\(\int (a+\frac {b}{x^2})^p (c x)^m \, dx\) [1964]

Optimal result

Integrand size = 15, antiderivative size = 70 \[ \int \left (a+\frac {b}{x^2}\right )^p (c x)^m \, dx=\frac {\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} (c x)^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-m),-p,\frac {1-m}{2},-\frac {b}{a x^2}\right )}{c (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {346, 372, 371} \[ \int \left (a+\frac {b}{x^2}\right )^p (c x)^m \, dx=\frac {(c x)^{m+1} \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-1),-p,\frac {1-m}{2},-\frac {b}{a x^2}\right )}{c (m+1)} \]

[In]

Int[(a + b/x^2)^p*(c*x)^m,x]

[Out]

((a + b/x^2)^p*(c*x)^(1 + m)*Hypergeometric2F1[(-1 - m)/2, -p, (1 - m)/2, -(b/(a*x^2))])/(c*(1 + m)*(1 + b/(a*
x^2))^p)

Rule 346

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(-c^(-1))*(c*x)^(m + 1)*(1/x)^(m + 1),
Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[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])

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 {\left (\left (\frac {1}{x}\right )^{1+m} (c x)^{1+m}\right ) \text {Subst}\left (\int x^{-2-m} \left (a+b x^2\right )^p \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {\left (\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} \left (\frac {1}{x}\right )^{1+m} (c x)^{1+m}\right ) \text {Subst}\left (\int x^{-2-m} \left (1+\frac {b x^2}{a}\right )^p \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} (c x)^{1+m} \, _2F_1\left (\frac {1}{2} (-1-m),-p;\frac {1-m}{2};-\frac {b}{a x^2}\right )}{c (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.04 \[ \int \left (a+\frac {b}{x^2}\right )^p (c x)^m \, dx=\frac {\left (a+\frac {b}{x^2}\right )^p x (c x)^m \left (1+\frac {a x^2}{b}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+m-2 p),-p,1+\frac {1}{2} (1+m-2 p),-\frac {a x^2}{b}\right )}{1+m-2 p} \]

[In]

Integrate[(a + b/x^2)^p*(c*x)^m,x]

[Out]

((a + b/x^2)^p*x*(c*x)^m*Hypergeometric2F1[(1 + m - 2*p)/2, -p, 1 + (1 + m - 2*p)/2, -((a*x^2)/b)])/((1 + m -
2*p)*(1 + (a*x^2)/b)^p)

Maple [F]

\[\int \left (a +\frac {b}{x^{2}}\right )^{p} \left (c x \right )^{m}d x\]

[In]

int((a+b/x^2)^p*(c*x)^m,x)

[Out]

int((a+b/x^2)^p*(c*x)^m,x)

Fricas [F]

\[ \int \left (a+\frac {b}{x^2}\right )^p (c x)^m \, dx=\int { \left (c x\right )^{m} {\left (a + \frac {b}{x^{2}}\right )}^{p} \,d x } \]

[In]

integrate((a+b/x^2)^p*(c*x)^m,x, algorithm="fricas")

[Out]

integral((c*x)^m*((a*x^2 + b)/x^2)^p, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.78 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \left (a+\frac {b}{x^2}\right )^p (c x)^m \, dx=- \frac {a^{p} c^{m} x^{m + 1} \Gamma \left (- \frac {m}{2} - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - \frac {m}{2} - \frac {1}{2} \\ \frac {1}{2} - \frac {m}{2} \end {matrix}\middle | {\frac {b e^{i \pi }}{a x^{2}}} \right )}}{2 \Gamma \left (\frac {1}{2} - \frac {m}{2}\right )} \]

[In]

integrate((a+b/x**2)**p*(c*x)**m,x)

[Out]

-a**p*c**m*x**(m + 1)*gamma(-m/2 - 1/2)*hyper((-p, -m/2 - 1/2), (1/2 - m/2,), b*exp_polar(I*pi)/(a*x**2))/(2*g
amma(1/2 - m/2))

Maxima [F]

\[ \int \left (a+\frac {b}{x^2}\right )^p (c x)^m \, dx=\int { \left (c x\right )^{m} {\left (a + \frac {b}{x^{2}}\right )}^{p} \,d x } \]

[In]

integrate((a+b/x^2)^p*(c*x)^m,x, algorithm="maxima")

[Out]

integrate((c*x)^m*(a + b/x^2)^p, x)

Giac [F]

\[ \int \left (a+\frac {b}{x^2}\right )^p (c x)^m \, dx=\int { \left (c x\right )^{m} {\left (a + \frac {b}{x^{2}}\right )}^{p} \,d x } \]

[In]

integrate((a+b/x^2)^p*(c*x)^m,x, algorithm="giac")

[Out]

integrate((c*x)^m*(a + b/x^2)^p, x)

Mupad [F(-1)]

Timed out. \[ \int \left (a+\frac {b}{x^2}\right )^p (c x)^m \, dx=\int {\left (c\,x\right )}^m\,{\left (a+\frac {b}{x^2}\right )}^p \,d x \]

[In]

int((c*x)^m*(a + b/x^2)^p,x)

[Out]

int((c*x)^m*(a + b/x^2)^p, x)