\(\int (g x)^m (a+c x^2)^p \, dx\) [433]

Optimal result

Integrand size = 15, antiderivative size = 66 \[ \int (g x)^m \left (a+c x^2\right )^p \, dx=\frac {(g x)^{1+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {c x^2}{a}\right )}{g (1+m)} \]

[Out]

(g*x)^(1+m)*(c*x^2+a)^p*hypergeom([-p, 1/2+1/2*m],[3/2+1/2*m],-c*x^2/a)/g/(1+m)/((1+c*x^2/a)^p)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {372, 371} \[ \int (g x)^m \left (a+c x^2\right )^p \, dx=\frac {(g x)^{m+1} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {c x^2}{a}\right )}{g (m+1)} \]

[In]

Int[(g*x)^m*(a + c*x^2)^p,x]

[Out]

((g*x)^(1 + m)*(a + c*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((c*x^2)/a)])/(g*(1 + m)*(1 + (c*x^2
)/a)^p)

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}& = \left (\left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int (g x)^m \left (1+\frac {c x^2}{a}\right )^p \, dx \\ & = \frac {(g x)^{1+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};-\frac {c x^2}{a}\right )}{g (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int (g x)^m \left (a+c x^2\right )^p \, dx=\frac {x (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,1+\frac {1+m}{2},-\frac {c x^2}{a}\right )}{1+m} \]

[In]

Integrate[(g*x)^m*(a + c*x^2)^p,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((c*x^2 + a)^p*(g*x)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.52 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int (g x)^m \left (a+c x^2\right )^p \, dx=\frac {a^{p} g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]

[In]

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

[Out]

a**p*g**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-p, m/2 + 1/2), (m/2 + 3/2,), c*x**2*exp_polar(I*pi)/a)/(2*gamma(
m/2 + 3/2))

Maxima [F]

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

[In]

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

[Out]

integrate((c*x^2 + a)^p*(g*x)^m, x)

Giac [F]

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

[In]

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

[Out]

integrate((c*x^2 + a)^p*(g*x)^m, x)

Mupad [F(-1)]

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

[In]

int((g*x)^m*(a + c*x^2)^p,x)

[Out]

int((g*x)^m*(a + c*x^2)^p, x)