\(\int (g x)^m (d+e x) (d^2-e^2 x^2)^{5/2} \, dx\) [228]

Optimal result

Integrand size = 27, antiderivative size = 162 \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^5 (g x)^{1+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{g (1+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {d^4 e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[Out]

d^5*(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)+d^4*e*(g*x)^(2+m)*hypergeom([-5/2, 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 = 0.06 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {822, 372, 371} \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^5 \sqrt {d^2-e^2 x^2} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{g (m+1) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {d^4 e \sqrt {d^2-e^2 x^2} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[In]

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

[Out]

(d^5*(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]) + (d^4*e*(g*x)^(2 + m)*Sqrt[d^2 - e^2*x^2]*Hypergeometric2F1[-5/2, (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 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 822

Int[((e_.)*(x_))^(m_)*((f_) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[f, Int[(e*x)^m*(a + c*
x^2)^p, x], x] + Dist[g/e, Int[(e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, p}, x] &&  !Ration
alQ[m] &&  !IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = d \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx+\frac {e \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{5/2} \, dx}{g} \\ & = \frac {\left (d^5 \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 {\left (d^4 e \sqrt {d^2-e^2 x^2}\right ) \int (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^{5/2} \, dx}{g \sqrt {1-\frac {e^2 x^2}{d^2}}} \\ & = \frac {d^5 (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}}}+\frac {d^4 e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{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 = 0.63 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.75 \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^4 x (g x)^m \sqrt {d^2-e^2 x^2} \left (d (2+m) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )+e (1+m) x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )\right )}{(1+m) (2+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((e^5*x^5 + d*e^4*x^4 - 2*d^2*e^3*x^3 - 2*d^3*e^2*x^2 + d^4*e*x + d^5)*sqrt(-e^2*x^2 + d^2)*(g*x)^m, x
)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 8.75 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.26 \[ \int (g x)^m (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^{6} g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d^{5} e g^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} - \frac {d^{4} e^{2} g^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {d^{3} e^{3} g^{m} x^{m + 4} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac {m}{2} + 3\right )} + \frac {d^{2} e^{4} g^{m} x^{m + 5} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {5}{2} \\ \frac {m}{2} + \frac {7}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {d e^{5} g^{m} x^{m + 6} \Gamma \left (\frac {m}{2} + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 3 \\ \frac {m}{2} + 4 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 4\right )} \]

[In]

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

[Out]

d**6*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**5*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**4*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)/gamma(m/2 + 5/2) - d**3*e**3*g**m*x**(m + 4)*gamma(m/2 + 2)*hy
per((-1/2, m/2 + 2), (m/2 + 3,), e**2*x**2*exp_polar(2*I*pi)/d**2)/gamma(m/2 + 3) + d**2*e**4*g**m*x**(m + 5)*
gamma(m/2 + 5/2)*hyper((-1/2, m/2 + 5/2), (m/2 + 7/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/(2*gamma(m/2 + 7/2))
+ d*e**5*g**m*x**(m + 6)*gamma(m/2 + 3)*hyper((-1/2, m/2 + 3), (m/2 + 4,), e**2*x**2*exp_polar(2*I*pi)/d**2)/
(2*gamma(m/2 + 4))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)