\(\int \frac {x^2}{(b x^2)^{3/2}} \, dx\) [49]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {x^2}{\left (b x^2\right )^{3/2}} \, dx=\frac {x \log (x)}{b \sqrt {b x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, 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} \[ \int \frac {x^2}{\left (b x^2\right )^{3/2}} \, dx=\frac {x \log (x)}{b \sqrt {b x^2}} \]

[In]

Int[x^2/(b*x^2)^(3/2),x]

[Out]

(x*Log[x])/(b*Sqrt[b*x^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 29

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{\left (b x^2\right )^{3/2}} \, dx=\frac {x^3 \log (x)}{\left (b x^2\right )^{3/2}} \]

[In]

Integrate[x^2/(b*x^2)^(3/2),x]

[Out]

(x^3*Log[x])/(b*x^2)^(3/2)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {x^{3} \ln \left (x \right )}{\left (b \,x^{2}\right )^{\frac {3}{2}}}\)	\(14\)
risch	\(\frac {x \ln \left (x \right )}{b \sqrt {b \,x^{2}}}\)	\(15\)

[In]

int(x^2/(b*x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/(b*x^2)^(3/2)*x^3*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\left (b x^2\right )^{3/2}} \, dx=\frac {\sqrt {b x^{2}} \log \left (x\right )}{b^{2} x} \]

[In]

integrate(x^2/(b*x^2)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\left (b x^2\right )^{3/2}} \, dx=\frac {x^{3} \log {\left (x \right )}}{\left (b x^{2}\right )^{\frac {3}{2}}} \]

[In]

integrate(x**2/(b*x**2)**(3/2),x)

[Out]

x**3*log(x)/(b*x**2)**(3/2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.38 \[ \int \frac {x^2}{\left (b x^2\right )^{3/2}} \, dx=\frac {\log \left (x\right )}{b^{\frac {3}{2}}} \]

[In]

integrate(x^2/(b*x^2)^(3/2),x, algorithm="maxima")

[Out]

log(x)/b^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int \frac {x^2}{\left (b x^2\right )^{3/2}} \, dx=\frac {\log \left ({\left | x \right |}\right )}{b^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x^2/(b*x^2)^(3/2),x, algorithm="giac")

[Out]

log(abs(x))/(b^(3/2)*sgn(x))

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.31 \[ \int \frac {x^2}{\left (b x^2\right )^{3/2}} \, dx=-\frac {x-\ln \left (2\,\sqrt {b}\,\sqrt {x^2}+2\,\sqrt {b}\,x\right )\,\sqrt {x^2}}{b^{3/2}\,\sqrt {x^2}} \]

[In]

int(x^2/(b*x^2)^(3/2),x)

[Out]

-(x - log(2*b^(1/2)*(x^2)^(1/2) + 2*b^(1/2)*x)*(x^2)^(1/2))/(b^(3/2)*(x^2)^(1/2))