Integrand size = 25, antiderivative size = 172 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx=\frac {e^5 \sqrt {d^2-e^2 x^2}}{16 d^2 x^2}-\frac {e^3 \left (d^2-e^2 x^2\right )^{3/2}}{24 d^2 x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}-\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{6 d^2 x^6}-\frac {2 e^2 \left (d^2-e^2 x^2\right )^{5/2}}{35 d^3 x^5}-\frac {e^7 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^3} \]
-1/24*e^3*(-e^2*x^2+d^2)^(3/2)/d^2/x^4-1/7*(-e^2*x^2+d^2)^(5/2)/d/x^7-1/6* e*(-e^2*x^2+d^2)^(5/2)/d^2/x^6-2/35*e^2*(-e^2*x^2+d^2)^(5/2)/d^3/x^5-1/16* e^7*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^3+1/16*e^5*(-e^2*x^2+d^2)^(1/2)/d^2/ x^2
Time = 0.36 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.89 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-240 d^6-280 d^5 e x+384 d^4 e^2 x^2+490 d^3 e^3 x^3-48 d^2 e^4 x^4-105 d e^5 x^5-96 e^6 x^6\right )}{1680 d^3 x^7}-\frac {\sqrt {d^2} e^7 \log (x)}{16 d^4}+\frac {\sqrt {d^2} e^7 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{16 d^4} \]
(Sqrt[d^2 - e^2*x^2]*(-240*d^6 - 280*d^5*e*x + 384*d^4*e^2*x^2 + 490*d^3*e ^3*x^3 - 48*d^2*e^4*x^4 - 105*d*e^5*x^5 - 96*e^6*x^6))/(1680*d^3*x^7) - (S qrt[d^2]*e^7*Log[x])/(16*d^4) + (Sqrt[d^2]*e^7*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(16*d^4)
Time = 0.30 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.06, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {539, 25, 27, 539, 25, 27, 534, 243, 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.
| \(\displaystyle \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle -\frac {\int -\frac {d e (7 d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7}dx}{7 d^2}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d e (7 d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7}dx}{7 d^2}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int \frac {(7 d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7}dx}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {e \left (-\frac {\int -\frac {d e (12 d+7 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx}{6 d^2}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \left (\frac {\int \frac {d e (12 d+7 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx}{6 d^2}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \left (\frac {e \int \frac {(12 d+7 e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx}{6 d}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {e \left (\frac {e \left (7 e \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5}dx-\frac {12 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {e \left (\frac {e \left (\frac {7}{2} e \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx^2-\frac {12 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {e \left (\frac {e \left (\frac {7}{2} e \left (-\frac {3}{4} e^2 \int \frac {\sqrt {d^2-e^2 x^2}}{x^4}dx^2-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {12 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {e \left (\frac {e \left (\frac {7}{2} e \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {12 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {e \left (\frac {e \left (\frac {7}{2} e \left (-\frac {3}{4} e^2 \left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {12 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {e \left (\frac {e \left (\frac {7}{2} e \left (-\frac {3}{4} e^2 \left (\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {12 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {7 \left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\right )}{7 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{7 d x^7}\) |
-1/7*(d^2 - e^2*x^2)^(5/2)/(d*x^7) + (e*((-7*(d^2 - e^2*x^2)^(5/2))/(6*d*x ^6) + (e*((-12*(d^2 - e^2*x^2)^(5/2))/(5*d*x^5) + (7*e*(-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))/2))/(6*d)))/(7*d)
3.1.14.3.1 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]
Time = 0.42 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.77
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (96 e^{6} x^{6}+105 d \,e^{5} x^{5}+48 d^{2} e^{4} x^{4}-490 d^{3} x^{3} e^{3}-384 d^{4} e^{2} x^{2}+280 d^{5} e x +240 d^{6}\right )}{1680 x^{7} d^{3}}-\frac {e^{7} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 d^{2} \sqrt {d^{2}}}\) | \(132\) |
| default | \(e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 d^{2} x^{6}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{4 d^{2} x^{4}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )+d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 d^{2} x^{7}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 d^{4} x^{5}}\right )\) | \(225\) |
-1/1680*(-e^2*x^2+d^2)^(1/2)*(96*e^6*x^6+105*d*e^5*x^5+48*d^2*e^4*x^4-490* d^3*e^3*x^3-384*d^4*e^2*x^2+280*d^5*e*x+240*d^6)/x^7/d^3-1/16/d^2*e^7/(d^2 )^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx=\frac {105 \, e^{7} x^{7} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (96 \, e^{6} x^{6} + 105 \, d e^{5} x^{5} + 48 \, d^{2} e^{4} x^{4} - 490 \, d^{3} e^{3} x^{3} - 384 \, d^{4} e^{2} x^{2} + 280 \, d^{5} e x + 240 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{1680 \, d^{3} x^{7}} \]
1/1680*(105*e^7*x^7*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (96*e^6*x^6 + 105 *d*e^5*x^5 + 48*d^2*e^4*x^4 - 490*d^3*e^3*x^3 - 384*d^4*e^2*x^2 + 280*d^5* e*x + 240*d^6)*sqrt(-e^2*x^2 + d^2))/(d^3*x^7)
Result contains complex when optimal does not.
Time = 8.68 (sec) , antiderivative size = 1037, normalized size of antiderivative = 6.03 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx=\text {Too large to display} \]
d**3*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*sqrt(-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* *2*e*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))/(1 6*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**5/(16*d**4*x*sqrt(-d**2/(e**2*x** 2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) - d*e**2*Piecewise((3*I* d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**2 *x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I*e**6* x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e** 4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e** 2*x**2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e* *2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2 *x**7) + 2*e**6*x**6*sqrt(1 - e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*...
Time = 0.28 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx=-\frac {e^{7} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{3}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{7}}{16 \, d^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}}{48 \, d^{6}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}}{48 \, d^{6} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}}{24 \, d^{4} x^{4}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{35 \, d^{3} x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{6 \, d^{2} x^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{7 \, d x^{7}} \]
-1/16*e^7*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^3 + 1/16*s qrt(-e^2*x^2 + d^2)*e^7/d^4 + 1/48*(-e^2*x^2 + d^2)^(3/2)*e^7/d^6 + 1/48*( -e^2*x^2 + d^2)^(5/2)*e^5/(d^6*x^2) - 1/24*(-e^2*x^2 + d^2)^(5/2)*e^3/(d^4 *x^4) - 2/35*(-e^2*x^2 + d^2)^(5/2)*e^2/(d^3*x^5) - 1/6*(-e^2*x^2 + d^2)^( 5/2)*e/(d^2*x^6) - 1/7*(-e^2*x^2 + d^2)^(5/2)/(d*x^7)
Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (148) = 296\).
Time = 0.29 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.01 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx=\frac {{\left (15 \, e^{8} + \frac {35 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{6}}{x} - \frac {21 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{4}}{x^{2}} - \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{2}}{x^{3}} - \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{x^{4}} - \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{2} x^{5}} + \frac {315 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{4} x^{6}}\right )} e^{14} x^{7}}{13440 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{3} {\left | e \right |}} - \frac {e^{8} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{16 \, d^{3} {\left | e \right |}} - \frac {\frac {315 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{18} e^{12}}{x} - \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{18} e^{10}}{x^{2}} - \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{18} e^{8}}{x^{3}} - \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{18} e^{6}}{x^{4}} - \frac {21 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{18} e^{4}}{x^{5}} + \frac {35 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{18} e^{2}}{x^{6}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{18}}{x^{7}}}{13440 \, d^{21} e^{6} {\left | e \right |}} \]
1/13440*(15*e^8 + 35*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^6/x - 21*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e^4/x^2 - 105*(d*e + sqrt(-e^2*x^2 + d^2)* abs(e))^3*e^2/x^3 - 105*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/x^4 - 105*(d *e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e^2*x^5) + 315*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/(e^4*x^6))*e^14*x^7/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7 *d^3*abs(e)) - 1/16*e^8*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e) )/(e^2*abs(x)))/(d^3*abs(e)) - 1/13440*(315*(d*e + sqrt(-e^2*x^2 + d^2)*ab s(e))*d^18*e^12/x - 105*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^18*e^10/x^ 2 - 105*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^18*e^8/x^3 - 105*(d*e + sq rt(-e^2*x^2 + d^2)*abs(e))^4*d^18*e^6/x^4 - 21*(d*e + sqrt(-e^2*x^2 + d^2) *abs(e))^5*d^18*e^4/x^5 + 35*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d^18*e^ 2/x^6 + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^7*d^18/x^7)/(d^21*e^6*abs(e ))
Time = 14.49 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^8} \, dx=\frac {8\,d\,e^2\,\sqrt {d^2-e^2\,x^2}}{35\,x^5}-\frac {d^3\,\sqrt {d^2-e^2\,x^2}}{7\,x^7}-\frac {e^4\,\sqrt {d^2-e^2\,x^2}}{35\,d\,x^3}-\frac {2\,e^6\,\sqrt {d^2-e^2\,x^2}}{35\,d^3\,x}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{3/2}}{6\,x^6}+\frac {d^2\,e\,\sqrt {d^2-e^2\,x^2}}{16\,x^6}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{16\,d^2\,x^6}+\frac {e^7\,\mathrm {atan}\left (\frac {\sqrt {d^2-e^2\,x^2}\,1{}\mathrm {i}}{d}\right )\,1{}\mathrm {i}}{16\,d^3} \]
(e^7*atan(((d^2 - e^2*x^2)^(1/2)*1i)/d)*1i)/(16*d^3) - (d^3*(d^2 - e^2*x^2 )^(1/2))/(7*x^7) - (e*(d^2 - e^2*x^2)^(3/2))/(6*x^6) - (e^4*(d^2 - e^2*x^2 )^(1/2))/(35*d*x^3) - (2*e^6*(d^2 - e^2*x^2)^(1/2))/(35*d^3*x) + (8*d*e^2* (d^2 - e^2*x^2)^(1/2))/(35*x^5) + (d^2*e*(d^2 - e^2*x^2)^(1/2))/(16*x^6) - (e*(d^2 - e^2*x^2)^(5/2))/(16*d^2*x^6)