3.133 \(\int \sqrt {b x^n} \, dx\)

Optimal. Leaf size=17 \[ \frac {2 x \sqrt {b x^n}}{n+2} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 17, 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} \[ \frac {2 x \sqrt {b x^n}}{n+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {2 x \sqrt {b x^n}}{n+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[b*x^n])/(2 + n)

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*x^n), x)

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maple [A]  time = 0.00, size = 16, normalized size = 0.94 \[ \frac {2 \sqrt {b \,x^{n}}\, x}{n +2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.31, size = 15, normalized size = 0.88 \[ \frac {2 \, \sqrt {b x^{n}} x}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(b*x^n)*x/(n + 2)

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mupad [B]  time = 1.48, size = 15, normalized size = 0.88 \[ \frac {2\,x\,\sqrt {b\,x^n}}{n+2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*x*(b*x^n)^(1/2))/(n + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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