Integrand size = 29, antiderivative size = 206 \[ \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}+\frac {d^6 (9+2 m) (g x)^{1+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{g (1+m) (8+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {2 d^5 e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
-(g*x)^(1+m)*(-e^2*x^2+d^2)^(7/2)/g/(8+m)+d^6*(9+2*m)*(g*x)^(1+m)*hypergeo m([-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)/ (8+m)/(1-e^2*x^2/d^2)^(1/2)+2*d^5*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 )
Time = 0.74 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.84 \[ \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^4 x (g x)^m \sqrt {d^2-e^2 x^2} \left (d^2 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )+e (1+m) x \left (2 d (3+m) \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )+e (2+m) x \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {3+m}{2},\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )\right )\right )}{(1+m) (2+m) (3+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
(d^4*x*(g*x)^m*Sqrt[d^2 - e^2*x^2]*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[ -5/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + e*(1 + m)*x*(2*d*(3 + m)*Hype rgeometric2F1[-5/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2] + e*(2 + m)*x*Hyp ergeometric2F1[-5/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])))/((1 + m)*(2 + m)*(3 + m)*Sqrt[1 - (e^2*x^2)/d^2])
Time = 0.34 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {559, 25, 27, 557, 279, 278}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} (g x)^m \, dx\) |
| \(\Big \downarrow \) 559 |
| \(\displaystyle -\frac {\int -d e^2 (g x)^m (d (2 m+9)+2 e (m+8) x) \left (d^2-e^2 x^2\right )^{5/2}dx}{e^2 (m+8)}-\frac {\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int d e^2 (g x)^m (d (2 m+9)+2 e (m+8) x) \left (d^2-e^2 x^2\right )^{5/2}dx}{e^2 (m+8)}-\frac {\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int (g x)^m (d (2 m+9)+2 e (m+8) x) \left (d^2-e^2 x^2\right )^{5/2}dx}{m+8}-\frac {\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle \frac {d \left (d (2 m+9) \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2}dx+\frac {2 e (m+8) \int (g x)^{m+1} \left (d^2-e^2 x^2\right )^{5/2}dx}{g}\right )}{m+8}-\frac {\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {d \left (\frac {d^5 (2 m+9) \sqrt {d^2-e^2 x^2} \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{5/2}dx}{\sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {2 d^4 e (m+8) \sqrt {d^2-e^2 x^2} \int (g x)^{m+1} \left (1-\frac {e^2 x^2}{d^2}\right )^{5/2}dx}{g \sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{m+8}-\frac {\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {d \left (\frac {d^5 (2 m+9) \sqrt {d^2-e^2 x^2} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{g (m+1) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {2 d^4 e (m+8) \sqrt {d^2-e^2 x^2} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt {1-\frac {e^2 x^2}{d^2}}}\right )}{m+8}-\frac {\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}\) |
-(((g*x)^(1 + m)*(d^2 - e^2*x^2)^(7/2))/(g*(8 + m))) + (d*((d^5*(9 + 2*m)* (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]) + (2*d^4*e*(8 + m)*(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])))/(8 + m)
3.3.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p]) Int[(c*x)^m* (1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1)) Int[(e*x)^m*(a + b* x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 )*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && IGtQ[n, 1] && !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
\[\int \left (g x \right )^{m} \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}d x\]
\[ \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{2} \left (g x\right )^{m} \,d x } \]
integral((e^6*x^6 + 2*d*e^5*x^5 - d^2*e^4*x^4 - 4*d^3*e^3*x^3 - d^4*e^2*x^ 2 + 2*d^5*e*x + d^6)*sqrt(-e^2*x^2 + d^2)*(g*x)^m, x)
Result contains complex when optimal does not.
Time = 11.94 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.10 \[ \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {d^{7} 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^{6} 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 )}}{\Gamma \left (\frac {m}{2} + 2\right )} - \frac {d^{5} 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 {2 d^{4} 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^{3} 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^{2} 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 )}}{\Gamma \left (\frac {m}{2} + 4\right )} + \frac {d e^{6} g^{m} x^{m + 7} \Gamma \left (\frac {m}{2} + \frac {7}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {7}{2} \\ \frac {m}{2} + \frac {9}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {9}{2}\right )} \]
d**7*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**6*e*g**m*x* *(m + 2)*gamma(m/2 + 1)*hyper((-1/2, m/2 + 1), (m/2 + 2,), e**2*x**2*exp_p olar(2*I*pi)/d**2)/gamma(m/2 + 2) - d**5*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)) - 2*d**4*e**3*g**m*x**(m + 4)*gamma(m/2 + 2)*hype r((-1/2, m/2 + 2), (m/2 + 3,), e**2*x**2*exp_polar(2*I*pi)/d**2)/gamma(m/2 + 3) - d**3*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 **2*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)/gamma(m/2 + 4) + d*e**6*g**m*x**(m + 7) *gamma(m/2 + 7/2)*hyper((-1/2, m/2 + 7/2), (m/2 + 9/2,), e**2*x**2*exp_pol ar(2*I*pi)/d**2)/(2*gamma(m/2 + 9/2))
\[ \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{2} \left (g x\right )^{m} \,d x } \]
\[ \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int { {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{2} \left (g x\right )^{m} \,d x } \]
Timed out. \[ \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^2 \,d x \]