x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\ln \left (x \right ) \sqrt {b \,x^{2}}}{x}\)	\(14\)
risch	\(\frac {\ln \left (x \right ) \sqrt {b \,x^{2}}}{x}\)	\(14\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {\sqrt {b x^{2}} \log \left (x\right )}{x} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\frac {\sqrt {b x^{2}} \log {\left (x \right )}}{x} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\sqrt {b} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

sqrt(b)*log(abs(x))*sgn(x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b x^2}}{x^2} \, dx=\int \frac {\sqrt {b\,x^2}}{x^2} \,d x \]

[In]

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

[Out]

int((b*x^2)^(1/2)/x^2, x)