Integrand size = 27, antiderivative size = 124 \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(g x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-4+m),\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{d g (1+m) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (g x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-3+m),\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^2 g^2 (2+m) \left (d^2-e^2 x^2\right )^{5/2}} \]
(g*x)^(1+m)*hypergeom([1, -2+1/2*m],[3/2+1/2*m],e^2*x^2/d^2)/d/g/(1+m)/(-e ^2*x^2+d^2)^(5/2)+e*(g*x)^(2+m)*hypergeom([1, -3/2+1/2*m],[2+1/2*m],e^2*x^ 2/d^2)/d^2/g^2/(2+m)/(-e^2*x^2+d^2)^(5/2)
Time = 0.78 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \left (d (2+m) \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )+e (1+m) x \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )\right )}{d^6 (1+m) (2+m) \sqrt {d^2-e^2 x^2}} \]
(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(d*(2 + m)*Hypergeometric2F1[7/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] + e*(1 + m)*x*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2]))/(d^6*(1 + m)*(2 + m)*Sqrt[d^2 - e^2*x^2] )
Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.31, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {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.
\(\displaystyle \int \frac {(d+e x) (g x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 557 |
\(\displaystyle d \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{7/2}}dx+\frac {e \int \frac {(g x)^{m+1}}{\left (d^2-e^2 x^2\right )^{7/2}}dx}{g}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^{m+1}}{\left (1-\frac {e^2 x^2}{d^2}\right )^{7/2}}dx}{d^6 g \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^m}{\left (1-\frac {e^2 x^2}{d^2}\right )^{7/2}}dx}{d^5 \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^6 g^2 (m+2) \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d^5 g (m+1) \sqrt {d^2-e^2 x^2}}\) |
((g*x)^(1 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[7/2, (1 + m)/2, ( 3 + m)/2, (e^2*x^2)/d^2])/(d^5*g*(1 + m)*Sqrt[d^2 - e^2*x^2]) + (e*(g*x)^( 2 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[7/2, (2 + m)/2, (4 + m)/2 , (e^2*x^2)/d^2])/(d^6*g^2*(2 + m)*Sqrt[d^2 - e^2*x^2])
3.3.35.3.1 Defintions of rubi rules used
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]
\[\int \frac {\left (g x \right )^{m} \left (e x +d \right )}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}d x\]
\[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
integral(sqrt(-e^2*x^2 + d^2)*(g*x)^m/(e^7*x^7 - d*e^6*x^6 - 3*d^2*e^5*x^5 + 3*d^3*e^4*x^4 + 3*d^4*e^3*x^3 - 3*d^5*e^2*x^2 - d^6*e*x + d^7), x)
Result contains complex when optimal does not.
Time = 26.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {g^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{6} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {e g^{m} x^{m + 2} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{7} \Gamma \left (\frac {m}{2} + 2\right )} \]
g**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((7/2, m/2 + 1/2), (m/2 + 3/2,), e** 2*x**2*exp_polar(2*I*pi)/d**2)/(2*d**6*gamma(m/2 + 3/2)) + e*g**m*x**(m + 2)*gamma(m/2 + 1)*hyper((7/2, m/2 + 1), (m/2 + 2,), e**2*x**2*exp_polar(2* I*pi)/d**2)/(2*d**7*gamma(m/2 + 2))
\[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (g\,x\right )}^m\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]