3.2.36 \(\int \frac {\sqrt {b x^n}}{x^3} \, dx\) [136]

Optimal. Leaf size=21 \[ -\frac {2 \sqrt {b x^n}}{(4-n) x^2} \]

[Out]

-2*(b*x^n)^(1/2)/(4-n)/x^2

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Rubi [A]
time = 0.00, 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} \begin {gather*} -\frac {2 \sqrt {b x^n}}{(4-n) x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^n]/x^3,x]

[Out]

(-2*Sqrt[b*x^n])/((4 - n)*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 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^3} \, dx &=\left (x^{-n/2} \sqrt {b x^n}\right ) \int x^{-3+\frac {n}{2}} \, dx\\ &=-\frac {2 \sqrt {b x^n}}{(4-n) x^2}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 0.90 \begin {gather*} \frac {2 \sqrt {b x^n}}{(-4+n) x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^n]/x^3,x]

[Out]

(2*Sqrt[b*x^n])/((-4 + n)*x^2)

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Maple [A]
time = 0.01, size = 18, normalized size = 0.86

method	result	size
gosper	\(\frac {2 \sqrt {b \,x^{n}}}{x^{2} \left (-4+n \right )}\)	\(18\)
risch	\(\frac {2 b \,x^{n}}{\left (-4+n \right ) x^{2} \sqrt {b \,x^{n}}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(1/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

2/x^2/(-4+n)*(b*x^n)^(1/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(1/2)/x^3,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/2-3>0)', see `assume?` for m
ore details)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {2 \sqrt {b x^{n}}}{n x^{2} - 4 x^{2}} & \text {for}\: n \neq 4 \\\int \frac {\sqrt {b x^{4}}}{x^{3}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**n)**(1/2)/x**3,x)

[Out]

Piecewise((2*sqrt(b*x**n)/(n*x**2 - 4*x**2), Ne(n, 4)), (Integral(sqrt(b*x**4)/x**3, x), True))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(1/2)/x^3,x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n)/x^3, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {b\,x^n}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(1/2)/x^3,x)

[Out]

int((b*x^n)^(1/2)/x^3, x)

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