Integrand size = 27, antiderivative size = 204 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {e^6 (125 d+128 e x) \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {e^4 (125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {e^2 (125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{240 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}-e^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {125}{128} e^8 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
1/192*e^4*(64*e*x+125*d)*(-e^2*x^2+d^2)^(3/2)/x^4-1/240*e^2*(48*e*x+125*d) *(-e^2*x^2+d^2)^(5/2)/x^6-1/8*d*(-e^2*x^2+d^2)^(7/2)/x^8-3/7*e*(-e^2*x^2+d ^2)^(7/2)/x^7-e^8*arctan(e*x/(-e^2*x^2+d^2)^(1/2))+125/128*e^8*arctanh((-e ^2*x^2+d^2)^(1/2)/d)-1/128*e^6*(128*e*x+125*d)*(-e^2*x^2+d^2)^(1/2)/x^2
Time = 0.52 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-1680 d^7-5760 d^6 e x-1960 d^5 e^2 x^2+14592 d^4 e^3 x^3+17710 d^3 e^4 x^4-7424 d^2 e^5 x^5-27195 d e^6 x^6-14848 e^7 x^7\right )}{13440 x^8}+2 e^8 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+\frac {125 \sqrt {d^2} e^8 \log (x)}{128 d}-\frac {125 \sqrt {d^2} e^8 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{128 d} \]
(Sqrt[d^2 - e^2*x^2]*(-1680*d^7 - 5760*d^6*e*x - 1960*d^5*e^2*x^2 + 14592* d^4*e^3*x^3 + 17710*d^3*e^4*x^4 - 7424*d^2*e^5*x^5 - 27195*d*e^6*x^6 - 148 48*e^7*x^7))/(13440*x^8) + 2*e^8*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2* x^2])] + (125*Sqrt[d^2]*e^8*Log[x])/(128*d) - (125*Sqrt[d^2]*e^8*Log[Sqrt[ d^2] - Sqrt[d^2 - e^2*x^2]])/(128*d)
Time = 0.52 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.13, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {540, 25, 2338, 27, 537, 25, 537, 27, 537, 25, 538, 224, 216, 243, 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.
| \(\displaystyle \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle -\frac {\int -\frac {\left (d^2-e^2 x^2\right )^{5/2} \left (24 e d^4+25 e^2 x d^3+8 e^3 x^2 d^2\right )}{x^8}dx}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (24 e d^4+25 e^2 x d^3+8 e^3 x^2 d^2\right )}{x^8}dx}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {\int -\frac {7 d^4 e^2 (25 d+8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^7}dx}{7 d^2}-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 e^2 \int \frac {(25 d+8 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^7}dx-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {d^2 e^2 \left (\frac {1}{6} e^2 \int -\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5}dx-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \int \frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5}dx-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (\frac {1}{4} e^2 \int -\frac {3 (125 d+64 e x) \sqrt {d^2-e^2 x^2}}{x^3}dx-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \int \frac {(125 d+64 e x) \sqrt {d^2-e^2 x^2}}{x^3}dx-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \left (\frac {1}{2} e^2 \int -\frac {125 d+128 e x}{x \sqrt {d^2-e^2 x^2}}dx-\frac {(125 d+128 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \int \frac {125 d+128 e x}{x \sqrt {d^2-e^2 x^2}}dx-\frac {(125 d+128 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \left (128 e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+125 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx\right )-\frac {(125 d+128 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \left (125 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+128 e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {(125 d+128 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \left (125 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+128 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-\frac {(125 d+128 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \left (\frac {125}{2} d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2+128 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-\frac {(125 d+128 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \left (128 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {125 d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}\right )-\frac {(125 d+128 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {d^2 e^2 \left (-\frac {1}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \left (128 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-125 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )-\frac {(125 d+128 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(125 d+64 e x) \left (d^2-e^2 x^2\right )^{3/2}}{4 x^4}\right )-\frac {(125 d+48 e x) \left (d^2-e^2 x^2\right )^{5/2}}{30 x^6}\right )-\frac {24 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{7 x^7}}{8 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}\) |
-1/8*(d*(d^2 - e^2*x^2)^(7/2))/x^8 + ((-24*d^2*e*(d^2 - e^2*x^2)^(7/2))/(7 *x^7) + d^2*e^2*(-1/30*((125*d + 48*e*x)*(d^2 - e^2*x^2)^(5/2))/x^6 - (e^2 *(-1/4*((125*d + 64*e*x)*(d^2 - e^2*x^2)^(3/2))/x^4 - (3*e^2*(-1/2*((125*d + 128*e*x)*Sqrt[d^2 - e^2*x^2])/x^2 - (e^2*(128*ArcTan[(e*x)/Sqrt[d^2 - e ^2*x^2]] - 125*ArcTanh[Sqrt[d^2 - e^2*x^2]/d]))/2))/4))/6))/(8*d^2)
3.1.79.3.1 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] :> 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), x] - Simp[2*b*(p/((m + 1)*(m + 2))) Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) *x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && GtQ[p, 0] && !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
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 = 0.49 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (14848 e^{7} x^{7}+27195 d \,e^{6} x^{6}+7424 d^{2} e^{5} x^{5}-17710 d^{3} e^{4} x^{4}-14592 d^{4} e^{3} x^{3}+1960 d^{5} e^{2} x^{2}+5760 d^{6} e x +1680 d^{7}\right )}{13440 x^{8}}-\frac {e^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}+\frac {125 e^{8} d \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}}\) | \(170\) |
| default | \(e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{5 d^{2} x^{5}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{3 d^{2} x^{3}}-\frac {4 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{d^{2} x}-\frac {6 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{d^{2}}\right )}{3 d^{2}}\right )}{5 d^{2}}\right )+d^{3} \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 \,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 )-\frac {3 e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 x^{7}}\) | \(640\) |
-1/13440*(-e^2*x^2+d^2)^(1/2)*(14848*e^7*x^7+27195*d*e^6*x^6+7424*d^2*e^5* x^5-17710*d^3*e^4*x^4-14592*d^4*e^3*x^3+1960*d^5*e^2*x^2+5760*d^6*e*x+1680 *d^7)/x^8-e^9/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+125/1 28*e^8*d/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)
Time = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.80 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\frac {26880 \, e^{8} x^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 13125 \, e^{8} x^{8} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (14848 \, e^{7} x^{7} + 27195 \, d e^{6} x^{6} + 7424 \, d^{2} e^{5} x^{5} - 17710 \, d^{3} e^{4} x^{4} - 14592 \, d^{4} e^{3} x^{3} + 1960 \, d^{5} e^{2} x^{2} + 5760 \, d^{6} e x + 1680 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{13440 \, x^{8}} \]
1/13440*(26880*e^8*x^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 13125*e ^8*x^8*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (14848*e^7*x^7 + 27195*d*e^6*x ^6 + 7424*d^2*e^5*x^5 - 17710*d^3*e^4*x^4 - 14592*d^4*e^3*x^3 + 1960*d^5*e ^2*x^2 + 5760*d^6*e*x + 1680*d^7)*sqrt(-e^2*x^2 + d^2))/x^8
Result contains complex when optimal does not.
Time = 29.73 (sec) , antiderivative size = 1719, normalized size of antiderivative = 8.43 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\text {Too large to display} \]
d**7*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**2/(e**2*x**2) - 1)) - 5*e**7/(128*d** 6*x*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**8*acosh(d/(e*x))/(128*d**7), Abs(d* *2/(e**2*x**2)) > 1), (I*d**2/(8*e*x**9*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I *e/(48*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(192*d**2*x**5*sqrt(-d** 2/(e**2*x**2) + 1)) - 5*I*e**5/(384*d**4*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 5*I*e**7/(128*d**6*x*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e**8*asin(d/(e* x))/(128*d**7), True)) + 3*d**6*e*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1) /(7*x**6) + e**3*sqrt(d**2/(e**2*x**2) - 1)/(35*d**2*x**4) + 4*e**5*sqrt(d **2/(e**2*x**2) - 1)/(105*d**4*x**2) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/( 105*d**6), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/( 7*x**6) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(35*d**2*x**4) + 4*I*e**5*sqr t(-d**2/(e**2*x**2) + 1)/(105*d**4*x**2) + 8*I*e**7*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**6), True)) + d**5*e**2*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/ (e**2*x**2) - 1)) + 5*e/(24*x**5*sqrt(d**2/(e**2*x**2) - 1)) + e**3/(48*d* *2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2 ) - 1)) + e**6*acosh(d/(e*x))/(16*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d* *2/(6*e*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e/(24*x**5*sqrt(-d**2/(e** 2*x**2) + 1)) - I*e**3/(48*d**2*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e...
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (178) = 356\).
Time = 0.28 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.78 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=-\frac {e^{9} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} + \frac {125}{128} \, e^{8} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{9} x}{d^{2}} - \frac {125 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{8}}{128 \, d} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9} x}{3 \, d^{4}} - \frac {125 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{8}}{384 \, d^{3}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}}{128 \, d^{5}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}}{15 \, d^{4} x} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{6}}{128 \, d^{5} x^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{15 \, d^{4} x^{3}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{4}}{192 \, d^{3} x^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{5 \, d^{2} x^{5}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{48 \, d x^{6}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{7 \, x^{7}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{8 \, x^{8}} \]
-e^9*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 125/128*e^8*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) - sqrt(-e^2*x^2 + d^2)*e^9*x/d^2 - 125/ 128*sqrt(-e^2*x^2 + d^2)*e^8/d - 2/3*(-e^2*x^2 + d^2)^(3/2)*e^9*x/d^4 - 12 5/384*(-e^2*x^2 + d^2)^(3/2)*e^8/d^3 - 25/128*(-e^2*x^2 + d^2)^(5/2)*e^8/d ^5 - 8/15*(-e^2*x^2 + d^2)^(5/2)*e^7/(d^4*x) - 25/128*(-e^2*x^2 + d^2)^(7/ 2)*e^6/(d^5*x^2) + 2/15*(-e^2*x^2 + d^2)^(7/2)*e^5/(d^4*x^3) + 25/192*(-e^ 2*x^2 + d^2)^(7/2)*e^4/(d^3*x^4) - 1/5*(-e^2*x^2 + d^2)^(7/2)*e^3/(d^2*x^5 ) - 25/48*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^6) - 3/7*(-e^2*x^2 + d^2)^(7/2)* e/x^7 - 1/8*(-e^2*x^2 + d^2)^(7/2)*d/x^8
Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (178) = 356\).
Time = 0.31 (sec) , antiderivative size = 584, normalized size of antiderivative = 2.86 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\frac {{\left (105 \, e^{9} + \frac {720 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{7}}{x} + \frac {1120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{5}}{x^{2}} - \frac {3696 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{3}}{x^{3}} - \frac {14280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e}{x^{4}} - \frac {560 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e x^{5}} + \frac {77280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{3} x^{6}} + \frac {122640 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{5} x^{7}}\right )} e^{16} x^{8}}{215040 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} {\left | e \right |}} - \frac {e^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {125 \, e^{9} \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 )}{128 \, {\left | e \right |}} - \frac {\frac {122640 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{13} {\left | e \right |}}{x} + \frac {77280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{11} {\left | e \right |}}{x^{2}} - \frac {560 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{9} {\left | e \right |}}{x^{3}} - \frac {14280 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e^{7} {\left | e \right |}}{x^{4}} - \frac {3696 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} e^{5} {\left | e \right |}}{x^{5}} + \frac {1120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} e^{3} {\left | e \right |}}{x^{6}} + \frac {720 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} e {\left | e \right |}}{x^{7}} + \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} {\left | e \right |}}{e x^{8}}}{215040 \, e^{8}} \]
1/215040*(105*e^9 + 720*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^7/x + 1120*( d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e^5/x^2 - 3696*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e^3/x^3 - 14280*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*e/x^ 4 - 560*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e*x^5) + 77280*(d*e + sqrt( -e^2*x^2 + d^2)*abs(e))^6/(e^3*x^6) + 122640*(d*e + sqrt(-e^2*x^2 + d^2)*a bs(e))^7/(e^5*x^7))*e^16*x^8/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8*abs(e) ) - e^9*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 125/128*e^9*log(1/2*abs(-2*d* e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) - 1/215040*(122640 *(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^13*abs(e)/x + 77280*(d*e + sqrt(-e^ 2*x^2 + d^2)*abs(e))^2*e^11*abs(e)/x^2 - 560*(d*e + sqrt(-e^2*x^2 + d^2)*a bs(e))^3*e^9*abs(e)/x^3 - 14280*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*e^7* abs(e)/x^4 - 3696*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*e^5*abs(e)/x^5 + 1 120*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*e^3*abs(e)/x^6 + 720*(d*e + sqrt (-e^2*x^2 + d^2)*abs(e))^7*e*abs(e)/x^7 + 105*(d*e + sqrt(-e^2*x^2 + d^2)* abs(e))^8*abs(e)/(e*x^8))/e^8
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^9} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^9} \,d x \]