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

Optimal. Leaf size=37 \[ \frac {x \sqrt {a x^{2 n}} \, _2F_1\left (\frac {1}{2},1+\frac {1}{n};2+\frac {1}{n};-x^n\right )}{n+1} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {15, 364} \[ \frac {x \sqrt {a x^{2 n}} \, _2F_1\left (\frac {1}{2},1+\frac {1}{n};2+\frac {1}{n};-x^n\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^(2*n)]/Sqrt[1 + x^n],x]

[Out]

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

Rubi steps

\begin {align*} \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 {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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 37, normalized size = 1.00 \[ \frac {x \sqrt {a x^{2 n}} \, _2F_1\left (\frac {1}{2},1+\frac {1}{n};2+\frac {1}{n};-x^n\right )}{n+1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^(2*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)

________________________________________________________________________________________

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),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \,x^{2 n}}}{\sqrt {x^{n}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {a\,x^{2\,n}}}{\sqrt {x^n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________