Integrand size = 24, antiderivative size = 100 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{143 d^2 e (d+e x)^{10}}-\frac {2 \left (d^2-e^2 x^2\right )^{9/2}}{1287 d^3 e (d+e x)^9} \] Output:
-1/13*(-e^2*x^2+d^2)^(9/2)/d/e/(e*x+d)^11-2/143*(-e^2*x^2+d^2)^(9/2)/d^2/e /(e*x+d)^10-2/1287*(-e^2*x^2+d^2)^(9/2)/d^3/e/(e*x+d)^9
Time = 0.68 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.60 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=-\frac {(d-e x)^4 \sqrt {d^2-e^2 x^2} \left (119 d^2+22 d e x+2 e^2 x^2\right )}{1287 d^3 e (d+e x)^7} \] Input:
Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]
Output:
-1/1287*((d - e*x)^4*Sqrt[d^2 - e^2*x^2]*(119*d^2 + 22*d*e*x + 2*e^2*x^2)) /(d^3*e*(d + e*x)^7)
Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {461, 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.
\(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {2 \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}}dx}{13 d}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9}dx}{11 d}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}\right )}{13 d}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {2 \left (-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}\right )}{13 d}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{13 d e (d+e x)^{11}}\) |
Input:
Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x]
Output:
-1/13*(d^2 - e^2*x^2)^(9/2)/(d*e*(d + e*x)^11) + (2*(-1/11*(d^2 - e^2*x^2) ^(9/2)/(d*e*(d + e*x)^10) - (d^2 - e^2*x^2)^(9/2)/(99*d^2*e*(d + e*x)^9))) /(13*d)
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])
Time = 4.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (2 e^{2} x^{2}+22 d e x +119 d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{1287 \left (e x +d \right )^{10} d^{3} e}\) | \(55\) |
orering | \(-\frac {\left (-e x +d \right ) \left (2 e^{2} x^{2}+22 d e x +119 d^{2}\right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{1287 \left (e x +d \right )^{10} d^{3} e}\) | \(55\) |
trager | \(-\frac {\left (2 e^{6} x^{6}+14 d \,e^{5} x^{5}+43 d^{2} e^{4} x^{4}-352 d^{3} e^{3} x^{3}+628 d^{4} e^{2} x^{2}-454 d^{5} e x +119 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{1287 d^{3} \left (e x +d \right )^{7} e}\) | \(93\) |
default | \(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{13 d e \left (x +\frac {d}{e}\right )^{11}}+\frac {2 e \left (-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 d e \left (x +\frac {d}{e}\right )^{10}}-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 d^{2} \left (x +\frac {d}{e}\right )^{9}}\right )}{13 d}}{e^{11}}\) | \(145\) |
Input:
int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^11,x,method=_RETURNVERBOSE)
Output:
-1/1287*(-e*x+d)*(2*e^2*x^2+22*d*e*x+119*d^2)*(-e^2*x^2+d^2)^(7/2)/(e*x+d) ^10/d^3/e
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (88) = 176\).
Time = 0.21 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.36 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=-\frac {119 \, e^{7} x^{7} + 833 \, d e^{6} x^{6} + 2499 \, d^{2} e^{5} x^{5} + 4165 \, d^{3} e^{4} x^{4} + 4165 \, d^{4} e^{3} x^{3} + 2499 \, d^{5} e^{2} x^{2} + 833 \, d^{6} e x + 119 \, d^{7} + {\left (2 \, e^{6} x^{6} + 14 \, d e^{5} x^{5} + 43 \, d^{2} e^{4} x^{4} - 352 \, d^{3} e^{3} x^{3} + 628 \, d^{4} e^{2} x^{2} - 454 \, d^{5} e x + 119 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1287 \, {\left (d^{3} e^{8} x^{7} + 7 \, d^{4} e^{7} x^{6} + 21 \, d^{5} e^{6} x^{5} + 35 \, d^{6} e^{5} x^{4} + 35 \, d^{7} e^{4} x^{3} + 21 \, d^{8} e^{3} x^{2} + 7 \, d^{9} e^{2} x + d^{10} e\right )}} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^11,x, algorithm="fricas")
Output:
-1/1287*(119*e^7*x^7 + 833*d*e^6*x^6 + 2499*d^2*e^5*x^5 + 4165*d^3*e^4*x^4 + 4165*d^4*e^3*x^3 + 2499*d^5*e^2*x^2 + 833*d^6*e*x + 119*d^7 + (2*e^6*x^ 6 + 14*d*e^5*x^5 + 43*d^2*e^4*x^4 - 352*d^3*e^3*x^3 + 628*d^4*e^2*x^2 - 45 4*d^5*e*x + 119*d^6)*sqrt(-e^2*x^2 + d^2))/(d^3*e^8*x^7 + 7*d^4*e^7*x^6 + 21*d^5*e^6*x^5 + 35*d^6*e^5*x^4 + 35*d^7*e^4*x^3 + 21*d^8*e^3*x^2 + 7*d^9* e^2*x + d^10*e)
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \] Input:
integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**11,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (88) = 176\).
Time = 0.04 (sec) , antiderivative size = 790, normalized size of antiderivative = 7.90 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^11,x, algorithm="maxima")
Output:
-1/3*(-e^2*x^2 + d^2)^(7/2)/(e^11*x^10 + 10*d*e^10*x^9 + 45*d^2*e^9*x^8 + 120*d^3*e^8*x^7 + 210*d^4*e^7*x^6 + 252*d^5*e^6*x^5 + 210*d^6*e^5*x^4 + 12 0*d^7*e^4*x^3 + 45*d^8*e^3*x^2 + 10*d^9*e^2*x + d^10*e) + 7/12*(-e^2*x^2 + d^2)^(5/2)*d/(e^10*x^9 + 9*d*e^9*x^8 + 36*d^2*e^8*x^7 + 84*d^3*e^7*x^6 + 126*d^4*e^6*x^5 + 126*d^5*e^5*x^4 + 84*d^6*e^4*x^3 + 36*d^7*e^3*x^2 + 9*d^ 8*e^2*x + d^9*e) - 7/12*(-e^2*x^2 + d^2)^(3/2)*d^2/(e^9*x^8 + 8*d*e^8*x^7 + 28*d^2*e^7*x^6 + 56*d^3*e^6*x^5 + 70*d^4*e^5*x^4 + 56*d^5*e^4*x^3 + 28*d ^6*e^3*x^2 + 8*d^7*e^2*x + d^8*e) + 7/26*sqrt(-e^2*x^2 + d^2)*d^3/(e^8*x^7 + 7*d*e^7*x^6 + 21*d^2*e^6*x^5 + 35*d^3*e^5*x^4 + 35*d^4*e^4*x^3 + 21*d^5 *e^3*x^2 + 7*d^6*e^2*x + d^7*e) - 7/572*sqrt(-e^2*x^2 + d^2)*d^2/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3*x^2 + 6*d^5*e ^2*x + d^6*e) - 35/5148*sqrt(-e^2*x^2 + d^2)*d/(e^6*x^5 + 5*d*e^5*x^4 + 10 *d^2*e^4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*x + d^5*e) - 5/1287*sqrt(-e^2*x^ 2 + d^2)/(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 /429*sqrt(-e^2*x^2 + d^2)/(d*e^4*x^3 + 3*d^2*e^3*x^2 + 3*d^3*e^2*x + d^4*e ) - 2/1287*sqrt(-e^2*x^2 + d^2)/(d^2*e^3*x^2 + 2*d^3*e^2*x + d^4*e) - 2/12 87*sqrt(-e^2*x^2 + d^2)/(d^3*e^2*x + d^4*e)
Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (88) = 176\).
Time = 0.15 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.13 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\frac {2 \, {\left (\frac {260 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} + \frac {6708 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {11726 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} + \frac {52481 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {61776 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{10} x^{5}} + \frac {120120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{12} x^{6}} + \frac {84084 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{14} x^{7}} + \frac {91377 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{16} x^{8}} + \frac {32604 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9}}{e^{18} x^{9}} + \frac {22308 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10}}{e^{20} x^{10}} + \frac {2574 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{11}}{e^{22} x^{11}} + \frac {1287 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{12}}{e^{24} x^{12}} + 119\right )}}{1287 \, d^{3} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{13} {\left | e \right |}} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^11,x, algorithm="giac")
Output:
2/1287*(260*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 6708*(d*e + sqrt (-e^2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 11726*(d*e + sqrt(-e^2*x^2 + d^2)*a bs(e))^3/(e^6*x^3) + 52481*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^8*x^4) + 61776*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e^10*x^5) + 120120*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/(e^12*x^6) + 84084*(d*e + sqrt(-e^2*x^2 + d ^2)*abs(e))^7/(e^14*x^7) + 91377*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e^ 16*x^8) + 32604*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9/(e^18*x^9) + 22308*( d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^10/(e^20*x^10) + 2574*(d*e + sqrt(-e^2* x^2 + d^2)*abs(e))^11/(e^22*x^11) + 1287*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e ))^12/(e^24*x^12) + 119)/(d^3*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 1)^13*abs(e))
Time = 8.62 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.99 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\frac {424\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^4}-\frac {1832\,d\,\sqrt {d^2-e^2\,x^2}}{1287\,e\,{\left (d+e\,x\right )}^5}-\frac {\sqrt {d^2-e^2\,x^2}}{429\,d\,e\,{\left (d+e\,x\right )}^3}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{1287\,d^2\,e\,{\left (d+e\,x\right )}^2}-\frac {2\,\sqrt {d^2-e^2\,x^2}}{1287\,d^3\,e\,\left (d+e\,x\right )}+\frac {320\,d^2\,\sqrt {d^2-e^2\,x^2}}{143\,e\,{\left (d+e\,x\right )}^6}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{13\,e\,{\left (d+e\,x\right )}^7} \] Input:
int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^11,x)
Output:
(424*(d^2 - e^2*x^2)^(1/2))/(1287*e*(d + e*x)^4) - (1832*d*(d^2 - e^2*x^2) ^(1/2))/(1287*e*(d + e*x)^5) - (d^2 - e^2*x^2)^(1/2)/(429*d*e*(d + e*x)^3) - (2*(d^2 - e^2*x^2)^(1/2))/(1287*d^2*e*(d + e*x)^2) - (2*(d^2 - e^2*x^2) ^(1/2))/(1287*d^3*e*(d + e*x)) + (320*d^2*(d^2 - e^2*x^2)^(1/2))/(143*e*(d + e*x)^6) - (16*d^3*(d^2 - e^2*x^2)^(1/2))/(13*e*(d + e*x)^7)
Time = 0.21 (sec) , antiderivative size = 462, normalized size of antiderivative = 4.62 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\frac {198 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}+20 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e x +1813 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{2} x^{2}+1228 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{3} x^{3}+1228 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{4} x^{4}+488 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{5} x^{5}+81 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{6} x^{6}-198 d^{7}+20 d^{6} e x -2741 d^{5} e^{2} x^{2}-1785 d^{4} e^{3} x^{3}-3160 d^{3} e^{4} x^{4}-1630 d^{2} e^{5} x^{5}-541 d \,e^{6} x^{6}-77 e^{7} x^{7}}{1287 d^{3} e \left (\sqrt {-e^{2} x^{2}+d^{2}}\, d^{6}+6 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e x +15 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{2} x^{2}+20 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{3} x^{3}+15 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{4} x^{4}+6 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{5} x^{5}+\sqrt {-e^{2} x^{2}+d^{2}}\, e^{6} x^{6}-d^{7}-7 d^{6} e x -21 d^{5} e^{2} x^{2}-35 d^{4} e^{3} x^{3}-35 d^{3} e^{4} x^{4}-21 d^{2} e^{5} x^{5}-7 d \,e^{6} x^{6}-e^{7} x^{7}\right )} \] Input:
int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^11,x)
Output:
(198*sqrt(d**2 - e**2*x**2)*d**6 + 20*sqrt(d**2 - e**2*x**2)*d**5*e*x + 18 13*sqrt(d**2 - e**2*x**2)*d**4*e**2*x**2 + 1228*sqrt(d**2 - e**2*x**2)*d** 3*e**3*x**3 + 1228*sqrt(d**2 - e**2*x**2)*d**2*e**4*x**4 + 488*sqrt(d**2 - e**2*x**2)*d*e**5*x**5 + 81*sqrt(d**2 - e**2*x**2)*e**6*x**6 - 198*d**7 + 20*d**6*e*x - 2741*d**5*e**2*x**2 - 1785*d**4*e**3*x**3 - 3160*d**3*e**4* x**4 - 1630*d**2*e**5*x**5 - 541*d*e**6*x**6 - 77*e**7*x**7)/(1287*d**3*e* (sqrt(d**2 - e**2*x**2)*d**6 + 6*sqrt(d**2 - e**2*x**2)*d**5*e*x + 15*sqrt (d**2 - e**2*x**2)*d**4*e**2*x**2 + 20*sqrt(d**2 - e**2*x**2)*d**3*e**3*x* *3 + 15*sqrt(d**2 - e**2*x**2)*d**2*e**4*x**4 + 6*sqrt(d**2 - e**2*x**2)*d *e**5*x**5 + sqrt(d**2 - e**2*x**2)*e**6*x**6 - d**7 - 7*d**6*e*x - 21*d** 5*e**2*x**2 - 35*d**4*e**3*x**3 - 35*d**3*e**4*x**4 - 21*d**2*e**5*x**5 - 7*d*e**6*x**6 - e**7*x**7))