3.2.60 \(\int \frac {x^m}{\sqrt {b x^n}} \, dx\) [160]

Optimal. Leaf size=26 \[ \frac {2 x^{1+m}}{(2+2 m-n) \sqrt {b x^n}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 26, 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 x^{m+1}}{(2 m-n+2) \sqrt {b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 25, normalized size = 0.96 \begin {gather*} \frac {x^{1+m}}{\left (1+m-\frac {n}{2}\right ) \sqrt {b x^n}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m)/((1 + m - n/2)*Sqrt[b*x^n])

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 24, normalized size = 0.92

method	result	size
risch	\(\frac {2 x \,x^{m}}{\left (2+2 m -n \right ) \sqrt {b \,x^{n}}}\)	\(24\)
gosper	\(\frac {2 x^{1+m}}{\left (2+2 m -n \right ) \sqrt {b \,x^{n}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.32, size = 24, normalized size = 0.92 \begin {gather*} \frac {2 \, x x^{m}}{\sqrt {b} {\left (2 \, m - n + 2\right )} \sqrt {x^{n}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*x^m/(sqrt(b)*(2*m - n + 2)*sqrt(x^n))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*x*x**m/(2*m*sqrt(b*x**n) - n*sqrt(b*x**n) + 2*sqrt(b*x**n)), Ne(m, n/2 - 1)), (Integral(x**(n/2 -
1)/sqrt(b*x**n), x), True))

________________________________________________________________________________________

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

[Out]

integrate(x^m/sqrt(b*x^n), x)

________________________________________________________________________________________

Mupad [B]
time = 1.09, size = 30, normalized size = 1.15 \begin {gather*} \frac {2\,x^{m-n+1}\,\sqrt {b\,x^n}}{b\,\left (2\,m-n+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________