3.448 \(\int \sqrt{x^{2 (-1+n)} (a+b x^n)} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{3 n-2}\right )^{3/2}}{3 b n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0181042, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1979, 2000} \[ \frac{2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{3 n-2}\right )^{3/2}}{3 b n} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x^(2*(-1 + n))*(a + b*x^n)],x]

[Out]

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

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0306752, size = 36, normalized size = 0.82 \[ \frac{2 x^{3-3 n} \left (x^{2 n-2} \left (a+b x^n\right )\right )^{3/2}}{3 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x^(2*(-1 + n))*(a + b*x^n)],x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.028, size = 40, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,a+2\,b{x}^{n} \right ) x}{3\,b{x}^{n}n}\sqrt{{\frac{ \left ({x}^{n} \right ) ^{2} \left ( a+b{x}^{n} \right ) }{{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.17945, size = 23, normalized size = 0.52 \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^(-2+2*n)*(a+b*x^n))^(1/2),x, algorithm="maxima")

[Out]

2/3*(b*x^n + a)^(3/2)/(b*n)

________________________________________________________________________________________

Fricas [A]  time = 0.775232, size = 88, normalized size = 2. \begin{align*} \frac{2 \,{\left (b x x^{n} + a x\right )} \sqrt{\frac{b x^{3 \, n} + a x^{2 \, n}}{x^{2}}}}{3 \, b n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{{\left (b x^{n} + a\right )} x^{2 \, n - 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt((b*x^n + a)*x^(2*n - 2)), x)