3.2.48 \(\int \frac {1}{x^3 \sqrt {b x^n}} \, dx\) [148]

Optimal. Leaf size=19 \[ -\frac {2}{(4+n) x^2 \sqrt {b x^n}} \]

[Out]

-2/(4+n)/x^2/(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} \begin {gather*} -\frac {2}{(n+4) x^2 \sqrt {b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^3*Sqrt[b*x^n]),x]

[Out]

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

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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} -\frac {2}{(4+n) x^2 \sqrt {b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*Sqrt[b*x^n]),x]

[Out]

-2/((4 + n)*x^2*Sqrt[b*x^n])

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Maple [A]
time = 0.02, size = 18, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.30, size = 17, normalized size = 0.89 \begin {gather*} -\frac {2}{\sqrt {b x^{n}} {\left (n + 4\right )} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(b*x^n)*(n + 4)*x^2)

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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(1/x^3/(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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} - \frac {2}{n x^{2} \sqrt {b x^{n}} + 4 x^{2} \sqrt {b x^{n}}} & \text {for}\: n \neq -4 \\\int \frac {1}{x^{3} \sqrt {\frac {b}{x^{4}}}}\, dx & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2/(n*x**2*sqrt(b*x**n) + 4*x**2*sqrt(b*x**n)), Ne(n, -4)), (Integral(1/(x**3*sqrt(b/x**4)), x), Tr
ue))

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

[Out]

integrate(1/(sqrt(b*x^n)*x^3), x)

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Mupad [B]
time = 0.98, size = 24, normalized size = 1.26 \begin {gather*} -\frac {2\,\sqrt {b\,x^n}}{b\,x^{n+2}\,\left (n+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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