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

Optimal. Leaf size=34 \[ \frac{2 x^{1-n} \sqrt{x^n+1} \sqrt{a x^{2 n}}}{n+2} \]

[Out]

(2*x^(1 - n)*Sqrt[a*x^(2*n)]*Sqrt[1 + x^n])/(2 + n)

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Rubi [C]  time = 0.0295837, antiderivative size = 80, normalized size of antiderivative = 2.35, number of steps used = 5, number of rules used = 3, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {15, 364, 245} \[ \frac{2 x^{1-n} \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-x^n\right )}{n+2}+\frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[a*x^(2*n)]*Hypergeometric2F1[1/2, 1 + n^(-1), 2 + n^(-1), -x^n])/(1 + n) + (2*x^(1 - n)*Sqrt[a*x^(2*n)
]*Hypergeometric2F1[1/2, n^(-1), 1 + n^(-1), -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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 245

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-p, 1/n, 1/n + 1, -((b*x^n)/a)],
x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p
] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \left (\frac{\sqrt{a x^{2 n}}}{\sqrt{1+x^n}}+\frac{2 x^{-n} \sqrt{a x^{2 n}}}{(2+n) \sqrt{1+x^n}}\right ) \, dx &=\frac{2 \int \frac{x^{-n} \sqrt{a x^{2 n}}}{\sqrt{1+x^n}} \, dx}{2+n}+\int \frac{\sqrt{a x^{2 n}}}{\sqrt{1+x^n}} \, dx\\ &=\left (x^{-n} \sqrt{a x^{2 n}}\right ) \int \frac{x^n}{\sqrt{1+x^n}} \, dx+\frac{\left (2 x^{-n} \sqrt{a x^{2 n}}\right ) \int \frac{1}{\sqrt{1+x^n}} \, dx}{2+n}\\ &=\frac{x \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},1+\frac{1}{n};2+\frac{1}{n};-x^n\right )}{1+n}+\frac{2 x^{1-n} \sqrt{a x^{2 n}} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-x^n\right )}{2+n}\\ \end{align*}

Mathematica [A]  time = 0.0292111, size = 33, normalized size = 0.97 \[ \frac{2 a x^{n+1} \sqrt{x^n+1}}{(n+2) \sqrt{a x^{2 n}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*x^(1 + n)*Sqrt[1 + x^n])/((2 + n)*Sqrt[a*x^(2*n)])

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Maple [A]  time = 0.023, size = 30, normalized size = 0.9 \begin{align*} 2\,{\frac{x\sqrt{1+{x}^{n}}\sqrt{a \left ({x}^{n} \right ) ^{2}}}{ \left ( 2+n \right ){x}^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.77577, size = 24, normalized size = 0.71 \begin{align*} \frac{2 \, \sqrt{a} \sqrt{x^{n} + 1} x}{n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(a)*sqrt(x^n + 1)*x/(n + 2)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{2 \sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx + \int \frac{n \sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx + \int \frac{2 x^{- n} \sqrt{a x^{2 n}}}{\sqrt{x^{n} + 1}}\, dx}{n + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(2*sqrt(a*x**(2*n))/sqrt(x**n + 1), x) + Integral(n*sqrt(a*x**(2*n))/sqrt(x**n + 1), x) + Integral(2*
x**(-n)*sqrt(a*x**(2*n))/sqrt(x**n + 1), x))/(n + 2)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{2 \, n}}}{\sqrt{x^{n} + 1}} + \frac{2 \, \sqrt{a x^{2 \, n}}}{{\left (n + 2\right )} \sqrt{x^{n} + 1} x^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x^(2*n))/sqrt(x^n + 1) + 2*sqrt(a*x^(2*n))/((n + 2)*sqrt(x^n + 1)*x^n), x)