\(\int \frac {(b x^2)^{3/2}}{x^4} \, dx\) [18]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {b \sqrt {b x^2} \log (x)}{x} \]

[Out]

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

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} \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {b \sqrt {b x^2} \log (x)}{x} \]

[In]

Int[(b*x^2)^(3/2)/x^4,x]

[Out]

(b*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 {\left (b \sqrt {b x^2}\right ) \int \frac {1}{x} \, dx}{x} \\ & = \frac {b \sqrt {b x^2} \log (x)}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {\left (b x^2\right )^{3/2} \log (x)}{x^3} \]

[In]

Integrate[(b*x^2)^(3/2)/x^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88

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

[In]

int((b*x^2)^(3/2)/x^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {b x^{2}} b \log \left (x\right )}{x} \]

[In]

integrate((b*x^2)^(3/2)/x^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\frac {\left (b x^{2}\right )^{\frac {3}{2}} \log {\left (x \right )}}{x^{3}} \]

[In]

integrate((b*x**2)**(3/2)/x**4,x)

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((b*x^2)^(3/2)/x^4,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 = 9, normalized size of antiderivative = 0.56 \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=b^{\frac {3}{2}} \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) \]

[In]

integrate((b*x^2)^(3/2)/x^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (b x^2\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (b\,x^2\right )}^{3/2}}{x^4} \,d x \]

[In]

int((b*x^2)^(3/2)/x^4,x)

[Out]

int((b*x^2)^(3/2)/x^4, x)