Integrand size = 29, antiderivative size = 217 \[ \int \frac {(g x)^m}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 (g x)^{1+m} (d-e x)}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}+\frac {(7-2 m) (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )}{9 d^8 g (1+m) \sqrt {d^2-e^2 x^2}}-\frac {2 e (7-m) (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )}{9 d^9 g^2 (2+m) \sqrt {d^2-e^2 x^2}} \]
2/9*(g*x)^(1+m)*(-e*x+d)/d/g/(-e^2*x^2+d^2)^(9/2)+1/9*(7-2*m)*(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^8/g/(1+m)/(-e^2*x^2+d^2)^(1/2)-2/9*e*(7-m)*(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^9/g^2/(2+m)/(-e^2*x ^2+d^2)^(1/2)
Time = 1.32 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.81 \[ \int \frac {(g x)^m}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \left (d^2 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {11}{2},\frac {1+m}{2},\frac {3+m}{2},\frac {e^2 x^2}{d^2}\right )-e (1+m) x \left (2 d (3+m) \operatorname {Hypergeometric2F1}\left (\frac {11}{2},\frac {2+m}{2},\frac {4+m}{2},\frac {e^2 x^2}{d^2}\right )-e (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {11}{2},\frac {3+m}{2},\frac {5+m}{2},\frac {e^2 x^2}{d^2}\right )\right )\right )}{d^{10} (1+m) (2+m) (3+m) \sqrt {d^2-e^2 x^2}} \]
(x*(g*x)^m*Sqrt[1 - (e^2*x^2)/d^2]*(d^2*(6 + 5*m + m^2)*Hypergeometric2F1[ 11/2, (1 + m)/2, (3 + m)/2, (e^2*x^2)/d^2] - e*(1 + m)*x*(2*d*(3 + m)*Hype rgeometric2F1[11/2, (2 + m)/2, (4 + m)/2, (e^2*x^2)/d^2] - e*(2 + m)*x*Hyp ergeometric2F1[11/2, (3 + m)/2, (5 + m)/2, (e^2*x^2)/d^2])))/(d^10*(1 + m) *(2 + m)*(3 + m)*Sqrt[d^2 - e^2*x^2])
Time = 0.34 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {570, 558, 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.
| \(\displaystyle \int \frac {(g x)^m}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \int \frac {(d-e x)^2 (g x)^m}{\left (d^2-e^2 x^2\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 558 |
| \(\displaystyle \frac {2 (d-e x) (g x)^{m+1}}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}-\frac {\int -\frac {d (g x)^m (d (7-2 m)-2 e (7-m) x)}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{9 d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d (g x)^m (d (7-2 m)-2 e (7-m) x)}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{9 d^2}+\frac {2 (d-e x) (g x)^{m+1}}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(g x)^m (d (7-2 m)-2 e (7-m) x)}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{9 d}+\frac {2 (d-e x) (g x)^{m+1}}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle \frac {d (7-2 m) \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{9/2}}dx-\frac {2 e (7-m) \int \frac {(g x)^{m+1}}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{g}}{9 d}+\frac {2 (d-e x) (g x)^{m+1}}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 279 |
| \(\displaystyle \frac {\frac {(7-2 m) \sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^m}{\left (1-\frac {e^2 x^2}{d^2}\right )^{9/2}}dx}{d^7 \sqrt {d^2-e^2 x^2}}-\frac {2 e (7-m) \sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^{m+1}}{\left (1-\frac {e^2 x^2}{d^2}\right )^{9/2}}dx}{d^8 g \sqrt {d^2-e^2 x^2}}}{9 d}+\frac {2 (d-e x) (g x)^{m+1}}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle \frac {2 (d-e x) (g x)^{m+1}}{9 d g \left (d^2-e^2 x^2\right )^{9/2}}+\frac {\frac {(7-2 m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {m+1}{2},\frac {m+3}{2},\frac {e^2 x^2}{d^2}\right )}{d^7 g (m+1) \sqrt {d^2-e^2 x^2}}-\frac {2 e (7-m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},\frac {m+2}{2},\frac {m+4}{2},\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (m+2) \sqrt {d^2-e^2 x^2}}}{9 d}\) |
(2*(g*x)^(1 + m)*(d - e*x))/(9*d*g*(d^2 - e^2*x^2)^(9/2)) + (((7 - 2*m)*(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]) - (2*e*(7 - m)* (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]))/(9*d)
3.3.38.3.1 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] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, a + b*x^2, x], f = Coeff[PolynomialRemainder[(c + d*x)^n, a + b*x^2, x], x, 0], g = Coeff[Pol ynomialRemainder[(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(-(e*x)^(m + 1))* (f + g*x)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + f*(m + 2*p + 3) + g*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, m}, x] && IGt Q[n, 1] && !IntegerQ[m] && LtQ[p, -1]
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, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
\[\int \frac {\left (g x \right )^{m}}{\left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}d x\]
\[ \int \frac {(g x)^m}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{2}} \,d x } \]
integral(sqrt(-e^2*x^2 + d^2)*(g*x)^m/(e^10*x^10 + 2*d*e^9*x^9 - 3*d^2*e^8 *x^8 - 8*d^3*e^7*x^7 + 2*d^4*e^6*x^6 + 12*d^5*e^5*x^5 + 2*d^6*e^4*x^4 - 8* d^7*e^3*x^3 - 3*d^8*e^2*x^2 + 2*d^9*e*x + d^10), x)
Exception generated. \[ \int \frac {(g x)^m}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int \frac {(g x)^m}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{2}} \,d x } \]
\[ \int \frac {(g x)^m}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(g x)^m}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (g\,x\right )}^m}{{\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^2} \,d x \]