3.147 \(\int \frac {1}{x^2 \sqrt {b x^n}} \, dx\)

Optimal. Leaf size=19 \[ -\frac {2}{(n+2) x \sqrt {b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 19, 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} \[ -\frac {2}{(n+2) x \sqrt {b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[b*x^n]),x]

[Out]

-2/((2 + n)*x*Sqrt[b*x^n])

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 {1}{x^2 \sqrt {b x^n}} \, dx &=\frac {x^{n/2} \int x^{-2-\frac {n}{2}} \, dx}{\sqrt {b x^n}}\\ &=-\frac {2}{(2+n) x \sqrt {b x^n}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 1.00 \[ -\frac {2}{(n+2) x \sqrt {b x^n}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[b*x^n]),x]

[Out]

-2/((2 + n)*x*Sqrt[b*x^n])

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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(1/x^2/(b*x^n)^(1/2),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 \[ \int \frac {1}{\sqrt {b x^{n}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b*x^n)*x^2), x)

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maple [A]  time = 0.00, size = 18, normalized size = 0.95 \[ -\frac {2}{\left (n +2\right ) \sqrt {b \,x^{n}}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.33, size = 17, normalized size = 0.89 \[ -\frac {2}{\sqrt {b x^{n}} {\left (n + 2\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(b*x^n)*(n + 2)*x)

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mupad [B]  time = 0.97, size = 24, normalized size = 1.26 \[ -\frac {2\,\sqrt {b\,x^n}}{b\,x^{n+1}\,\left (n+2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} - \frac {2}{\sqrt {b} n x \sqrt {x^{n}} + 2 \sqrt {b} x \sqrt {x^{n}}} & \text {for}\: n \neq -2 \\\int \frac {1}{x^{2} \sqrt {\frac {b}{x^{2}}}}\, dx & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2/(sqrt(b)*n*x*sqrt(x**n) + 2*sqrt(b)*x*sqrt(x**n)), Ne(n, -2)), (Integral(1/(x**2*sqrt(b/x**2)),
x), True))

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