\(\int \frac {1}{x^2 \sqrt {a+b x^n}} \, dx\) [2505]

Optimal result

Integrand size = 15, antiderivative size = 49 \[ \int \frac {1}{x^2 \sqrt {a+b x^n}} \, dx=-\frac {\sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {372, 371} \[ \int \frac {1}{x^2 \sqrt {a+b x^n}} \, dx=-\frac {\sqrt {\frac {b x^n}{a}+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{x \sqrt {a+b x^n}} \]

[In]

Int[1/(x^2*Sqrt[a + b*x^n]),x]

[Out]

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

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])

Rule 372

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12 \[ \int \frac {1}{x^2 \sqrt {a+b x^n}} \, dx=-\frac {\sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{n},1-\frac {1}{n},-\frac {b x^n}{a}\right )}{x \sqrt {a+b x^n}} \]

[In]

Integrate[1/(x^2*Sqrt[a + b*x^n]),x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \sqrt {a+b x^n}} \, dx=\frac {a^{- \frac {1}{n}} a^{- \frac {1}{2} + \frac {1}{n}} \Gamma \left (- \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - \frac {1}{n} \\ 1 - \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n x \Gamma \left (1 - \frac {1}{n}\right )} \]

[In]

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

[Out]

a**(-1/2 + 1/n)*gamma(-1/n)*hyper((1/2, -1/n), (1 - 1/n,), b*x**n*exp_polar(I*pi)/a)/(a**(1/n)*n*x*gamma(1 - 1
/n))

Maxima [F]

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

[In]

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

[Out]

integrate(1/(sqrt(b*x^n + a)*x^2), x)

Giac [F]

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

[In]

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

[Out]

integrate(1/(sqrt(b*x^n + a)*x^2), x)

Mupad [F(-1)]

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

[In]

int(1/(x^2*(a + b*x^n)^(1/2)),x)

[Out]

int(1/(x^2*(a + b*x^n)^(1/2)), x)