\(\int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx\) [395]

Optimal result

Integrand size = 23, antiderivative size = 52 \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\frac {4 x \sqrt {a x^{n/2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1+\frac {4}{n}\right ),\frac {1}{4} \left (5+\frac {4}{n}\right ),-x^n\right )}{4+n} \]

[Out]

4*x*hypergeom([1/2, 1/4+1/n],[5/4+1/n],-x^n)*(a*x^(1/2*n))^(1/2)/(4+n)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {15, 371} \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\frac {4 x \sqrt {a x^{n/2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1+\frac {4}{n}\right ),\frac {1}{4} \left (5+\frac {4}{n}\right ),-x^n\right )}{n+4} \]

[In]

Int[Sqrt[a*x^(n/2)]/Sqrt[1 + x^n],x]

[Out]

(4*x*Sqrt[a*x^(n/2)]*Hypergeometric2F1[1/2, (1 + 4/n)/4, (5 + 4/n)/4, -x^n])/(4 + 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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps \begin{align*} \text {integral}& = \left (x^{-n/4} \sqrt {a x^{n/2}}\right ) \int \frac {x^{n/4}}{\sqrt {1+x^n}} \, dx \\ & = \frac {4 x \sqrt {a x^{n/2}} \, _2F_1\left (\frac {1}{2},\frac {1}{4} \left (1+\frac {4}{n}\right );\frac {1}{4} \left (5+\frac {4}{n}\right );-x^n\right )}{4+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\frac {4 x \sqrt {a x^{n/2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4}+\frac {1}{n},\frac {5}{4}+\frac {1}{n},-x^n\right )}{4+n} \]

[In]

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

[Out]

(4*x*Sqrt[a*x^(n/2)]*Hypergeometric2F1[1/2, 1/4 + n^(-1), 5/4 + n^(-1), -x^n])/(4 + n)

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.71

method	result	size
meijerg	\(\frac {4 x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},\frac {1}{4}+\frac {1}{n};\frac {5}{4}+\frac {1}{n};-x^{n}\right ) \sqrt {a \,x^{\frac {n}{2}}}}{4+n}\)	\(37\)

[In]

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

[Out]

4*x*hypergeom([1/2,1/4+1/n],[5/4+1/n],-x^n)*(a*x^(1/2*n))^(1/2)/(4+n)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F]

\[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\int \frac {\sqrt {a x^{\frac {n}{2}}}}{\sqrt {x^{n} + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\int { \frac {\sqrt {a x^{\frac {1}{2} \, n}}}{\sqrt {x^{n} + 1}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\int { \frac {\sqrt {a x^{\frac {1}{2} \, n}}}{\sqrt {x^{n} + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^{n/2}}}{\sqrt {1+x^n}} \, dx=\int \frac {\sqrt {a\,x^{n/2}}}{\sqrt {x^n+1}} \,d x \]

[In]

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

[Out]

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