Integrand size = 24, antiderivative size = 208 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {4 x}{65 d^3 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {2}{15 e (d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}}-\frac {11}{195 d e (d+e x) \left (d^2-e^2 x^2\right )^{11/2}}+\frac {8 x}{117 d^5 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {64 x}{819 d^7 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {128 x}{1365 d^9 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {512 x}{4095 d^{11} \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1024 x}{4095 d^{13} \sqrt {d^2-e^2 x^2}} \] Output:
4/65*x/d^3/(-e^2*x^2+d^2)^(11/2)-2/15/e/(e*x+d)^2/(-e^2*x^2+d^2)^(11/2)-11 /195/d/e/(e*x+d)/(-e^2*x^2+d^2)^(11/2)+8/117*x/d^5/(-e^2*x^2+d^2)^(9/2)+64 /819*x/d^7/(-e^2*x^2+d^2)^(7/2)+128/1365*x/d^9/(-e^2*x^2+d^2)^(5/2)+512/40 95*x/d^11/(-e^2*x^2+d^2)^(3/2)+1024/4095*x/d^13/(-e^2*x^2+d^2)^(1/2)
Time = 0.71 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.82 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-777 d^{12}+1764 d^{11} e x+7308 d^{10} e^2 x^2+840 d^9 e^3 x^3-17640 d^8 e^4 x^4-12096 d^7 e^5 x^5+17472 d^6 e^6 x^6+19584 d^5 e^7 x^7-5760 d^4 e^8 x^8-12800 d^3 e^9 x^9-1536 d^2 e^{10} x^{10}+3072 d e^{11} x^{11}+1024 e^{12} x^{12}\right )}{4095 d^{13} e (d-e x)^5 (d+e x)^8} \] Input:
Integrate[1/((d + e*x)^3*(d^2 - e^2*x^2)^(11/2)),x]
Output:
(Sqrt[d^2 - e^2*x^2]*(-777*d^12 + 1764*d^11*e*x + 7308*d^10*e^2*x^2 + 840* d^9*e^3*x^3 - 17640*d^8*e^4*x^4 - 12096*d^7*e^5*x^5 + 17472*d^6*e^6*x^6 + 19584*d^5*e^7*x^7 - 5760*d^4*e^8*x^8 - 12800*d^3*e^9*x^9 - 1536*d^2*e^10*x ^10 + 3072*d*e^11*x^11 + 1024*e^12*x^12))/(4095*d^13*e*(d - e*x)^5*(d + e* x)^8)
Time = 0.52 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.29, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {461, 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.
| \(\displaystyle \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {4 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{11/2}}dx}{5 d}-\frac {1}{15 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {4 \left (\frac {11 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}}dx}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{5 d}-\frac {1}{15 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 470 |
| \(\displaystyle \frac {4 \left (\frac {11 \left (\frac {10 \int \frac {1}{\left (d^2-e^2 x^2\right )^{11/2}}dx}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{5 d}-\frac {1}{15 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {4 \left (\frac {11 \left (\frac {10 \left (\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{5 d}-\frac {1}{15 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {4 \left (\frac {11 \left (\frac {10 \left (\frac {8 \left (\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}}dx}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{5 d}-\frac {1}{15 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {4 \left (\frac {11 \left (\frac {10 \left (\frac {8 \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{5 d}-\frac {1}{15 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {4 \left (\frac {11 \left (\frac {10 \left (\frac {8 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}\right )}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{5 d}-\frac {1}{15 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {4 \left (\frac {11 \left (\frac {10 \left (\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {8 \left (\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {6 \left (\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \left (\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^4 \sqrt {d^2-e^2 x^2}}\right )}{5 d^2}\right )}{7 d^2}\right )}{9 d^2}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\right )}{13 d}-\frac {1}{13 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{5 d}-\frac {1}{15 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{9/2}}\) |
Input:
Int[1/((d + e*x)^3*(d^2 - e^2*x^2)^(11/2)),x]
Output:
-1/15*1/(d*e*(d + e*x)^3*(d^2 - e^2*x^2)^(9/2)) + (4*(-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)))/(5*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 = 0.35 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.79
| method | result | size |
| gosper | \(-\frac {\left (-e x +d \right ) \left (-1024 e^{12} x^{12}-3072 e^{11} x^{11} d +1536 d^{2} e^{10} x^{10}+12800 e^{9} x^{9} d^{3}+5760 d^{4} e^{8} x^{8}-19584 e^{7} x^{7} d^{5}-17472 d^{6} e^{6} x^{6}+12096 e^{5} x^{5} d^{7}+17640 d^{8} e^{4} x^{4}-840 x^{3} e^{3} d^{9}-7308 d^{10} e^{2} x^{2}-1764 x \,d^{11} e +777 d^{12}\right )}{4095 \left (e x +d \right )^{2} d^{13} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\) | \(165\) |
| orering | \(-\frac {\left (-e x +d \right ) \left (-1024 e^{12} x^{12}-3072 e^{11} x^{11} d +1536 d^{2} e^{10} x^{10}+12800 e^{9} x^{9} d^{3}+5760 d^{4} e^{8} x^{8}-19584 e^{7} x^{7} d^{5}-17472 d^{6} e^{6} x^{6}+12096 e^{5} x^{5} d^{7}+17640 d^{8} e^{4} x^{4}-840 x^{3} e^{3} d^{9}-7308 d^{10} e^{2} x^{2}-1764 x \,d^{11} e +777 d^{12}\right )}{4095 \left (e x +d \right )^{2} d^{13} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\) | \(165\) |
| trager | \(-\frac {\left (-1024 e^{12} x^{12}-3072 e^{11} x^{11} d +1536 d^{2} e^{10} x^{10}+12800 e^{9} x^{9} d^{3}+5760 d^{4} e^{8} x^{8}-19584 e^{7} x^{7} d^{5}-17472 d^{6} e^{6} x^{6}+12096 e^{5} x^{5} d^{7}+17640 d^{8} e^{4} x^{4}-840 x^{3} e^{3} d^{9}-7308 d^{10} e^{2} x^{2}-1764 x \,d^{11} e +777 d^{12}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{4095 d^{13} \left (e x +d \right )^{8} \left (-e x +d \right )^{5} e}\) | \(167\) |
| default | \(\frac {-\frac {1}{15 d e \left (x +\frac {d}{e}\right )^{3} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {4 e \left (-\frac {1}{13 d e \left (x +\frac {d}{e}\right )^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {11 e \left (-\frac {1}{11 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {10 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{18 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{63 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}+\frac {8 \left (-\frac {3 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{35 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 \left (-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{7 d^{2}}\right )}{9 d^{2}}}{d^{2}}\right )}{11 d}\right )}{13 d}\right )}{5 d}}{e^{3}}\) | \(445\) |
Input:
int(1/(e*x+d)^3/(-e^2*x^2+d^2)^(11/2),x,method=_RETURNVERBOSE)
Output:
-1/4095*(-e*x+d)*(-1024*e^12*x^12-3072*d*e^11*x^11+1536*d^2*e^10*x^10+1280 0*d^3*e^9*x^9+5760*d^4*e^8*x^8-19584*d^5*e^7*x^7-17472*d^6*e^6*x^6+12096*d ^7*e^5*x^5+17640*d^8*e^4*x^4-840*d^9*e^3*x^3-7308*d^10*e^2*x^2-1764*d^11*e *x+777*d^12)/(e*x+d)^2/d^13/e/(-e^2*x^2+d^2)^(11/2)
Leaf count of result is larger than twice the leaf count of optimal. 435 vs. \(2 (176) = 352\).
Time = 2.06 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.09 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {777 \, e^{13} x^{13} + 2331 \, d e^{12} x^{12} - 1554 \, d^{2} e^{11} x^{11} - 10878 \, d^{3} e^{10} x^{10} - 3885 \, d^{4} e^{9} x^{9} + 19425 \, d^{5} e^{8} x^{8} + 15540 \, d^{6} e^{7} x^{7} - 15540 \, d^{7} e^{6} x^{6} - 19425 \, d^{8} e^{5} x^{5} + 3885 \, d^{9} e^{4} x^{4} + 10878 \, d^{10} e^{3} x^{3} + 1554 \, d^{11} e^{2} x^{2} - 2331 \, d^{12} e x - 777 \, d^{13} + {\left (1024 \, e^{12} x^{12} + 3072 \, d e^{11} x^{11} - 1536 \, d^{2} e^{10} x^{10} - 12800 \, d^{3} e^{9} x^{9} - 5760 \, d^{4} e^{8} x^{8} + 19584 \, d^{5} e^{7} x^{7} + 17472 \, d^{6} e^{6} x^{6} - 12096 \, d^{7} e^{5} x^{5} - 17640 \, d^{8} e^{4} x^{4} + 840 \, d^{9} e^{3} x^{3} + 7308 \, d^{10} e^{2} x^{2} + 1764 \, d^{11} e x - 777 \, d^{12}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{4095 \, {\left (d^{13} e^{14} x^{13} + 3 \, d^{14} e^{13} x^{12} - 2 \, d^{15} e^{12} x^{11} - 14 \, d^{16} e^{11} x^{10} - 5 \, d^{17} e^{10} x^{9} + 25 \, d^{18} e^{9} x^{8} + 20 \, d^{19} e^{8} x^{7} - 20 \, d^{20} e^{7} x^{6} - 25 \, d^{21} e^{6} x^{5} + 5 \, d^{22} e^{5} x^{4} + 14 \, d^{23} e^{4} x^{3} + 2 \, d^{24} e^{3} x^{2} - 3 \, d^{25} e^{2} x - d^{26} e\right )}} \] Input:
integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(11/2),x, algorithm="fricas")
Output:
-1/4095*(777*e^13*x^13 + 2331*d*e^12*x^12 - 1554*d^2*e^11*x^11 - 10878*d^3 *e^10*x^10 - 3885*d^4*e^9*x^9 + 19425*d^5*e^8*x^8 + 15540*d^6*e^7*x^7 - 15 540*d^7*e^6*x^6 - 19425*d^8*e^5*x^5 + 3885*d^9*e^4*x^4 + 10878*d^10*e^3*x^ 3 + 1554*d^11*e^2*x^2 - 2331*d^12*e*x - 777*d^13 + (1024*e^12*x^12 + 3072* d*e^11*x^11 - 1536*d^2*e^10*x^10 - 12800*d^3*e^9*x^9 - 5760*d^4*e^8*x^8 + 19584*d^5*e^7*x^7 + 17472*d^6*e^6*x^6 - 12096*d^7*e^5*x^5 - 17640*d^8*e^4* x^4 + 840*d^9*e^3*x^3 + 7308*d^10*e^2*x^2 + 1764*d^11*e*x - 777*d^12)*sqrt (-e^2*x^2 + d^2))/(d^13*e^14*x^13 + 3*d^14*e^13*x^12 - 2*d^15*e^12*x^11 - 14*d^16*e^11*x^10 - 5*d^17*e^10*x^9 + 25*d^18*e^9*x^8 + 20*d^19*e^8*x^7 - 20*d^20*e^7*x^6 - 25*d^21*e^6*x^5 + 5*d^22*e^5*x^4 + 14*d^23*e^4*x^3 + 2*d ^24*e^3*x^2 - 3*d^25*e^2*x - d^26*e)
\[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {11}{2}} \left (d + e x\right )^{3}}\, dx \] Input:
integrate(1/(e*x+d)**3/(-e**2*x**2+d**2)**(11/2),x)
Output:
Integral(1/((-(-d + e*x)*(d + e*x))**(11/2)*(d + e*x)**3), x)
Time = 0.05 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.50 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {1}{15 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d e^{4} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{2} e^{3} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{3} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{4} e\right )}} - \frac {4}{65 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{2} e^{3} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{3} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{4} e\right )}} - \frac {4}{65 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{3} e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{4} e\right )}} + \frac {8 \, x}{117 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{5}} + \frac {64 \, x}{819 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{7}} + \frac {128 \, x}{1365 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{9}} + \frac {512 \, x}{4095 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{11}} + \frac {1024 \, x}{4095 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{13}} \] Input:
integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(11/2),x, algorithm="maxima")
Output:
-1/15/((-e^2*x^2 + d^2)^(9/2)*d*e^4*x^3 + 3*(-e^2*x^2 + d^2)^(9/2)*d^2*e^3 *x^2 + 3*(-e^2*x^2 + d^2)^(9/2)*d^3*e^2*x + (-e^2*x^2 + d^2)^(9/2)*d^4*e) - 4/65/((-e^2*x^2 + d^2)^(9/2)*d^2*e^3*x^2 + 2*(-e^2*x^2 + d^2)^(9/2)*d^3* e^2*x + (-e^2*x^2 + d^2)^(9/2)*d^4*e) - 4/65/((-e^2*x^2 + d^2)^(9/2)*d^3*e ^2*x + (-e^2*x^2 + d^2)^(9/2)*d^4*e) + 8/117*x/((-e^2*x^2 + d^2)^(9/2)*d^5 ) + 64/819*x/((-e^2*x^2 + d^2)^(7/2)*d^7) + 128/1365*x/((-e^2*x^2 + d^2)^( 5/2)*d^9) + 512/4095*x/((-e^2*x^2 + d^2)^(3/2)*d^11) + 1024/4095*x/(sqrt(- e^2*x^2 + d^2)*d^13)
\[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {11}{2}} {\left (e x + d\right )}^{3}} \,d x } \] Input:
integrate(1/(e*x+d)^3/(-e^2*x^2+d^2)^(11/2),x, algorithm="giac")
Output:
integrate(1/((-e^2*x^2 + d^2)^(11/2)*(e*x + d)^3), x)
Time = 7.47 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {355\,x}{936\,d^5}-\frac {19}{52\,d^4\,e}\right )}{{\left (d+e\,x\right )}^5\,{\left (d-e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {x}{504\,d^7}-\frac {89}{416\,d^6\,e}\right )}{{\left (d+e\,x\right )}^4\,{\left (d-e\,x\right )}^4}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {128\,x}{1365\,d^9}-\frac {121}{8320\,d^8\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {\sqrt {d^2-e^2\,x^2}}{480\,d^6\,e\,{\left (d+e\,x\right )}^8}-\frac {89\,\sqrt {d^2-e^2\,x^2}}{12480\,d^7\,e\,{\left (d+e\,x\right )}^7}-\frac {121\,\sqrt {d^2-e^2\,x^2}}{8320\,d^8\,e\,{\left (d+e\,x\right )}^6}+\frac {512\,x\,\sqrt {d^2-e^2\,x^2}}{4095\,d^{11}\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {1024\,x\,\sqrt {d^2-e^2\,x^2}}{4095\,d^{13}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \] Input:
int(1/((d^2 - e^2*x^2)^(11/2)*(d + e*x)^3),x)
Output:
((d^2 - e^2*x^2)^(1/2)*((355*x)/(936*d^5) - 19/(52*d^4*e)))/((d + e*x)^5*( d - e*x)^5) - ((d^2 - e^2*x^2)^(1/2)*(x/(504*d^7) - 89/(416*d^6*e)))/((d + e*x)^4*(d - e*x)^4) + ((d^2 - e^2*x^2)^(1/2)*((128*x)/(1365*d^9) - 121/(8 320*d^8*e)))/((d + e*x)^3*(d - e*x)^3) - (d^2 - e^2*x^2)^(1/2)/(480*d^6*e* (d + e*x)^8) - (89*(d^2 - e^2*x^2)^(1/2))/(12480*d^7*e*(d + e*x)^7) - (121 *(d^2 - e^2*x^2)^(1/2))/(8320*d^8*e*(d + e*x)^6) + (512*x*(d^2 - e^2*x^2)^ (1/2))/(4095*d^11*(d + e*x)^2*(d - e*x)^2) + (1024*x*(d^2 - e^2*x^2)^(1/2) )/(4095*d^13*(d + e*x)*(d - e*x))
Time = 0.26 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.61 \[ \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {588 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{11}+1024 e^{12} x^{12}+1764 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{10} e x -588 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{9} e^{2} x^{2}-6468 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8} e^{3} x^{3}-3528 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} e^{4} x^{4}+8232 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6} e^{5} x^{5}+8232 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e^{6} x^{6}-3528 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{7} x^{7}-6468 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{8} x^{8}-588 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{9} x^{9}+1764 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{10} x^{10}+588 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{11} x^{11}+1764 d^{11} e x +7308 d^{10} e^{2} x^{2}+840 d^{9} e^{3} x^{3}-17640 d^{8} e^{4} x^{4}-12096 d^{7} e^{5} x^{5}+17472 d^{6} e^{6} x^{6}+19584 d^{5} e^{7} x^{7}-5760 d^{4} e^{8} x^{8}-12800 d^{3} e^{9} x^{9}-1536 d^{2} e^{10} x^{10}+3072 d \,e^{11} x^{11}-777 d^{12}}{4095 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{13} e \left (e^{11} x^{11}+3 d \,e^{10} x^{10}-d^{2} e^{9} x^{9}-11 d^{3} e^{8} x^{8}-6 d^{4} e^{7} x^{7}+14 d^{5} e^{6} x^{6}+14 d^{6} e^{5} x^{5}-6 d^{7} e^{4} x^{4}-11 d^{8} e^{3} x^{3}-d^{9} e^{2} x^{2}+3 d^{10} e x +d^{11}\right )} \] Input:
int(1/(e*x+d)^3/(-e^2*x^2+d^2)^(11/2),x)
Output:
(588*sqrt(d**2 - e**2*x**2)*d**11 + 1764*sqrt(d**2 - e**2*x**2)*d**10*e*x - 588*sqrt(d**2 - e**2*x**2)*d**9*e**2*x**2 - 6468*sqrt(d**2 - e**2*x**2)* d**8*e**3*x**3 - 3528*sqrt(d**2 - e**2*x**2)*d**7*e**4*x**4 + 8232*sqrt(d* *2 - e**2*x**2)*d**6*e**5*x**5 + 8232*sqrt(d**2 - e**2*x**2)*d**5*e**6*x** 6 - 3528*sqrt(d**2 - e**2*x**2)*d**4*e**7*x**7 - 6468*sqrt(d**2 - e**2*x** 2)*d**3*e**8*x**8 - 588*sqrt(d**2 - e**2*x**2)*d**2*e**9*x**9 + 1764*sqrt( d**2 - e**2*x**2)*d*e**10*x**10 + 588*sqrt(d**2 - e**2*x**2)*e**11*x**11 - 777*d**12 + 1764*d**11*e*x + 7308*d**10*e**2*x**2 + 840*d**9*e**3*x**3 - 17640*d**8*e**4*x**4 - 12096*d**7*e**5*x**5 + 17472*d**6*e**6*x**6 + 19584 *d**5*e**7*x**7 - 5760*d**4*e**8*x**8 - 12800*d**3*e**9*x**9 - 1536*d**2*e **10*x**10 + 3072*d*e**11*x**11 + 1024*e**12*x**12)/(4095*sqrt(d**2 - e**2 *x**2)*d**13*e*(d**11 + 3*d**10*e*x - d**9*e**2*x**2 - 11*d**8*e**3*x**3 - 6*d**7*e**4*x**4 + 14*d**6*e**5*x**5 + 14*d**5*e**6*x**6 - 6*d**4*e**7*x* *7 - 11*d**3*e**8*x**8 - d**2*e**9*x**9 + 3*d*e**10*x**10 + e**11*x**11))