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

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Rubi [C]  time = 0.03, 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 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])

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])

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*}

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Mathematica [A]  time = 0.03, 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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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((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: 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 \[ \int \frac {\sqrt {a x^{2 \, n}}}{\sqrt {x^{n} + 1}} + \frac {2 \, \sqrt {a x^{2 \, n}}}{{\left (n + 2\right )} \sqrt {x^{n} + 1} x^{n}}\,{d x} \]

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)

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maple [A]  time = 0.05, size = 30, normalized size = 0.88 \[ \frac {2 \sqrt {x^{n}+1}\, \sqrt {a \,x^{2 n}}\, x \,x^{-n}}{n +2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.51, size = 18, normalized size = 0.53 \[ \frac {2 \, \sqrt {a} \sqrt {x^{n} + 1} x}{n + 2} \]

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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mupad [B]  time = 2.89, size = 43, normalized size = 1.26 \[ \frac {\sqrt {a\,x^{2\,n}}\,\left (\frac {2\,x}{n+2}+\frac {2\,x^{n+1}}{n+2}\right )}{x^n\,\sqrt {x^n+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {2 \sqrt {a x^{2 n}}}{\sqrt {x^{n} + 1}}\, dx + \int \frac {n \sqrt {a x^{2 n}}}{\sqrt {x^{n} + 1}}\, dx + \int \frac {2 x^{- n} \sqrt {a x^{2 n}}}{\sqrt {x^{n} + 1}}\, dx}{n + 2} \]

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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