\(\int \frac {(g x)^m (d^2-e^2 x^2)^{5/2}}{d+e x} \, dx\) [230]

Optimal result

Integrand size = 29, antiderivative size = 163 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {d^3 (g x)^{1+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{g (1+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {d^2 e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {906, 83, 127, 372, 371} \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {d^3 \sqrt {d^2-e^2 x^2} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{g (m+1) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {d^2 e \sqrt {d^2-e^2 x^2} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[In]

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

[Out]

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

Rule 83

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

Rule 127

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[(a + b*x)^Frac
Part[m]*((c + d*x)^FracPart[m]/(a*c + b*d*x^2)^FracPart[m]), Int[(a*c + b*d*x^2)^m*(f*x)^p, x], x] /; FreeQ[{a
, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, 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])

Rule 906

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + c*x^
2)^FracPart[p]/((d + e*x)^FracPart[p]*(a/d + (c*x)/e)^FracPart[p]), Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (
c/e)*x)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c*d^2 + a*e^2, 0] &&  !Int
egerQ[p] &&  !IGtQ[m, 0] &&  !IGtQ[n, 0]

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

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.75 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {d^2 x (g x)^m \sqrt {d^2-e^2 x^2} \left (-e (1+m) x \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1+\frac {m}{2},2+\frac {m}{2},\frac {e^2 x^2}{d^2}\right )+d (2+m) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )\right )}{(1+m) (2+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((e^3*x^3 - d*e^2*x^2 - d^2*e*x + d^3)*sqrt(-e^2*x^2 + d^2)*(g*x)^m, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 10.03 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.49 \[ \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {d^{4} g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {d^{3} e g^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} - \frac {d^{2} e^{2} g^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {d e^{3} g^{m} x^{m + 4} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} \]

[In]

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

[Out]

d**4*g**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-1/2, 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)) - d**3*e*g**m*x**(m + 2)*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), e**2*x**2*exp
_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 2)) - d**2*e**2*g**m*x**(m + 3)*gamma(m/2 + 3/2)*hyper((-1/2, m/2 + 3/2),
(m/2 + 5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 5/2)) + d*e**3*g**m*x**(m + 4)*gamma(m/2 + 2)*h
yper((-1/2, m/2 + 2), (m/2 + 3,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 3))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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