3.2652 \(\int x^{-1+n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=21 \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b n} \]

[Out]

(2*(a + b*x^n)^(3/2))/(3*b*n)

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Rubi [A]  time = 0.0055118, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {261} \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n)*Sqrt[a + b*x^n],x]

[Out]

(2*(a + b*x^n)^(3/2))/(3*b*n)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^{-1+n} \sqrt{a+b x^n} \, dx &=\frac{2 \left (a+b x^n\right )^{3/2}}{3 b n}\\ \end{align*}

Mathematica [A]  time = 0.0059373, size = 21, normalized size = 1. \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n)*Sqrt[a + b*x^n],x]

[Out]

(2*(a + b*x^n)^(3/2))/(3*b*n)

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Maple [A]  time = 0.017, size = 18, normalized size = 0.9 \begin{align*}{\frac{2}{3\,bn} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(a+b*x^n)^(3/2)/b/n

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Maxima [A]  time = 0.950077, size = 23, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{3 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x^n + a)^(3/2)/(b*n)

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Fricas [A]  time = 1.01527, size = 39, normalized size = 1.86 \begin{align*} \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{3 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x^n + a)^(3/2)/(b*n)

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Sympy [B]  time = 4.54024, size = 48, normalized size = 2.29 \begin{align*} \frac{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{n}}{a}}}{3 b n} + \frac{2 \sqrt{a} x^{n} \sqrt{1 + \frac{b x^{n}}{a}}}{3 n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+n)*(a+b*x**n)**(1/2),x)

[Out]

2*a**(3/2)*sqrt(1 + b*x**n/a)/(3*b*n) + 2*sqrt(a)*x**n*sqrt(1 + b*x**n/a)/(3*n)

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Giac [A]  time = 1.13554, size = 23, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{3 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x^n + a)^(3/2)/(b*n)