3.135 \(\int \frac {\sqrt {b x^n}}{x^2} \, dx\)

Optimal. Leaf size=21 \[ -\frac {2 \sqrt {b x^n}}{(2-n) x} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {15, 30} \[ -\frac {2 \sqrt {b x^n}}{(2-n) x} \]

Antiderivative was successfully verified.

[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*} \int \frac {\sqrt {b x^n}}{x^2} \, dx &=\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*}

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Mathematica [A]  time = 0.00, size = 19, normalized size = 0.90 \[ \frac {2 \sqrt {b x^n}}{(n-2) x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[b*x^n])/((-2 + n)*x)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b x^{n}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x^n)/x^2, x)

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maple [A]  time = 0.00, size = 18, normalized size = 0.86 \[ \frac {2 \sqrt {b \,x^{n}}}{\left (n -2\right ) x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)Is n/2-2 equal to -1?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\sqrt {b\,x^n}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((b*x^n)^(1/2)/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {2 \sqrt {b} \sqrt {x^{n}}}{n x - 2 x} & \text {for}\: n \neq 2 \\\int \frac {\sqrt {b x^{2}}}{x^{2}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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