3.1.89 \(\int \frac {1}{\sqrt {b x^3}} \, dx\) [89]

Optimal. Leaf size=12 \[ -\frac {2 x}{\sqrt {b x^3}} \]

[Out]

-2*x/(b*x^3)^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \begin {gather*} -\frac {2 x}{\sqrt {b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[b*x^3]

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}{\sqrt {b x^3}} \, dx &=\frac {x^{3/2} \int \frac {1}{x^{3/2}} \, dx}{\sqrt {b x^3}}\\ &=-\frac {2 x}{\sqrt {b x^3}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {2 x}{\sqrt {b x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*x)/Sqrt[b*x^3]

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Maple [A]
time = 0.02, size = 11, normalized size = 0.92

method	result	size
gosper	\(-\frac {2 x}{\sqrt {b \,x^{3}}}\)	\(11\)
default	\(-\frac {2 x}{\sqrt {b \,x^{3}}}\)	\(11\)
risch	\(-\frac {2 x}{\sqrt {b \,x^{3}}}\)	\(11\)
trager	\(-\frac {2 \sqrt {b \,x^{3}}}{b \,x^{2}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/(b*x^3)^(1/2)

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Maxima [A]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} -\frac {2 \, x}{\sqrt {b x^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/sqrt(b*x^3)

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Fricas [A]
time = 0.35, size = 15, normalized size = 1.25 \begin {gather*} -\frac {2 \, \sqrt {b x^{3}}}{b x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(b*x^3)/(b*x^2)

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Sympy [A]
time = 0.17, size = 12, normalized size = 1.00 \begin {gather*} - \frac {2 x}{\sqrt {b x^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x/sqrt(b*x**3)

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Giac [A]
time = 1.32, size = 11, normalized size = 0.92 \begin {gather*} -\frac {2}{\sqrt {b x} \mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(b*x)*sgn(x))

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Mupad [B]
time = 0.94, size = 15, normalized size = 1.25 \begin {gather*} -\frac {2\,\sqrt {b\,x^3}}{b\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(b*x^3)^(1/2))/(b*x^2)

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