Integrand size = 34, antiderivative size = 180 \[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5} \, dx=-\frac {\left (C d^2-B d e+A e^2\right ) \left (d^2-e^2 x^2\right )^{3/2}}{7 d e^3 (d+e x)^5}+\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}-\frac {\left (23 C d^2+e (5 B d+2 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{35 d^2 e^3 (d+e x)^4}-\frac {\left (23 C d^2+e (5 B d+2 A e)\right ) \left (d^2-e^2 x^2\right )^{3/2}}{105 d^3 e^3 (d+e x)^3} \]
-1/7*(A*e^2-B*d*e+C*d^2)*(-e^2*x^2+d^2)^(3/2)/d/e^3/(e*x+d)^5+C*(-e^2*x^2+ d^2)^(3/2)/e^3/(e*x+d)^4-1/35*(23*C*d^2+e*(2*A*e+5*B*d))*(-e^2*x^2+d^2)^(3 /2)/d^2/e^3/(e*x+d)^4-1/105*(23*C*d^2+e*(2*A*e+5*B*d))*(-e^2*x^2+d^2)^(3/2 )/d^3/e^3/(e*x+d)^3
Time = 0.79 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.61 \[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5} \, dx=-\frac {(d-e x) \sqrt {d^2-e^2 x^2} \left (C d^2 \left (2 d^2+10 d e x+23 e^2 x^2\right )+e \left (5 B d \left (d^2+5 d e x+e^2 x^2\right )+A e \left (23 d^2+10 d e x+2 e^2 x^2\right )\right )\right )}{105 d^3 e^3 (d+e x)^4} \]
-1/105*((d - e*x)*Sqrt[d^2 - e^2*x^2]*(C*d^2*(2*d^2 + 10*d*e*x + 23*e^2*x^ 2) + e*(5*B*d*(d^2 + 5*d*e*x + e^2*x^2) + A*e*(23*d^2 + 10*d*e*x + 2*e^2*x ^2))))/(d^3*e^3*(d + e*x)^4)
Time = 0.38 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2170, 27, 671, 461, 460}
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.
| \(\displaystyle \int \frac {\sqrt {d^2-e^2 x^2} \left (A+B x+C x^2\right )}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 2170 |
| \(\displaystyle \frac {\int \frac {e^2 \left (4 C d^2+A e^2+e (3 C d+B e) x\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5}dx}{e^4}+\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (4 C d^2+A e^2+e (3 C d+B e) x\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5}dx}{e^2}+\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}\) |
| \(\Big \downarrow \) 671 |
| \(\displaystyle \frac {\frac {\left (e (2 A e+5 B d)+23 C d^2\right ) \int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^4}dx}{7 d}+\frac {\left (d^2-e^2 x^2\right )^{3/2} \left (-\frac {A e}{d}+B-\frac {C d}{e}\right )}{7 (d+e x)^5}}{e^2}+\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {\frac {\left (e (2 A e+5 B d)+23 C d^2\right ) \left (\frac {\int \frac {\sqrt {d^2-e^2 x^2}}{(d+e x)^3}dx}{5 d}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}\right )}{7 d}+\frac {\left (d^2-e^2 x^2\right )^{3/2} \left (-\frac {A e}{d}+B-\frac {C d}{e}\right )}{7 (d+e x)^5}}{e^2}+\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}\) |
| \(\Big \downarrow \) 460 |
| \(\displaystyle \frac {\frac {\left (d^2-e^2 x^2\right )^{3/2} \left (-\frac {A e}{d}+B-\frac {C d}{e}\right )}{7 (d+e x)^5}+\frac {\left (-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{15 d^2 e (d+e x)^3}-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{5 d e (d+e x)^4}\right ) \left (e (2 A e+5 B d)+23 C d^2\right )}{7 d}}{e^2}+\frac {C \left (d^2-e^2 x^2\right )^{3/2}}{e^3 (d+e x)^4}\) |
(C*(d^2 - e^2*x^2)^(3/2))/(e^3*(d + e*x)^4) + (((B - (C*d)/e - (A*e)/d)*(d ^2 - e^2*x^2)^(3/2))/(7*(d + e*x)^5) + ((23*C*d^2 + e*(5*B*d + 2*A*e))*(-1 /5*(d^2 - e^2*x^2)^(3/2)/(d*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(3/2)/(15*d^2 *e*(d + e*x)^3)))/(7*d))/e^2
3.1.8.3.1 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_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 2, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl ify[n + 2*p + 2]/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d*(m + p + 1))), x] + Simp[(m*(g*c*d + c*e*f) + 2*e*c*f*(p + 1))/(e*(2*c*d)*(m + p + 1)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && ((LtQ[m, -1] && !IGtQ[m + p + 1, 0]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) ^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si mp[1/(b*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - 2*e*f*(m + p + q)*(d + e*x)^(q - 2)*(a*e - b*d*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d, e, m, p}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && !IGtQ[m, 0]
Time = 0.66 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.64
| method | result | size |
| gosper | \(-\frac {\left (-e x +d \right ) \left (2 A \,e^{4} x^{2}+5 x^{2} d B \,e^{3}+23 C \,d^{2} e^{2} x^{2}+10 A d \,e^{3} x +25 x B \,d^{2} e^{2}+10 C \,d^{3} x e +23 A \,d^{2} e^{2}+5 B \,d^{3} e +2 C \,d^{4}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{105 \left (e x +d \right )^{4} d^{3} e^{3}}\) | \(116\) |
| trager | \(-\frac {\left (-2 A \,e^{5} x^{3}-5 x^{3} d B \,e^{4}-23 C \,d^{2} e^{3} x^{3}-8 A d \,e^{4} x^{2}-20 x^{2} d^{2} B \,e^{3}+13 C \,d^{3} e^{2} x^{2}-13 A \,d^{2} e^{3} x +20 x \,d^{3} B \,e^{2}+8 C \,d^{4} e x +23 A \,d^{3} e^{2}+5 B \,d^{4} e +2 C \,d^{5}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{105 d^{3} \left (e x +d \right )^{4} e^{3}}\) | \(146\) |
| default | \(-\frac {C \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{3 e^{6} d \left (x +\frac {d}{e}\right )^{3}}+\frac {\left (B e -2 C d \right ) \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{e^{6}}+\frac {\left (A \,e^{2}-B d e +C \,d^{2}\right ) \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 d e \left (x +\frac {d}{e}\right )^{5}}+\frac {2 e \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 d e \left (x +\frac {d}{e}\right )^{4}}-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 d^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 d}\right )}{e^{7}}\) | \(308\) |
-1/105*(-e*x+d)*(2*A*e^4*x^2+5*B*d*e^3*x^2+23*C*d^2*e^2*x^2+10*A*d*e^3*x+2 5*B*d^2*e^2*x+10*C*d^3*e*x+23*A*d^2*e^2+5*B*d^3*e+2*C*d^4)*(-e^2*x^2+d^2)^ (1/2)/(e*x+d)^4/d^3/e^3
Time = 0.38 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.78 \[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5} \, dx=-\frac {2 \, C d^{6} + 5 \, B d^{5} e + 23 \, A d^{4} e^{2} + {\left (2 \, C d^{2} e^{4} + 5 \, B d e^{5} + 23 \, A e^{6}\right )} x^{4} + 4 \, {\left (2 \, C d^{3} e^{3} + 5 \, B d^{2} e^{4} + 23 \, A d e^{5}\right )} x^{3} + 6 \, {\left (2 \, C d^{4} e^{2} + 5 \, B d^{3} e^{3} + 23 \, A d^{2} e^{4}\right )} x^{2} + 4 \, {\left (2 \, C d^{5} e + 5 \, B d^{4} e^{2} + 23 \, A d^{3} e^{3}\right )} x + {\left (2 \, C d^{5} + 5 \, B d^{4} e + 23 \, A d^{3} e^{2} - {\left (23 \, C d^{2} e^{3} + 5 \, B d e^{4} + 2 \, A e^{5}\right )} x^{3} + {\left (13 \, C d^{3} e^{2} - 20 \, B d^{2} e^{3} - 8 \, A d e^{4}\right )} x^{2} + {\left (8 \, C d^{4} e + 20 \, B d^{3} e^{2} - 13 \, A d^{2} e^{3}\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (d^{3} e^{7} x^{4} + 4 \, d^{4} e^{6} x^{3} + 6 \, d^{5} e^{5} x^{2} + 4 \, d^{6} e^{4} x + d^{7} e^{3}\right )}} \]
-1/105*(2*C*d^6 + 5*B*d^5*e + 23*A*d^4*e^2 + (2*C*d^2*e^4 + 5*B*d*e^5 + 23 *A*e^6)*x^4 + 4*(2*C*d^3*e^3 + 5*B*d^2*e^4 + 23*A*d*e^5)*x^3 + 6*(2*C*d^4* e^2 + 5*B*d^3*e^3 + 23*A*d^2*e^4)*x^2 + 4*(2*C*d^5*e + 5*B*d^4*e^2 + 23*A* d^3*e^3)*x + (2*C*d^5 + 5*B*d^4*e + 23*A*d^3*e^2 - (23*C*d^2*e^3 + 5*B*d*e ^4 + 2*A*e^5)*x^3 + (13*C*d^3*e^2 - 20*B*d^2*e^3 - 8*A*d*e^4)*x^2 + (8*C*d ^4*e + 20*B*d^3*e^2 - 13*A*d^2*e^3)*x)*sqrt(-e^2*x^2 + d^2))/(d^3*e^7*x^4 + 4*d^4*e^6*x^3 + 6*d^5*e^5*x^2 + 4*d^6*e^4*x + d^7*e^3)
\[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (A + B x + C x^{2}\right )}{\left (d + e x\right )^{5}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 945 vs. \(2 (166) = 332\).
Time = 0.20 (sec) , antiderivative size = 945, normalized size of antiderivative = 5.25 \[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5} \, dx=-\frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C d^{2}}{7 \, {\left (e^{7} x^{4} + 4 \, d e^{6} x^{3} + 6 \, d^{2} e^{5} x^{2} + 4 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} C d^{2}}{35 \, {\left (d e^{6} x^{3} + 3 \, d^{2} e^{5} x^{2} + 3 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C d^{2}}{105 \, {\left (d^{2} e^{5} x^{2} + 2 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C d^{2}}{105 \, {\left (d^{3} e^{4} x + d^{4} e^{3}\right )}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} B d}{7 \, {\left (e^{6} x^{4} + 4 \, d e^{5} x^{3} + 6 \, d^{2} e^{4} x^{2} + 4 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} B d}{35 \, {\left (d e^{5} x^{3} + 3 \, d^{2} e^{4} x^{2} + 3 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} B d}{105 \, {\left (d^{2} e^{4} x^{2} + 2 \, d^{3} e^{3} x + d^{4} e^{2}\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} B d}{105 \, {\left (d^{3} e^{3} x + d^{4} e^{2}\right )}} + \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}} C d}{5 \, {\left (e^{6} x^{3} + 3 \, d e^{5} x^{2} + 3 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C d}{15 \, {\left (d e^{5} x^{2} + 2 \, d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C d}{15 \, {\left (d^{2} e^{4} x + d^{3} e^{3}\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} A}{7 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} A}{35 \, {\left (d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x + d^{4} e\right )}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} A}{105 \, {\left (d^{2} e^{3} x^{2} + 2 \, d^{3} e^{2} x + d^{4} e\right )}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} A}{105 \, {\left (d^{3} e^{2} x + d^{4} e\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} B}{5 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} B}{15 \, {\left (d e^{4} x^{2} + 2 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} B}{15 \, {\left (d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} C}{3 \, {\left (e^{5} x^{2} + 2 \, d e^{4} x + d^{2} e^{3}\right )}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} C}{3 \, {\left (d e^{4} x + d^{2} e^{3}\right )}} \]
-2/7*sqrt(-e^2*x^2 + d^2)*C*d^2/(e^7*x^4 + 4*d*e^6*x^3 + 6*d^2*e^5*x^2 + 4 *d^3*e^4*x + d^4*e^3) + 1/35*sqrt(-e^2*x^2 + d^2)*C*d^2/(d*e^6*x^3 + 3*d^2 *e^5*x^2 + 3*d^3*e^4*x + d^4*e^3) + 2/105*sqrt(-e^2*x^2 + d^2)*C*d^2/(d^2* e^5*x^2 + 2*d^3*e^4*x + d^4*e^3) + 2/105*sqrt(-e^2*x^2 + d^2)*C*d^2/(d^3*e ^4*x + d^4*e^3) + 2/7*sqrt(-e^2*x^2 + d^2)*B*d/(e^6*x^4 + 4*d*e^5*x^3 + 6* d^2*e^4*x^2 + 4*d^3*e^3*x + d^4*e^2) - 1/35*sqrt(-e^2*x^2 + d^2)*B*d/(d*e^ 5*x^3 + 3*d^2*e^4*x^2 + 3*d^3*e^3*x + d^4*e^2) - 2/105*sqrt(-e^2*x^2 + d^2 )*B*d/(d^2*e^4*x^2 + 2*d^3*e^3*x + d^4*e^2) - 2/105*sqrt(-e^2*x^2 + d^2)*B *d/(d^3*e^3*x + d^4*e^2) + 4/5*sqrt(-e^2*x^2 + d^2)*C*d/(e^6*x^3 + 3*d*e^5 *x^2 + 3*d^2*e^4*x + d^3*e^3) - 2/15*sqrt(-e^2*x^2 + d^2)*C*d/(d*e^5*x^2 + 2*d^2*e^4*x + d^3*e^3) - 2/15*sqrt(-e^2*x^2 + d^2)*C*d/(d^2*e^4*x + d^3*e ^3) - 2/7*sqrt(-e^2*x^2 + d^2)*A/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^2*x + d^4*e) + 1/35*sqrt(-e^2*x^2 + d^2)*A/(d*e^4*x^3 + 3*d^2*e^3* x^2 + 3*d^3*e^2*x + d^4*e) + 2/105*sqrt(-e^2*x^2 + d^2)*A/(d^2*e^3*x^2 + 2 *d^3*e^2*x + d^4*e) + 2/105*sqrt(-e^2*x^2 + d^2)*A/(d^3*e^2*x + d^4*e) - 2 /5*sqrt(-e^2*x^2 + d^2)*B/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) + 1/15*sqrt(-e^2*x^2 + d^2)*B/(d*e^4*x^2 + 2*d^2*e^3*x + d^3*e^2) + 1/15*s qrt(-e^2*x^2 + d^2)*B/(d^2*e^3*x + d^3*e^2) - 2/3*sqrt(-e^2*x^2 + d^2)*C/( e^5*x^2 + 2*d*e^4*x + d^2*e^3) + 1/3*sqrt(-e^2*x^2 + d^2)*C/(d*e^4*x + d^2 *e^3)
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 608, normalized size of antiderivative = 3.38 \[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5} \, dx=-\frac {1}{420} \, {\left (\frac {3 \, {\left (5 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {\frac {2 \, d}{e x + d} - 1}\right )} C \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 35 \, {\left (3 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\frac {2 \, d}{e x + d} - 1}\right )} C \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 280 \, {\left ({\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {\frac {2 \, d}{e x + d} - 1}\right )} C \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - \frac {3 \, {\left (5 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {\frac {2 \, d}{e x + d} - 1}\right )} B e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d} + \frac {21 \, {\left (3 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\frac {2 \, d}{e x + d} - 1}\right )} B e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d} - \frac {70 \, {\left ({\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 3 \, \sqrt {\frac {2 \, d}{e x + d} - 1}\right )} B e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d} + \frac {3 \, {\left (5 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {\frac {2 \, d}{e x + d} - 1}\right )} A e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{2}} - \frac {7 \, {\left (3 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\frac {2 \, d}{e x + d} - 1}\right )} A e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{2}} - 420 \, C \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d e^{4}} + \frac {4 \, {\left (23 i \, C d^{2} + 5 i \, B d e + 2 i \, A e^{2}\right )} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{3} e^{4}}\right )} {\left | e \right |} \]
-1/420*((3*(5*(2*d/(e*x + d) - 1)^(7/2) + 21*(2*d/(e*x + d) - 1)^(5/2) + 3 5*(2*d/(e*x + d) - 1)^(3/2) + 35*sqrt(2*d/(e*x + d) - 1))*C*sgn(1/(e*x + d ))*sgn(e) - 35*(3*(2*d/(e*x + d) - 1)^(5/2) + 10*(2*d/(e*x + d) - 1)^(3/2) + 15*sqrt(2*d/(e*x + d) - 1))*C*sgn(1/(e*x + d))*sgn(e) + 280*((2*d/(e*x + d) - 1)^(3/2) + 3*sqrt(2*d/(e*x + d) - 1))*C*sgn(1/(e*x + d))*sgn(e) - 3 *(5*(2*d/(e*x + d) - 1)^(7/2) + 21*(2*d/(e*x + d) - 1)^(5/2) + 35*(2*d/(e* x + d) - 1)^(3/2) + 35*sqrt(2*d/(e*x + d) - 1))*B*e*sgn(1/(e*x + d))*sgn(e )/d + 21*(3*(2*d/(e*x + d) - 1)^(5/2) + 10*(2*d/(e*x + d) - 1)^(3/2) + 15* sqrt(2*d/(e*x + d) - 1))*B*e*sgn(1/(e*x + d))*sgn(e)/d - 70*((2*d/(e*x + d ) - 1)^(3/2) + 3*sqrt(2*d/(e*x + d) - 1))*B*e*sgn(1/(e*x + d))*sgn(e)/d + 3*(5*(2*d/(e*x + d) - 1)^(7/2) + 21*(2*d/(e*x + d) - 1)^(5/2) + 35*(2*d/(e *x + d) - 1)^(3/2) + 35*sqrt(2*d/(e*x + d) - 1))*A*e^2*sgn(1/(e*x + d))*sg n(e)/d^2 - 7*(3*(2*d/(e*x + d) - 1)^(5/2) + 10*(2*d/(e*x + d) - 1)^(3/2) + 15*sqrt(2*d/(e*x + d) - 1))*A*e^2*sgn(1/(e*x + d))*sgn(e)/d^2 - 420*C*sqr t(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))*sgn(e))/(d*e^4) + 4*(23*I*C*d^2 + 5* I*B*d*e + 2*I*A*e^2)*sgn(1/(e*x + d))*sgn(e)/(d^3*e^4))*abs(e)
Time = 13.56 (sec) , antiderivative size = 601, normalized size of antiderivative = 3.34 \[ \int \frac {\left (A+B x+C x^2\right ) \sqrt {d^2-e^2 x^2}}{(d+e x)^5} \, dx=\frac {B\,\sqrt {d^2-e^2\,x^2}}{21\,\left (d^3\,e^2+x\,d^2\,e^3\right )}-\frac {3\,B\,\sqrt {d^2-e^2\,x^2}}{7\,\left (d^3\,e^2+3\,d^2\,e^3\,x+3\,d\,e^4\,x^2+e^5\,x^3\right )}+\frac {2\,A\,\sqrt {d^2-e^2\,x^2}}{105\,\left (d^4\,e+2\,d^3\,e^2\,x+d^2\,e^3\,x^2\right )}+\frac {B\,\sqrt {d^2-e^2\,x^2}}{21\,\left (d^3\,e^2+2\,d^2\,e^3\,x+d\,e^4\,x^2\right )}-\frac {82\,C\,\sqrt {d^2-e^2\,x^2}}{105\,\left (d^2\,e^3+2\,d\,e^4\,x+e^5\,x^2\right )}+\frac {2\,A\,\sqrt {d^2-e^2\,x^2}}{105\,\left (d^4\,e+x\,d^3\,e^2\right )}+\frac {23\,C\,\sqrt {d^2-e^2\,x^2}}{105\,\left (d^2\,e^3+x\,d\,e^4\right )}-\frac {2\,A\,\sqrt {d^2-e^2\,x^2}}{7\,\left (d^4\,e+4\,d^3\,e^2\,x+6\,d^2\,e^3\,x^2+4\,d\,e^4\,x^3+e^5\,x^4\right )}+\frac {A\,\sqrt {d^2-e^2\,x^2}}{35\,\left (d^4\,e+3\,d^3\,e^2\,x+3\,d^2\,e^3\,x^2+d\,e^4\,x^3\right )}-\frac {2\,C\,d^2\,\sqrt {d^2-e^2\,x^2}}{7\,\left (d^4\,e^3+4\,d^3\,e^4\,x+6\,d^2\,e^5\,x^2+4\,d\,e^6\,x^3+e^7\,x^4\right )}+\frac {2\,B\,d\,\sqrt {d^2-e^2\,x^2}}{7\,\left (d^4\,e^2+4\,d^3\,e^3\,x+6\,d^2\,e^4\,x^2+4\,d\,e^5\,x^3+e^6\,x^4\right )}+\frac {29\,C\,d\,\sqrt {d^2-e^2\,x^2}}{35\,\left (d^3\,e^3+3\,d^2\,e^4\,x+3\,d\,e^5\,x^2+e^6\,x^3\right )} \]
(B*(d^2 - e^2*x^2)^(1/2))/(21*(d^3*e^2 + d^2*e^3*x)) - (3*B*(d^2 - e^2*x^2 )^(1/2))/(7*(d^3*e^2 + e^5*x^3 + 3*d^2*e^3*x + 3*d*e^4*x^2)) + (2*A*(d^2 - e^2*x^2)^(1/2))/(105*(d^4*e + 2*d^3*e^2*x + d^2*e^3*x^2)) + (B*(d^2 - e^2 *x^2)^(1/2))/(21*(d^3*e^2 + 2*d^2*e^3*x + d*e^4*x^2)) - (82*C*(d^2 - e^2*x ^2)^(1/2))/(105*(d^2*e^3 + e^5*x^2 + 2*d*e^4*x)) + (2*A*(d^2 - e^2*x^2)^(1 /2))/(105*(d^4*e + d^3*e^2*x)) + (23*C*(d^2 - e^2*x^2)^(1/2))/(105*(d^2*e^ 3 + d*e^4*x)) - (2*A*(d^2 - e^2*x^2)^(1/2))/(7*(d^4*e + e^5*x^4 + 4*d^3*e^ 2*x + 4*d*e^4*x^3 + 6*d^2*e^3*x^2)) + (A*(d^2 - e^2*x^2)^(1/2))/(35*(d^4*e + 3*d^3*e^2*x + d*e^4*x^3 + 3*d^2*e^3*x^2)) - (2*C*d^2*(d^2 - e^2*x^2)^(1 /2))/(7*(d^4*e^3 + e^7*x^4 + 4*d^3*e^4*x + 4*d*e^6*x^3 + 6*d^2*e^5*x^2)) + (2*B*d*(d^2 - e^2*x^2)^(1/2))/(7*(d^4*e^2 + e^6*x^4 + 4*d^3*e^3*x + 4*d*e ^5*x^3 + 6*d^2*e^4*x^2)) + (29*C*d*(d^2 - e^2*x^2)^(1/2))/(35*(d^3*e^3 + e ^6*x^3 + 3*d^2*e^4*x + 3*d*e^5*x^2))