Integrand size = 24, antiderivative size = 175 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {x}{13 d^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {2}{13 e (d+e x) \left (d^2-e^2 x^2\right )^{11/2}}+\frac {10 x}{117 d^4 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {80 x}{819 d^6 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {32 x}{273 d^8 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{819 d^{10} \left (d^2-e^2 x^2\right )^{3/2}}+\frac {256 x}{819 d^{12} \sqrt {d^2-e^2 x^2}} \] Output:
1/13*x/d^2/(-e^2*x^2+d^2)^(11/2)-2/13/e/(e*x+d)/(-e^2*x^2+d^2)^(11/2)+10/1 17*x/d^4/(-e^2*x^2+d^2)^(9/2)+80/819*x/d^6/(-e^2*x^2+d^2)^(7/2)+32/273*x/d ^8/(-e^2*x^2+d^2)^(5/2)+128/819*x/d^10/(-e^2*x^2+d^2)^(3/2)+256/819*x/d^12 /(-e^2*x^2+d^2)^(1/2)
Time = 0.76 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-126 d^{11}+567 d^{10} e x+1260 d^9 e^2 x^2-1050 d^8 e^3 x^3-3360 d^7 e^4 x^4+336 d^6 e^5 x^5+4032 d^5 e^6 x^6+864 d^4 e^7 x^7-2304 d^3 e^8 x^8-896 d^2 e^9 x^9+512 d e^{10} x^{10}+256 e^{11} x^{11}\right )}{819 d^{12} e (d-e x)^5 (d+e x)^7} \] Input:
Integrate[1/((d + e*x)^2*(d^2 - e^2*x^2)^(11/2)),x]
Output:
(Sqrt[d^2 - e^2*x^2]*(-126*d^11 + 567*d^10*e*x + 1260*d^9*e^2*x^2 - 1050*d ^8*e^3*x^3 - 3360*d^7*e^4*x^4 + 336*d^6*e^5*x^5 + 4032*d^5*e^6*x^6 + 864*d ^4*e^7*x^7 - 2304*d^3*e^8*x^8 - 896*d^2*e^9*x^9 + 512*d*e^10*x^10 + 256*e^ 11*x^11))/(819*d^12*e*(d - e*x)^5*(d + e*x)^7)
Time = 0.46 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.30, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {461, 470, 209, 209, 209, 209, 208}
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.
| \(\displaystyle \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {11 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}}dx}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 470 |
| \(\displaystyle \frac {11 \left (\frac {10 \int \frac {1}{\left (d^2-e^2 x^2\right )^{11/2}}dx}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {8 \left (\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}}dx}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {8 \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {8 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}\right )}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {11 \left (\frac {10 \left (\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {8 \left (\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {6 \left (\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \left (\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^4 \sqrt {d^2-e^2 x^2}}\right )}{5 d^2}\right )}{7 d^2}\right )}{9 d^2}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\) |
Input:
Int[1/((d + e*x)^2*(d^2 - e^2*x^2)^(11/2)),x]
Output:
-1/13*1/(d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(9/2)) + (11*(-1/11*1/(d*e*(d + e *x)*(d^2 - e^2*x^2)^(9/2)) + (10*(x/(9*d^2*(d^2 - e^2*x^2)^(9/2)) + (8*(x/ (7*d^2*(d^2 - e^2*x^2)^(7/2)) + (6*(x/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (4*( x/(3*d^2*(d^2 - e^2*x^2)^(3/2)) + (2*x)/(3*d^4*Sqrt[d^2 - e^2*x^2])))/(5*d ^2)))/(7*d^2)))/(9*d^2)))/(11*d)))/(13*d)
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 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[((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[(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, p}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[n, 0] && NeQ[n + p + 1, 0] && IntegerQ[2*p]
Time = 0.32 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.88
| method | result | size |
| gosper | \(-\frac {\left (-e x +d \right ) \left (-256 e^{11} x^{11}-512 d \,e^{10} x^{10}+896 d^{2} e^{9} x^{9}+2304 d^{3} e^{8} x^{8}-864 d^{4} e^{7} x^{7}-4032 d^{5} e^{6} x^{6}-336 d^{6} e^{5} x^{5}+3360 d^{7} e^{4} x^{4}+1050 d^{8} e^{3} x^{3}-1260 d^{9} e^{2} x^{2}-567 d^{10} e x +126 d^{11}\right )}{819 \left (e x +d \right ) d^{12} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\) | \(154\) |
| orering | \(-\frac {\left (-e x +d \right ) \left (-256 e^{11} x^{11}-512 d \,e^{10} x^{10}+896 d^{2} e^{9} x^{9}+2304 d^{3} e^{8} x^{8}-864 d^{4} e^{7} x^{7}-4032 d^{5} e^{6} x^{6}-336 d^{6} e^{5} x^{5}+3360 d^{7} e^{4} x^{4}+1050 d^{8} e^{3} x^{3}-1260 d^{9} e^{2} x^{2}-567 d^{10} e x +126 d^{11}\right )}{819 \left (e x +d \right ) d^{12} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\) | \(154\) |
| trager | \(-\frac {\left (-256 e^{11} x^{11}-512 d \,e^{10} x^{10}+896 d^{2} e^{9} x^{9}+2304 d^{3} e^{8} x^{8}-864 d^{4} e^{7} x^{7}-4032 d^{5} e^{6} x^{6}-336 d^{6} e^{5} x^{5}+3360 d^{7} e^{4} x^{4}+1050 d^{8} e^{3} x^{3}-1260 d^{9} e^{2} x^{2}-567 d^{10} e x +126 d^{11}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{819 d^{12} \left (e x +d \right )^{7} \left (-e x +d \right )^{5} e}\) | \(156\) |
| default | \(\frac {-\frac {1}{13 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {11 e \left (-\frac {1}{11 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {10 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{18 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{63 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}+\frac {8 \left (-\frac {3 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{35 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 \left (-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{7 d^{2}}\right )}{9 d^{2}}}{d^{2}}\right )}{11 d}\right )}{13 d}}{e^{2}}\) | \(393\) |
Input:
int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(11/2),x,method=_RETURNVERBOSE)
Output:
-1/819*(-e*x+d)*(-256*e^11*x^11-512*d*e^10*x^10+896*d^2*e^9*x^9+2304*d^3*e ^8*x^8-864*d^4*e^7*x^7-4032*d^5*e^6*x^6-336*d^6*e^5*x^5+3360*d^7*e^4*x^4+1 050*d^8*e^3*x^3-1260*d^9*e^2*x^2-567*d^10*e*x+126*d^11)/(e*x+d)/d^12/e/(-e ^2*x^2+d^2)^(11/2)
Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (147) = 294\).
Time = 1.28 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.17 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {126 \, e^{12} x^{12} + 252 \, d e^{11} x^{11} - 504 \, d^{2} e^{10} x^{10} - 1260 \, d^{3} e^{9} x^{9} + 630 \, d^{4} e^{8} x^{8} + 2520 \, d^{5} e^{7} x^{7} - 2520 \, d^{7} e^{5} x^{5} - 630 \, d^{8} e^{4} x^{4} + 1260 \, d^{9} e^{3} x^{3} + 504 \, d^{10} e^{2} x^{2} - 252 \, d^{11} e x - 126 \, d^{12} + {\left (256 \, e^{11} x^{11} + 512 \, d e^{10} x^{10} - 896 \, d^{2} e^{9} x^{9} - 2304 \, d^{3} e^{8} x^{8} + 864 \, d^{4} e^{7} x^{7} + 4032 \, d^{5} e^{6} x^{6} + 336 \, d^{6} e^{5} x^{5} - 3360 \, d^{7} e^{4} x^{4} - 1050 \, d^{8} e^{3} x^{3} + 1260 \, d^{9} e^{2} x^{2} + 567 \, d^{10} e x - 126 \, d^{11}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{819 \, {\left (d^{12} e^{13} x^{12} + 2 \, d^{13} e^{12} x^{11} - 4 \, d^{14} e^{11} x^{10} - 10 \, d^{15} e^{10} x^{9} + 5 \, d^{16} e^{9} x^{8} + 20 \, d^{17} e^{8} x^{7} - 20 \, d^{19} e^{6} x^{5} - 5 \, d^{20} e^{5} x^{4} + 10 \, d^{21} e^{4} x^{3} + 4 \, d^{22} e^{3} x^{2} - 2 \, d^{23} e^{2} x - d^{24} e\right )}} \] Input:
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(11/2),x, algorithm="fricas")
Output:
-1/819*(126*e^12*x^12 + 252*d*e^11*x^11 - 504*d^2*e^10*x^10 - 1260*d^3*e^9 *x^9 + 630*d^4*e^8*x^8 + 2520*d^5*e^7*x^7 - 2520*d^7*e^5*x^5 - 630*d^8*e^4 *x^4 + 1260*d^9*e^3*x^3 + 504*d^10*e^2*x^2 - 252*d^11*e*x - 126*d^12 + (25 6*e^11*x^11 + 512*d*e^10*x^10 - 896*d^2*e^9*x^9 - 2304*d^3*e^8*x^8 + 864*d ^4*e^7*x^7 + 4032*d^5*e^6*x^6 + 336*d^6*e^5*x^5 - 3360*d^7*e^4*x^4 - 1050* d^8*e^3*x^3 + 1260*d^9*e^2*x^2 + 567*d^10*e*x - 126*d^11)*sqrt(-e^2*x^2 + d^2))/(d^12*e^13*x^12 + 2*d^13*e^12*x^11 - 4*d^14*e^11*x^10 - 10*d^15*e^10 *x^9 + 5*d^16*e^9*x^8 + 20*d^17*e^8*x^7 - 20*d^19*e^6*x^5 - 5*d^20*e^5*x^4 + 10*d^21*e^4*x^3 + 4*d^22*e^3*x^2 - 2*d^23*e^2*x - d^24*e)
\[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {11}{2}} \left (d + e x\right )^{2}}\, dx \] Input:
integrate(1/(e*x+d)**2/(-e**2*x**2+d**2)**(11/2),x)
Output:
Integral(1/((-(-d + e*x)*(d + e*x))**(11/2)*(d + e*x)**2), x)
Time = 0.04 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{3} e\right )}} - \frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{2} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{3} e\right )}} + \frac {10 \, x}{117 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{4}} + \frac {80 \, x}{819 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{6}} + \frac {32 \, x}{273 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{8}} + \frac {128 \, x}{819 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{10}} + \frac {256 \, x}{819 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{12}} \] Input:
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(11/2),x, algorithm="maxima")
Output:
-1/13/((-e^2*x^2 + d^2)^(9/2)*d*e^3*x^2 + 2*(-e^2*x^2 + d^2)^(9/2)*d^2*e^2 *x + (-e^2*x^2 + d^2)^(9/2)*d^3*e) - 1/13/((-e^2*x^2 + d^2)^(9/2)*d^2*e^2* x + (-e^2*x^2 + d^2)^(9/2)*d^3*e) + 10/117*x/((-e^2*x^2 + d^2)^(9/2)*d^4) + 80/819*x/((-e^2*x^2 + d^2)^(7/2)*d^6) + 32/273*x/((-e^2*x^2 + d^2)^(5/2) *d^8) + 128/819*x/((-e^2*x^2 + d^2)^(3/2)*d^10) + 256/819*x/(sqrt(-e^2*x^2 + d^2)*d^12)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.27 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {e^{11} {\left (\frac {13 \, {\left (20790 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{4} + 3465 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{3} + 693 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{2} + \frac {198 \, d}{e x + d} - 92\right )}}{d^{12} e^{11} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {63 \, d^{144} e^{132} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {13}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{12} \mathrm {sgn}\left (e\right )^{12} + 819 \, d^{144} e^{132} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{12} \mathrm {sgn}\left (e\right )^{12} + 5005 \, d^{144} e^{132} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{12} \mathrm {sgn}\left (e\right )^{12} + 19305 \, d^{144} e^{132} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{12} \mathrm {sgn}\left (e\right )^{12} + 54054 \, d^{144} e^{132} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{12} \mathrm {sgn}\left (e\right )^{12} + 126126 \, d^{144} e^{132} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{12} \mathrm {sgn}\left (e\right )^{12} + 378378 \, d^{144} e^{132} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{12} \mathrm {sgn}\left (e\right )^{12}}{d^{156} e^{143} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{13} \mathrm {sgn}\left (e\right )^{13}}\right )} + \frac {524288 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{12}}}{1677312 \, {\left | e \right |}} \] Input:
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(11/2),x, algorithm="giac")
Output:
1/1677312*(e^11*(13*(20790*(2*d/(e*x + d) - 1)^4 + 3465*(2*d/(e*x + d) - 1 )^3 + 693*(2*d/(e*x + d) - 1)^2 + 198*d/(e*x + d) - 92)/(d^12*e^11*(2*d/(e *x + d) - 1)^(9/2)*sgn(1/(e*x + d))*sgn(e)) - (63*d^144*e^132*(2*d/(e*x + d) - 1)^(13/2)*sgn(1/(e*x + d))^12*sgn(e)^12 + 819*d^144*e^132*(2*d/(e*x + d) - 1)^(11/2)*sgn(1/(e*x + d))^12*sgn(e)^12 + 5005*d^144*e^132*(2*d/(e*x + d) - 1)^(9/2)*sgn(1/(e*x + d))^12*sgn(e)^12 + 19305*d^144*e^132*(2*d/(e *x + d) - 1)^(7/2)*sgn(1/(e*x + d))^12*sgn(e)^12 + 54054*d^144*e^132*(2*d/ (e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))^12*sgn(e)^12 + 126126*d^144*e^132*(2 *d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))^12*sgn(e)^12 + 378378*d^144*e^132 *sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))^12*sgn(e)^12)/(d^156*e^143*sgn(1 /(e*x + d))^13*sgn(e)^13)) + 524288*I*sgn(1/(e*x + d))*sgn(e)/d^12)/abs(e)
Time = 7.34 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {25\,x}{504\,d^6}+\frac {7}{52\,d^5\,e}\right )}{{\left (d+e\,x\right )}^4\,{\left (d-e\,x\right )}^4}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {139\,x}{468\,d^4}-\frac {7}{26\,d^3\,e}\right )}{{\left (d+e\,x\right )}^5\,{\left (d-e\,x\right )}^5}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {32\,x}{273\,d^8}-\frac {7}{832\,d^7\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{416\,d^6\,e\,{\left (d+e\,x\right )}^7}-\frac {7\,\sqrt {d^2-e^2\,x^2}}{832\,d^7\,e\,{\left (d+e\,x\right )}^6}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{819\,d^{10}\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {256\,x\,\sqrt {d^2-e^2\,x^2}}{819\,d^{12}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \] Input:
int(1/((d^2 - e^2*x^2)^(11/2)*(d + e*x)^2),x)
Output:
((d^2 - e^2*x^2)^(1/2)*((25*x)/(504*d^6) + 7/(52*d^5*e)))/((d + e*x)^4*(d - e*x)^4) + ((d^2 - e^2*x^2)^(1/2)*((139*x)/(468*d^4) - 7/(26*d^3*e)))/((d + e*x)^5*(d - e*x)^5) + ((d^2 - e^2*x^2)^(1/2)*((32*x)/(273*d^8) - 7/(832 *d^7*e)))/((d + e*x)^3*(d - e*x)^3) - (d^2 - e^2*x^2)^(1/2)/(416*d^6*e*(d + e*x)^7) - (7*(d^2 - e^2*x^2)^(1/2))/(832*d^7*e*(d + e*x)^6) + (128*x*(d^ 2 - e^2*x^2)^(1/2))/(819*d^10*(d + e*x)^2*(d - e*x)^2) + (256*x*(d^2 - e^2 *x^2)^(1/2))/(819*d^12*(d + e*x)*(d - e*x))
Time = 0.24 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.83 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {567 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{10}+1134 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{9} e x -1701 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8} e^{2} x^{2}-4536 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} e^{3} x^{3}+1134 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6} e^{4} x^{4}+6804 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e^{5} x^{5}+1134 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{6} x^{6}-4536 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{7} x^{7}-1701 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{8} x^{8}+1134 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{9} x^{9}+567 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{10} x^{10}-252 d^{11}+1134 d^{10} e x +2520 d^{9} e^{2} x^{2}-2100 d^{8} e^{3} x^{3}-6720 d^{7} e^{4} x^{4}+672 d^{6} e^{5} x^{5}+8064 d^{5} e^{6} x^{6}+1728 d^{4} e^{7} x^{7}-4608 d^{3} e^{8} x^{8}-1792 d^{2} e^{9} x^{9}+1024 d \,e^{10} x^{10}+512 e^{11} x^{11}}{1638 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{12} e \left (e^{10} x^{10}+2 d \,e^{9} x^{9}-3 d^{2} e^{8} x^{8}-8 d^{3} e^{7} x^{7}+2 d^{4} e^{6} x^{6}+12 d^{5} e^{5} x^{5}+2 d^{6} e^{4} x^{4}-8 d^{7} e^{3} x^{3}-3 d^{8} e^{2} x^{2}+2 d^{9} e x +d^{10}\right )} \] Input:
int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(11/2),x)
Output:
(567*sqrt(d**2 - e**2*x**2)*d**10 + 1134*sqrt(d**2 - e**2*x**2)*d**9*e*x - 1701*sqrt(d**2 - e**2*x**2)*d**8*e**2*x**2 - 4536*sqrt(d**2 - e**2*x**2)* d**7*e**3*x**3 + 1134*sqrt(d**2 - e**2*x**2)*d**6*e**4*x**4 + 6804*sqrt(d* *2 - e**2*x**2)*d**5*e**5*x**5 + 1134*sqrt(d**2 - e**2*x**2)*d**4*e**6*x** 6 - 4536*sqrt(d**2 - e**2*x**2)*d**3*e**7*x**7 - 1701*sqrt(d**2 - e**2*x** 2)*d**2*e**8*x**8 + 1134*sqrt(d**2 - e**2*x**2)*d*e**9*x**9 + 567*sqrt(d** 2 - e**2*x**2)*e**10*x**10 - 252*d**11 + 1134*d**10*e*x + 2520*d**9*e**2*x **2 - 2100*d**8*e**3*x**3 - 6720*d**7*e**4*x**4 + 672*d**6*e**5*x**5 + 806 4*d**5*e**6*x**6 + 1728*d**4*e**7*x**7 - 4608*d**3*e**8*x**8 - 1792*d**2*e **9*x**9 + 1024*d*e**10*x**10 + 512*e**11*x**11)/(1638*sqrt(d**2 - e**2*x* *2)*d**12*e*(d**10 + 2*d**9*e*x - 3*d**8*e**2*x**2 - 8*d**7*e**3*x**3 + 2* d**6*e**4*x**4 + 12*d**5*e**5*x**5 + 2*d**4*e**6*x**6 - 8*d**3*e**7*x**7 - 3*d**2*e**8*x**8 + 2*d*e**9*x**9 + e**10*x**10))