\(\int \frac {x^4}{(b x^2)^{5/2}} \, dx\) [58]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {x^4}{\left (b x^2\right )^{5/2}} \, dx=\frac {x \log (x)}{b^2 \sqrt {b x^2}} \]

[Out]

x*ln(x)/b^2/(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} \[ \int \frac {x^4}{\left (b x^2\right )^{5/2}} \, dx=\frac {x \log (x)}{b^2 \sqrt {b x^2}} \]

[In]

Int[x^4/(b*x^2)^(5/2),x]

[Out]

(x*Log[x])/(b^2*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^2 \sqrt {b x^2}} \\ & = \frac {x \log (x)}{b^2 \sqrt {b x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\left (b x^2\right )^{5/2}} \, dx=\frac {x^5 \log (x)}{\left (b x^2\right )^{5/2}} \]

[In]

Integrate[x^4/(b*x^2)^(5/2),x]

[Out]

(x^5*Log[x])/(b*x^2)^(5/2)

Maple [A] (verified)

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

method	result	size
default	\(\frac {x^{5} \ln \left (x \right )}{\left (b \,x^{2}\right )^{\frac {5}{2}}}\)	\(14\)
risch	\(\frac {x \ln \left (x \right )}{b^{2} \sqrt {b \,x^{2}}}\)	\(15\)

[In]

int(x^4/(b*x^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/(b*x^2)^(5/2)*x^5*ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (b x^2\right )^{5/2}} \, dx=\frac {\sqrt {b x^{2}} \log \left (x\right )}{b^{3} x} \]

[In]

integrate(x^4/(b*x^2)^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\left (b x^2\right )^{5/2}} \, dx=\frac {x^{5} \log {\left (x \right )}}{\left (b x^{2}\right )^{\frac {5}{2}}} \]

[In]

integrate(x**4/(b*x**2)**(5/2),x)

[Out]

x**5*log(x)/(b*x**2)**(5/2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.38 \[ \int \frac {x^4}{\left (b x^2\right )^{5/2}} \, dx=\frac {\log \left (x\right )}{b^{\frac {5}{2}}} \]

[In]

integrate(x^4/(b*x^2)^(5/2),x, algorithm="maxima")

[Out]

log(x)/b^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int \frac {x^4}{\left (b x^2\right )^{5/2}} \, dx=\frac {\log \left ({\left | x \right |}\right )}{b^{\frac {5}{2}} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(x^4/(b*x^2)^(5/2),x, algorithm="giac")

[Out]

log(abs(x))/(b^(5/2)*sgn(x))

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (b x^2\right )^{5/2}} \, dx=\int \frac {x^4}{{\left (b\,x^2\right )}^{5/2}} \,d x \]

[In]

int(x^4/(b*x^2)^(5/2),x)

[Out]

int(x^4/(b*x^2)^(5/2), x)