3.1.10 \(\int \frac {\sqrt {b x^2}}{x^5} \, dx\) [10]

Optimal. Leaf size=16 \[ -\frac {\sqrt {b x^2}}{3 x^4} \]

[Out]

-1/3*(b*x^2)^(1/2)/x^4

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Rubi [A]
time = 0.00, antiderivative size = 16, 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 {\sqrt {b x^2}}{3 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^2]/x^5,x]

[Out]

-1/3*Sqrt[b*x^2]/x^4

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^2}}{x^5} \, dx &=\frac {\sqrt {b x^2} \int \frac {1}{x^4} \, dx}{x}\\ &=-\frac {\sqrt {b x^2}}{3 x^4}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {b x^2}}{3 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^2]/x^5,x]

[Out]

-1/3*Sqrt[b*x^2]/x^4

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Maple [A]
time = 0.02, size = 13, normalized size = 0.81

method	result	size
gosper	\(-\frac {\sqrt {b \,x^{2}}}{3 x^{4}}\)	\(13\)
default	\(-\frac {\sqrt {b \,x^{2}}}{3 x^{4}}\)	\(13\)
risch	\(-\frac {\sqrt {b \,x^{2}}}{3 x^{4}}\)	\(13\)
trager	\(\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \sqrt {b \,x^{2}}}{3 x^{4}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(1/2)/x^5,x,method=_RETURNVERBOSE)

[Out]

-1/3*(b*x^2)^(1/2)/x^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(1/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [A]
time = 0.34, size = 12, normalized size = 0.75 \begin {gather*} -\frac {\sqrt {b x^{2}}}{3 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(1/2)/x^5,x, algorithm="fricas")

[Out]

-1/3*sqrt(b*x^2)/x^4

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Sympy [A]
time = 0.18, size = 14, normalized size = 0.88 \begin {gather*} - \frac {\sqrt {b x^{2}}}{3 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2)**(1/2)/x**5,x)

[Out]

-sqrt(b*x**2)/(3*x**4)

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Giac [A]
time = 1.06, size = 10, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {b} \mathrm {sgn}\left (x\right )}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2)^(1/2)/x^5,x, algorithm="giac")

[Out]

-1/3*sqrt(b)*sgn(x)/x^3

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Mupad [B]
time = 0.95, size = 10, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {b}}{3\,{\left (x^2\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2)^(1/2)/x^5,x)

[Out]

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

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