\(\int \frac {\sqrt {a+b x^n}}{x^3} \, dx\) [2491]

Optimal result

Integrand size = 15, antiderivative size = 51 \[ \int \frac {\sqrt {a+b x^n}}{x^3} \, dx=-\frac {\left (a+b x^n\right )^{3/2} \operatorname {Hypergeometric2F1}\left (1,\frac {3}{2}-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 a x^2} \]

[Out]

-1/2*(a+b*x^n)^(3/2)*hypergeom([1, 3/2-2/n],[(-2+n)/n],-b*x^n/a)/a/x^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 60, 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} \[ \int \frac {\sqrt {a+b x^n}}{x^3} \, dx=-\frac {\sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{n},-\frac {2-n}{n},-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {\frac {b x^n}{a}+1}} \]

[In]

Int[Sqrt[a + b*x^n]/x^3,x]

[Out]

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

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 {a+b x^n} \int \frac {\sqrt {1+\frac {b x^n}{a}}}{x^3} \, dx}{\sqrt {1+\frac {b x^n}{a}}} \\ & = -\frac {\sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},-\frac {2}{n};-\frac {2-n}{n};-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {1+\frac {b x^n}{a}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {a+b x^n}}{x^3} \, dx=-\frac {\sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {2}{n},1-\frac {2}{n},-\frac {b x^n}{a}\right )}{2 x^2 \sqrt {1+\frac {b x^n}{a}}} \]

[In]

Integrate[Sqrt[a + b*x^n]/x^3,x]

[Out]

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

Maple [F]

\[\int \frac {\sqrt {a +b \,x^{n}}}{x^{3}}d x\]

[In]

int((a+b*x^n)^(1/2)/x^3,x)

[Out]

int((a+b*x^n)^(1/2)/x^3,x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x^n}}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*x^n)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.76 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {a+b x^n}}{x^3} \, dx=\frac {a^{- \frac {2}{n}} a^{\frac {1}{2} + \frac {2}{n}} \Gamma \left (- \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {2}{n} \\ 1 - \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n x^{2} \Gamma \left (1 - \frac {2}{n}\right )} \]

[In]

integrate((a+b*x**n)**(1/2)/x**3,x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {a+b x^n}}{x^3} \, dx=\int { \frac {\sqrt {b x^{n} + a}}{x^{3}} \,d x } \]

[In]

integrate((a+b*x^n)^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^n + a)/x^3, x)

Giac [F]

\[ \int \frac {\sqrt {a+b x^n}}{x^3} \, dx=\int { \frac {\sqrt {b x^{n} + a}}{x^{3}} \,d x } \]

[In]

integrate((a+b*x^n)^(1/2)/x^3,x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n + a)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^n}}{x^3} \, dx=\int \frac {\sqrt {a+b\,x^n}}{x^3} \,d x \]

[In]

int((a + b*x^n)^(1/2)/x^3,x)

[Out]

int((a + b*x^n)^(1/2)/x^3, x)