Integrand size = 29, antiderivative size = 163 \[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^7 g (1+m) \sqrt {d^2-e^2 x^2}}-\frac {e (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (2+m) \sqrt {d^2-e^2 x^2}} \]
(g*x)^(1+m)*hypergeom([9/2, 1/2+1/2*m],[3/2+1/2*m],e^2*x^2/d^2)*(1-e^2*x^2 /d^2)^(1/2)/d^7/g/(1+m)/(-e^2*x^2+d^2)^(1/2)-e*(g*x)^(2+m)*hypergeom([9/2, 1+1/2*m],[2+1/2*m],e^2*x^2/d^2)*(1-e^2*x^2/d^2)^(1/2)/d^8/g^2/(2+m)/(-e^2 *x^2+d^2)^(1/2)
Time = 1.11 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.75 \[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \left (-e (1+m) x \operatorname {Hypergeometric2F1}\left (\frac {9}{2},1+\frac {m}{2},2+\frac {m}{2},\frac {e^2 x^2}{d^2}\right )+d (2+m) \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )\right )}{d^8 (1+m) (2+m) \sqrt {d^2-e^2 x^2}} \]
(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(-(e*(1 + m)*x*Hypergeometric2F1[9/2, 1 + m/2, 2 + m/2, (e^2*x^2)/d^2]) + d*(2 + m)*Hypergeometric2F1[9/2, (1 + m )/2, (3 + m)/2, (e^2*x^2)/d^2]))/(d^8*(1 + m)*(2 + m)*Sqrt[d^2 - e^2*x^2])
Time = 0.28 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {583, 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.
\(\displaystyle \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 583 |
\(\displaystyle \int \frac {(d-e x) (g x)^m}{\left (d^2-e^2 x^2\right )^{9/2}}dx\) |
\(\Big \downarrow \) 557 |
\(\displaystyle d \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{9/2}}dx-\frac {e \int \frac {(g x)^{m+1}}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{g}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^m}{\left (1-\frac {e^2 x^2}{d^2}\right )^{9/2}}dx}{d^7 \sqrt {d^2-e^2 x^2}}-\frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^{m+1}}{\left (1-\frac {e^2 x^2}{d^2}\right )^{9/2}}dx}{d^8 g \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d^7 g (m+1) \sqrt {d^2-e^2 x^2}}-\frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (m+2) \sqrt {d^2-e^2 x^2}}\) |
((g*x)^(1 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[9/2, (1 + m)/2, ( 3 + m)/2, (e^2*x^2)/d^2])/(d^7*g*(1 + m)*Sqrt[d^2 - e^2*x^2]) - (e*(g*x)^( 2 + m)*Sqrt[1 - (e^2*x^2)/d^2]*Hypergeometric2F1[9/2, (2 + m)/2, (4 + m)/2 , (e^2*x^2)/d^2])/(d^8*g^2*(2 + m)*Sqrt[d^2 - e^2*x^2])
3.3.37.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, 0]
\[\int \frac {\left (g x \right )^{m}}{\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}d x\]
\[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}} \,d x } \]
integral(sqrt(-e^2*x^2 + d^2)*(g*x)^m/(e^9*x^9 + d*e^8*x^8 - 4*d^2*e^7*x^7 - 4*d^3*e^6*x^6 + 6*d^4*e^5*x^5 + 6*d^5*e^4*x^4 - 4*d^6*e^3*x^3 - 4*d^7*e ^2*x^2 + d^8*e*x + d^9), x)
\[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (g x\right )^{m}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )}\, dx \]
\[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}} \,d x } \]
\[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}} \,d x } \]
Timed out. \[ \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (g\,x\right )}^m}{{\left (d^2-e^2\,x^2\right )}^{7/2}\,\left (d+e\,x\right )} \,d x \]