Integrand size = 27, antiderivative size = 182 \[ \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (5 d+6 e x)}{5 d^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 e^2 (10 d+11 e x)}{5 d^7 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^6 x^2}-\frac {2 e \sqrt {d^2-e^2 x^2}}{d^7 x}-\frac {9 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^7} \]
2/5*e^2*(e*x+d)/d^3/(-e^2*x^2+d^2)^(5/2)+1/5*e^2*(6*e*x+5*d)/d^5/(-e^2*x^2 +d^2)^(3/2)-9/2*e^2*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^7+2/5*e^2*(11*e*x+10 *d)/d^7/(-e^2*x^2+d^2)^(1/2)-1/2*(-e^2*x^2+d^2)^(1/2)/d^6/x^2-2*e*(-e^2*x^ 2+d^2)^(1/2)/d^7/x
Time = 0.47 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (5 d^5+10 d^4 e x-94 d^3 e^2 x^2+58 d^2 e^3 x^3+83 d e^4 x^4-64 e^5 x^5\right )}{x^2 (-d+e x)^3 (d+e x)}+90 e^2 \text {arctanh}\left (\frac {\sqrt {-e^2} x-\sqrt {d^2-e^2 x^2}}{d}\right )}{10 d^7} \]
((Sqrt[d^2 - e^2*x^2]*(5*d^5 + 10*d^4*e*x - 94*d^3*e^2*x^2 + 58*d^2*e^3*x^ 3 + 83*d*e^4*x^4 - 64*e^5*x^5))/(x^2*(-d + e*x)^3*(d + e*x)) + 90*e^2*ArcT anh[(Sqrt[-e^2]*x - Sqrt[d^2 - e^2*x^2])/d])/(10*d^7)
Time = 0.65 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.07, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {532, 25, 2336, 27, 2336, 27, 2338, 25, 27, 534, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int -\frac {\frac {8 e^3 x^3}{d}+10 e^2 x^2+10 d e x+5 d^2}{x^3 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {8 e^3 x^3}{d}+10 e^2 x^2+10 d e x+5 d^2}{x^3 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int -\frac {3 \left (\frac {12 e^3 x^3}{d}+15 e^2 x^2+10 d e x+5 d^2\right )}{x^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {12 e^3 x^3}{d}+15 e^2 x^2+10 d e x+5 d^2}{x^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {5 \left (d^2+2 e x d+4 e^2 x^2\right )}{x^3 \sqrt {d^2-e^2 x^2}}dx}{d^2}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {5 \int \frac {d^2+2 e x d+4 e^2 x^2}{x^3 \sqrt {d^2-e^2 x^2}}dx}{d^2}+\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {\frac {5 \left (-\frac {\int -\frac {d^2 e (4 d+9 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {5 \left (\frac {\int \frac {d^2 e (4 d+9 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {5 \left (\frac {1}{2} e \int \frac {4 d+9 e x}{x^2 \sqrt {d^2-e^2 x^2}}dx-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {\frac {5 \left (\frac {1}{2} e \left (9 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\frac {4 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {\frac {5 \left (\frac {1}{2} e \left (\frac {9}{2} e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {4 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {5 \left (\frac {1}{2} e \left (-\frac {9 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-\frac {4 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {5 \left (\frac {1}{2} e \left (-\frac {9 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {4 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {\sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {2 e^2 (10 d+11 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}+\frac {e^2 (5 d+6 e x)}{d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {2 e^2 (d+e x)}{5 d^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
(2*e^2*(d + e*x))/(5*d^3*(d^2 - e^2*x^2)^(5/2)) + ((e^2*(5*d + 6*e*x))/(d^ 3*(d^2 - e^2*x^2)^(3/2)) + ((2*e^2*(10*d + 11*e*x))/(d^3*Sqrt[d^2 - e^2*x^ 2]) + (5*(-1/2*Sqrt[d^2 - e^2*x^2]/x^2 + (e*((-4*Sqrt[d^2 - e^2*x^2])/(d*x ) - (9*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d))/2))/d^2)/d^2)/(5*d^2)
3.1.52.3.1 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[(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_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
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[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
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 = 0.40 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.41
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (4 e x +d \right )}{2 d^{7} x^{2}}-\frac {9 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{6} \sqrt {d^{2}}}+\frac {e \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{8 d^{7} \left (x +\frac {d}{e}\right )}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{10 d^{5} e \left (x -\frac {d}{e}\right )^{3}}+\frac {13 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{20 d^{6} \left (x -\frac {d}{e}\right )^{2}}-\frac {181 e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{40 d^{7} \left (x -\frac {d}{e}\right )}\) | \(257\) |
| default | \(e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )+d^{2} \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )}{2 d^{2}}\right )+2 d e \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )\) | \(361\) |
-1/2*(-e^2*x^2+d^2)^(1/2)*(4*e*x+d)/d^7/x^2-9/2/d^6*e^2/(d^2)^(1/2)*ln((2* d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)+1/8/d^7*e/(x+d/e)*(-(x+d/e)^2*e ^2+2*d*e*(x+d/e))^(1/2)-1/10/d^5/e/(x-d/e)^3*(-(x-d/e)^2*e^2-2*d*e*(x-d/e) )^(1/2)+13/20/d^6/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-181/40/d^ 7*e/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)
Time = 0.27 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {54 \, e^{6} x^{6} - 108 \, d e^{5} x^{5} + 108 \, d^{3} e^{3} x^{3} - 54 \, d^{4} e^{2} x^{2} + 45 \, {\left (e^{6} x^{6} - 2 \, d e^{5} x^{5} + 2 \, d^{3} e^{3} x^{3} - d^{4} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (64 \, e^{5} x^{5} - 83 \, d e^{4} x^{4} - 58 \, d^{2} e^{3} x^{3} + 94 \, d^{3} e^{2} x^{2} - 10 \, d^{4} e x - 5 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{10 \, {\left (d^{7} e^{4} x^{6} - 2 \, d^{8} e^{3} x^{5} + 2 \, d^{10} e x^{3} - d^{11} x^{2}\right )}} \]
1/10*(54*e^6*x^6 - 108*d*e^5*x^5 + 108*d^3*e^3*x^3 - 54*d^4*e^2*x^2 + 45*( e^6*x^6 - 2*d*e^5*x^5 + 2*d^3*e^3*x^3 - d^4*e^2*x^2)*log(-(d - sqrt(-e^2*x ^2 + d^2))/x) - (64*e^5*x^5 - 83*d*e^4*x^4 - 58*d^2*e^3*x^3 + 94*d^3*e^2*x ^2 - 10*d^4*e*x - 5*d^5)*sqrt(-e^2*x^2 + d^2))/(d^7*e^4*x^6 - 2*d^8*e^3*x^ 5 + 2*d^10*e*x^3 - d^11*x^2)
\[ \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{2}}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.20 \[ \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {12 \, e^{3} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}} + \frac {9 \, e^{2}}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}} + \frac {16 \, e^{3} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}} + \frac {3 \, e^{2}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} - \frac {2 \, e}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x} + \frac {32 \, e^{3} x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7}} - \frac {9 \, e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{7}} + \frac {9 \, e^{2}}{2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}} - \frac {1}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{2}} \]
12/5*e^3*x/((-e^2*x^2 + d^2)^(5/2)*d^3) + 9/10*e^2/((-e^2*x^2 + d^2)^(5/2) *d^2) + 16/5*e^3*x/((-e^2*x^2 + d^2)^(3/2)*d^5) + 3/2*e^2/((-e^2*x^2 + d^2 )^(3/2)*d^4) - 2*e/((-e^2*x^2 + d^2)^(5/2)*d*x) + 32/5*e^3*x/(sqrt(-e^2*x^ 2 + d^2)*d^7) - 9/2*e^2*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x) )/d^7 + 9/2*e^2/(sqrt(-e^2*x^2 + d^2)*d^6) - 1/2/((-e^2*x^2 + d^2)^(5/2)*x ^2)
\[ \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(d+e x)^2}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{x^3\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]