3.2.59 \(\int x^m \sqrt {b x^n} \, dx\) [159]

Optimal. Leaf size=24 \[ \frac {2 x^{1+m} \sqrt {b x^n}}{2+2 m+n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 24, 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} \sqrt {b x^n}}{2 m+n+2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 25, normalized size = 1.04 \begin {gather*} \frac {x^{1+m} \sqrt {b x^n}}{1+m+\frac {n}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*Sqrt[b*x^n])/(1 + m + n/2)

________________________________________________________________________________________

Maple [A]
time = 0.03, size = 23, normalized size = 0.96

method	result	size
gosper	\(\frac {2 x^{1+m} \sqrt {b \,x^{n}}}{2+2 m +n}\)	\(23\)
risch	\(\frac {2 b x \,x^{m} x^{n}}{\left (2+2 m +n \right ) \sqrt {b \,x^{n}}}\)	\(26\)

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

________________________________________________________________________________________

Maxima [A]
time = 0.31, size = 22, normalized size = 0.92 \begin {gather*} \frac {2 \, \sqrt {b} x x^{m} \sqrt {x^{n}}}{2 \, m + n + 2} \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*sqrt(b)*x*x^m*sqrt(x^n)/(2*m + n + 2)

________________________________________________________________________________________

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} \sqrt {b x^{n}}}{2 m + n + 2} & \text {for}\: m \neq - \frac {n}{2} - 1 \\\int 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*sqrt(b*x**n)/(2*m + n + 2), Ne(m, -n/2 - 1)), (Integral(x**(-n/2 - 1)*sqrt(b*x**n), x), Tr
ue))

________________________________________________________________________________________

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(sqrt(b*x^n)*x^m, x)

________________________________________________________________________________________

Mupad [B]
time = 1.04, size = 22, normalized size = 0.92 \begin {gather*} \frac {2\,x^{m+1}\,\sqrt {b\,x^n}}{2\,m+n+2} \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 + 1)*(b*x^n)^(1/2))/(2*m + n + 2)

________________________________________________________________________________________