\(\int \frac {\sqrt {b x^n}}{x^2} \, dx\) [135]

Optimal result

Integrand size = 13, antiderivative size = 21 \[ \int \frac {\sqrt {b x^n}}{x^2} \, dx=-\frac {2 \sqrt {b x^n}}{(2-n) x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, 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, 30} \[ \int \frac {\sqrt {b x^n}}{x^2} \, dx=-\frac {2 \sqrt {b x^n}}{(2-n) x} \]

[In]

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

[Out]

(-2*Sqrt[b*x^n])/((2 - n)*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 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}& = \left (x^{-n/2} \sqrt {b x^n}\right ) \int x^{-2+\frac {n}{2}} \, dx \\ & = -\frac {2 \sqrt {b x^n}}{(2-n) x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {b x^n}}{x^2} \, dx=\frac {2 \sqrt {b x^n}}{(-2+n) x} \]

[In]

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

[Out]

(2*Sqrt[b*x^n])/((-2 + n)*x)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {2 \sqrt {b \,x^{n}}}{x \left (-2+n \right )}\)	\(18\)
risch	\(\frac {2 b \,x^{n}}{\left (-2+n \right ) x \sqrt {b \,x^{n}}}\)	\(22\)

[In]

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

[Out]

2/x/(-2+n)*(b*x^n)^(1/2)

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {b x^n}}{x^2} \, dx=\begin {cases} \frac {2 \sqrt {b x^{n}}}{n x - 2 x} & \text {for}\: n \neq 2 \\\frac {\sqrt {b x^{2}} \log {\left (x \right )}}{x} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*sqrt(b*x**n)/(n*x - 2*x), Ne(n, 2)), (sqrt(b*x**2)*log(x)/x, True))

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(n/2-2>0)', see `assume?` for m
ore details)

Giac [F]

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

[In]

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

[Out]

integrate(sqrt(b*x^n)/x^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((b*x^n)^(1/2)/x^2, x)