Integrand size = 27, antiderivative size = 225 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=-\frac {33 e^8 \sqrt {d^2-e^2 x^2}}{256 d x^2}+\frac {11 e^6 \left (d^2-e^2 x^2\right )^{3/2}}{128 d x^4}-\frac {11 e^4 \left (d^2-e^2 x^2\right )^{5/2}}{160 d x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}-\frac {33 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{80 d x^8}-\frac {5 e^3 \left (d^2-e^2 x^2\right )^{7/2}}{21 d^2 x^7}+\frac {33 e^{10} \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{256 d^2} \]
11/128*e^6*(-e^2*x^2+d^2)^(3/2)/d/x^4-11/160*e^4*(-e^2*x^2+d^2)^(5/2)/d/x^ 6-1/10*d*(-e^2*x^2+d^2)^(7/2)/x^10-1/3*e*(-e^2*x^2+d^2)^(7/2)/x^9-33/80*e^ 2*(-e^2*x^2+d^2)^(7/2)/d/x^8-5/21*e^3*(-e^2*x^2+d^2)^(7/2)/d^2/x^7+33/256* e^10*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^2-33/256*e^8*(-e^2*x^2+d^2)^(1/2)/d /x^2
Time = 0.52 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.83 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2688 d^9-8960 d^8 e x-3024 d^7 e^2 x^2+20480 d^6 e^3 x^3+23352 d^5 e^4 x^4-7680 d^4 e^5 x^5-24570 d^3 e^6 x^6-10240 d^2 e^7 x^7+3465 d e^8 x^8+6400 e^9 x^9\right )}{26880 d^2 x^{10}}+\frac {33 \sqrt {d^2} e^{10} \log (x)}{256 d^3}-\frac {33 \sqrt {d^2} e^{10} \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{256 d^3} \]
(Sqrt[d^2 - e^2*x^2]*(-2688*d^9 - 8960*d^8*e*x - 3024*d^7*e^2*x^2 + 20480* d^6*e^3*x^3 + 23352*d^5*e^4*x^4 - 7680*d^4*e^5*x^5 - 24570*d^3*e^6*x^6 - 1 0240*d^2*e^7*x^7 + 3465*d*e^8*x^8 + 6400*e^9*x^9))/(26880*d^2*x^10) + (33* Sqrt[d^2]*e^10*Log[x])/(256*d^3) - (33*Sqrt[d^2]*e^10*Log[Sqrt[d^2] - Sqrt [d^2 - e^2*x^2]])/(256*d^3)
Time = 0.47 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.11, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {540, 25, 2338, 27, 539, 25, 27, 534, 243, 51, 51, 51, 73, 221}
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.
| \(\displaystyle \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {\int -\frac {\left (d^2-e^2 x^2\right )^{5/2} \left (30 e d^4+33 e^2 x d^3+10 e^3 x^2 d^2\right )}{x^{10}}dx}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (30 e d^4+33 e^2 x d^3+10 e^3 x^2 d^2\right )}{x^{10}}dx}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 d^4 e^2 (99 d+50 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^9}dx}{9 d^2}-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \int \frac {(99 d+50 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^9}dx-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (-\frac {\int -\frac {d e (400 d+99 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^8}dx}{8 d^2}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {\int \frac {d e (400 d+99 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^8}dx}{8 d^2}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {e \int \frac {(400 d+99 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^8}dx}{8 d}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {e \left (99 e \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7}dx-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )}{8 d}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {e \left (\frac {99}{2} e \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8}dx^2-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )}{8 d}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {e \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx^2-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )}{8 d}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {e \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \left (-\frac {3}{4} e^2 \int \frac {\sqrt {d^2-e^2 x^2}}{x^4}dx^2-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )}{8 d}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {e \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )}{8 d}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {e \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \left (-\frac {3}{4} e^2 \left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )}{8 d}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {e \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \left (-\frac {3}{4} e^2 \left (\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )}{8 d}-\frac {99 \left (d^2-e^2 x^2\right )^{7/2}}{8 d x^8}\right )-\frac {10 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{3 x^9}}{10 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{10 x^{10}}\) |
-1/10*(d*(d^2 - e^2*x^2)^(7/2))/x^10 + ((-10*d^2*e*(d^2 - e^2*x^2)^(7/2))/ (3*x^9) + (d^2*e^2*((-99*(d^2 - e^2*x^2)^(7/2))/(8*d*x^8) + (e*((-400*(d^2 - e^2*x^2)^(7/2))/(7*d*x^7) + (99*e*(-1/3*(d^2 - e^2*x^2)^(5/2)/x^6 - (5* e^2*(-1/2*(d^2 - e^2*x^2)^(3/2)/x^4 - (3*e^2*(-(Sqrt[d^2 - e^2*x^2]/x^2) + (e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d))/4))/6))/2))/(8*d)))/3)/(10*d^2)
3.1.81.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.55 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-6400 e^{9} x^{9}-3465 d \,e^{8} x^{8}+10240 d^{2} e^{7} x^{7}+24570 d^{3} e^{6} x^{6}+7680 d^{4} e^{5} x^{5}-23352 d^{5} e^{4} x^{4}-20480 d^{6} e^{3} x^{3}+3024 x^{2} d^{7} e^{2}+8960 x \,d^{8} e +2688 d^{9}\right )}{26880 d^{2} x^{10}}+\frac {33 e^{10} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{256 d \sqrt {d^{2}}}\) | \(165\) |
| default | \(-\frac {e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 d^{2} x^{7}}+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 d^{2} x^{10}}+\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )}{10 d^{2}}\right )+3 d \,e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 d^{2} x^{9}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 d^{4} x^{7}}\right )\) | \(568\) |
-1/26880*(-e^2*x^2+d^2)^(1/2)*(-6400*e^9*x^9-3465*d*e^8*x^8+10240*d^2*e^7* x^7+24570*d^3*e^6*x^6+7680*d^4*e^5*x^5-23352*d^5*e^4*x^4-20480*d^6*e^3*x^3 +3024*d^7*e^2*x^2+8960*d^8*e*x+2688*d^9)/d^2/x^10+33/256/d*e^10/(d^2)^(1/2 )*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)
Time = 0.33 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.68 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=-\frac {3465 \, e^{10} x^{10} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (6400 \, e^{9} x^{9} + 3465 \, d e^{8} x^{8} - 10240 \, d^{2} e^{7} x^{7} - 24570 \, d^{3} e^{6} x^{6} - 7680 \, d^{4} e^{5} x^{5} + 23352 \, d^{5} e^{4} x^{4} + 20480 \, d^{6} e^{3} x^{3} - 3024 \, d^{7} e^{2} x^{2} - 8960 \, d^{8} e x - 2688 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{26880 \, d^{2} x^{10}} \]
-1/26880*(3465*e^10*x^10*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (6400*e^9*x^ 9 + 3465*d*e^8*x^8 - 10240*d^2*e^7*x^7 - 24570*d^3*e^6*x^6 - 7680*d^4*e^5* x^5 + 23352*d^5*e^4*x^4 + 20480*d^6*e^3*x^3 - 3024*d^7*e^2*x^2 - 8960*d^8* e*x - 2688*d^9)*sqrt(-e^2*x^2 + d^2))/(d^2*x^10)
Result contains complex when optimal does not.
Time = 136.08 (sec) , antiderivative size = 2159, normalized size of antiderivative = 9.60 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\text {Too large to display} \]
d**7*Piecewise((-d**2/(10*e*x**11*sqrt(d**2/(e**2*x**2) - 1)) + 9*e/(80*x* *9*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(480*d**2*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**5/(1920*d**4*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 7*e**7/(768* d**6*x**3*sqrt(d**2/(e**2*x**2) - 1)) - 7*e**9/(256*d**8*x*sqrt(d**2/(e**2 *x**2) - 1)) + 7*e**10*acosh(d/(e*x))/(256*d**9), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(10*e*x**11*sqrt(-d**2/(e**2*x**2) + 1)) - 9*I*e/(80*x**9*sqrt (-d**2/(e**2*x**2) + 1)) - I*e**3/(480*d**2*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**5/(1920*d**4*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e**7/(76 8*d**6*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 7*I*e**9/(256*d**8*x*sqrt(-d**2 /(e**2*x**2) + 1)) - 7*I*e**10*asin(d/(e*x))/(256*d**9), True)) + 3*d**6*e *Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(9*x**8) + e**3*sqrt(d**2/(e**2* x**2) - 1)/(63*d**2*x**6) + 2*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x* *4) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(315*d**6*x**2) + 16*e**9*sqrt(d** 2/(e**2*x**2) - 1)/(315*d**8), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d** 2/(e**2*x**2) + 1)/(9*x**8) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(63*d**2* x**6) + 2*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**4) + 8*I*e**7*sq rt(-d**2/(e**2*x**2) + 1)/(315*d**6*x**2) + 16*I*e**9*sqrt(-d**2/(e**2*x** 2) + 1)/(315*d**8), True)) + d**5*e**2*Piecewise((-d**2/(8*e*x**9*sqrt(d** 2/(e**2*x**2) - 1)) + 7*e/(48*x**7*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(192 *d**2*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**5/(384*d**4*x**3*sqrt(d**...
Time = 0.28 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.21 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\frac {33 \, e^{10} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{256 \, d^{2}} - \frac {33 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{10}}{256 \, d^{3}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{10}}{256 \, d^{5}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{10}}{1280 \, d^{7}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{8}}{1280 \, d^{7} x^{2}} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{640 \, d^{5} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{160 \, d^{3} x^{6}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{21 \, d^{2} x^{7}} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{80 \, d x^{8}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{3 \, x^{9}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{10 \, x^{10}} \]
33/256*e^10*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^2 - 33/2 56*sqrt(-e^2*x^2 + d^2)*e^10/d^3 - 11/256*(-e^2*x^2 + d^2)^(3/2)*e^10/d^5 - 33/1280*(-e^2*x^2 + d^2)^(5/2)*e^10/d^7 - 33/1280*(-e^2*x^2 + d^2)^(7/2) *e^8/(d^7*x^2) + 11/640*(-e^2*x^2 + d^2)^(7/2)*e^6/(d^5*x^4) - 11/160*(-e^ 2*x^2 + d^2)^(7/2)*e^4/(d^3*x^6) - 5/21*(-e^2*x^2 + d^2)^(7/2)*e^3/(d^2*x^ 7) - 33/80*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^8) - 1/3*(-e^2*x^2 + d^2)^(7/2) *e/x^9 - 1/10*(-e^2*x^2 + d^2)^(7/2)*d/x^10
Leaf count of result is larger than twice the leaf count of optimal. 731 vs. \(2 (193) = 386\).
Time = 0.31 (sec) , antiderivative size = 731, normalized size of antiderivative = 3.25 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\frac {{\left (42 \, e^{11} + \frac {280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{9}}{x} + \frac {525 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{7}}{x^{2}} - \frac {600 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{5}}{x^{3}} - \frac {3570 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e^{3}}{x^{4}} - \frac {3360 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} e}{x^{5}} + \frac {5880 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e x^{6}} + \frac {16800 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{3} x^{7}} + \frac {10500 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{5} x^{8}} - \frac {31920 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9}}{e^{7} x^{9}}\right )} e^{20} x^{10}}{430080 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10} d^{2} {\left | e \right |}} + \frac {33 \, e^{11} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{256 \, d^{2} {\left | e \right |}} + \frac {\frac {31920 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{18} e^{17} {\left | e \right |}}{x} - \frac {10500 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{18} e^{15} {\left | e \right |}}{x^{2}} - \frac {16800 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{18} e^{13} {\left | e \right |}}{x^{3}} - \frac {5880 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{18} e^{11} {\left | e \right |}}{x^{4}} + \frac {3360 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{18} e^{9} {\left | e \right |}}{x^{5}} + \frac {3570 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{18} e^{7} {\left | e \right |}}{x^{6}} + \frac {600 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{18} e^{5} {\left | e \right |}}{x^{7}} - \frac {525 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} d^{18} e^{3} {\left | e \right |}}{x^{8}} - \frac {280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9} d^{18} e {\left | e \right |}}{x^{9}} - \frac {42 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10} d^{18} {\left | e \right |}}{e x^{10}}}{430080 \, d^{20} e^{10}} \]
1/430080*(42*e^11 + 280*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^9/x + 525*(d *e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e^7/x^2 - 600*(d*e + sqrt(-e^2*x^2 + d ^2)*abs(e))^3*e^5/x^3 - 3570*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*e^3/x^4 - 3360*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*e/x^5 + 5880*(d*e + sqrt(-e^ 2*x^2 + d^2)*abs(e))^6/(e*x^6) + 16800*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) ^7/(e^3*x^7) + 10500*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e^5*x^8) - 319 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9/(e^7*x^9))*e^20*x^10/((d*e + sqrt (-e^2*x^2 + d^2)*abs(e))^10*d^2*abs(e)) + 33/256*e^11*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^2*abs(e)) + 1/430080*(319 20*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^18*e^17*abs(e)/x - 10500*(d*e + s qrt(-e^2*x^2 + d^2)*abs(e))^2*d^18*e^15*abs(e)/x^2 - 16800*(d*e + sqrt(-e^ 2*x^2 + d^2)*abs(e))^3*d^18*e^13*abs(e)/x^3 - 5880*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^18*e^11*abs(e)/x^4 + 3360*(d*e + sqrt(-e^2*x^2 + d^2)*abs (e))^5*d^18*e^9*abs(e)/x^5 + 3570*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d^ 18*e^7*abs(e)/x^6 + 600*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d^18*e^5*abs (e)/x^7 - 525*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8*d^18*e^3*abs(e)/x^8 - 280*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9*d^18*e*abs(e)/x^9 - 42*(d*e + sq rt(-e^2*x^2 + d^2)*abs(e))^10*d^18*abs(e)/(e*x^10))/(d^20*e^10)
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{11}} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{11}} \,d x \]