\(\int (g x)^m (d^2-e^2 x^2)^{5/2} \, dx\) [229]

Optimal result

Integrand size = 22, antiderivative size = 80 \[ \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^4 (g x)^{1+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{g (1+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[Out]

d^4*(g*x)^(1+m)*hypergeom([-5/2, 1/2+1/2*m],[3/2+1/2*m],e^2*x^2/d^2)*(-e^2*x^2+d^2)^(1/2)/g/(1+m)/(1-e^2*x^2/d
^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {372, 371} \[ \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^4 \sqrt {d^2-e^2 x^2} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{g (m+1) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[In]

Int[(g*x)^m*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(d^4*(g*x)^(1 + m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2])/(g*(1 + m
)*Sqrt[1 - (e^2*x^2)/d^2])

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 (d^4 \sqrt {d^2-e^2 x^2}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{5/2} \, dx}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \\ & = \frac {d^4 (g x)^{1+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^4 x (g x)^m \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{2},1+\frac {1+m}{2},\frac {e^2 x^2}{d^2}\right )}{(1+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[In]

Integrate[(g*x)^m*(d^2 - e^2*x^2)^(5/2),x]

[Out]

(d^4*x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (1 + m)/2, 1 + (1 + m)/2, (e^2*x^2)/d^2])/((1 + m)*
Sqrt[1 - (e^2*x^2)/d^2])

Maple [F]

\[\int \left (g x \right )^{m} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}d x\]

[In]

int((g*x)^m*(-e^2*x^2+d^2)^(5/2),x)

[Out]

int((g*x)^m*(-e^2*x^2+d^2)^(5/2),x)

Fricas [F]

\[ \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m} \,d x } \]

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^(5/2),x, algorithm="fricas")

[Out]

integral((e^4*x^4 - 2*d^2*e^2*x^2 + d^4)*sqrt(-e^2*x^2 + d^2)*(g*x)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.76 \[ \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^{5} g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} \]

[In]

integrate((g*x)**m*(-e**2*x**2+d**2)**(5/2),x)

[Out]

d**5*g**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-5/2, m/2 + 1/2), (m/2 + 3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)
/(2*gamma(m/2 + 3/2))

Maxima [F]

\[ \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m} \,d x } \]

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m, x)

Giac [F]

\[ \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m} \,d x } \]

[In]

integrate((g*x)^m*(-e^2*x^2+d^2)^(5/2),x, algorithm="giac")

[Out]

integrate((-e^2*x^2 + d^2)^(5/2)*(g*x)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m \,d x \]

[In]

int((d^2 - e^2*x^2)^(5/2)*(g*x)^m,x)

[Out]

int((d^2 - e^2*x^2)^(5/2)*(g*x)^m, x)