3.2.34 \(\int \frac {\sqrt {b x^n}}{x} \, dx\) [134]

Optimal. Leaf size=14 \[ \frac {2 \sqrt {b x^n}}{n} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 14, 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}}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b x^n}}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.01, size = 13, normalized size = 0.93

method	result	size
gosper	\(\frac {2 \sqrt {b \,x^{n}}}{n}\)	\(13\)
derivativedivides	\(\frac {2 \sqrt {b \,x^{n}}}{n}\)	\(13\)
default	\(\frac {2 \sqrt {b \,x^{n}}}{n}\)	\(13\)
risch	\(\frac {2 b \,x^{n}}{n \sqrt {b \,x^{n}}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {b x^{n}}}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^n)/n

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Fricas [A]
time = 0.35, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {b x^{n}}}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^n)/n

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Sympy [A]
time = 0.10, size = 19, normalized size = 1.36 \begin {gather*} \begin {cases} \frac {2 \sqrt {b x^{n}}}{n} & \text {for}\: n \neq 0 \\\sqrt {b} \log {\left (x \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*sqrt(b*x**n)/n, Ne(n, 0)), (sqrt(b)*log(x), True))

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Giac [A]
time = 1.87, size = 12, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {b x^{n}}}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^n)/n

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Mupad [B]
time = 0.93, size = 12, normalized size = 0.86 \begin {gather*} \frac {2\,\sqrt {b\,x^n}}{n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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