Integrand size = 27, antiderivative size = 182 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e^2 (25 d+31 e x)}{15 d^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {e^2 (90 d+107 e x)}{15 d^6 \sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{2 d^5 x^2}-\frac {3 e \sqrt {d^2-e^2 x^2}}{d^6 x}-\frac {13 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^6} \]
4/5*e^2*(e*x+d)/d^2/(-e^2*x^2+d^2)^(5/2)+1/15*e^2*(31*e*x+25*d)/d^4/(-e^2* x^2+d^2)^(3/2)-13/2*e^2*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^6+1/15*e^2*(107* e*x+90*d)/d^6/(-e^2*x^2+d^2)^(1/2)-1/2*(-e^2*x^2+d^2)^(1/2)/d^5/x^2-3*e*(- e^2*x^2+d^2)^(1/2)/d^6/x
Time = 0.44 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (15 d^4+45 d^3 e x-479 d^2 e^2 x^2+717 d e^3 x^3-304 e^4 x^4\right )}{x^2 (-d+e x)^3}-195 \sqrt {d^2} e^2 \log (x)+195 \sqrt {d^2} e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{30 d^7} \]
((d*Sqrt[d^2 - e^2*x^2]*(15*d^4 + 45*d^3*e*x - 479*d^2*e^2*x^2 + 717*d*e^3 *x^3 - 304*e^4*x^4))/(x^2*(-d + e*x)^3) - 195*Sqrt[d^2]*e^2*Log[x] + 195*S qrt[d^2]*e^2*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(30*d^7)
Time = 0.70 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.10, 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, 25, 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.
| \(\displaystyle \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int -\frac {5 d^3+15 e x d^2+20 e^2 x^2 d+16 e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {5 d^3+15 e x d^2+20 e^2 x^2 d+16 e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int -\frac {15 d^3+45 e x d^2+75 e^2 x^2 d+62 e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {15 d^3+45 e x d^2+75 e^2 x^2 d+62 e^3 x^3}{x^3 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {15 \left (d^3+3 e x d^2+6 e^2 x^2 d\right )}{x^3 \sqrt {d^2-e^2 x^2}}dx}{d^2}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {15 \int \frac {d^3+3 e x d^2+6 e^2 x^2 d}{x^3 \sqrt {d^2-e^2 x^2}}dx}{d^2}+\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {\frac {15 \left (-\frac {\int -\frac {d^3 e (6 d+13 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {d \sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {15 \left (\frac {\int \frac {d^3 e (6 d+13 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {d \sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {15 \left (\frac {1}{2} d e \int \frac {6 d+13 e x}{x^2 \sqrt {d^2-e^2 x^2}}dx-\frac {d \sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {\frac {15 \left (\frac {1}{2} d e \left (13 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\frac {6 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {d \sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {\frac {15 \left (\frac {1}{2} d e \left (\frac {13}{2} e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {6 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {d \sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {15 \left (\frac {1}{2} d e \left (-\frac {13 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-\frac {6 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {d \sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {15 \left (\frac {1}{2} d e \left (-\frac {13 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {6 \sqrt {d^2-e^2 x^2}}{d x}\right )-\frac {d \sqrt {d^2-e^2 x^2}}{2 x^2}\right )}{d^2}+\frac {e^2 (90 d+107 e x)}{d^2 \sqrt {d^2-e^2 x^2}}}{3 d^2}+\frac {e^2 (25 d+31 e x)}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {4 e^2 (d+e x)}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
(4*e^2*(d + e*x))/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + ((e^2*(25*d + 31*e*x))/( 3*d^2*(d^2 - e^2*x^2)^(3/2)) + ((e^2*(90*d + 107*e*x))/(d^2*Sqrt[d^2 - e^2 *x^2]) + (15*(-1/2*(d*Sqrt[d^2 - e^2*x^2])/x^2 + (d*e*((-6*Sqrt[d^2 - e^2* x^2])/(d*x) - (13*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d))/2))/d^2)/(3*d^2))/ (5*d^2)
3.1.91.3.1 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 = 0.41 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (6 e x +d \right )}{2 d^{6} x^{2}}-\frac {13 e^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{5} \sqrt {d^{2}}}-\frac {107 e \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{15 d^{6} \left (x -\frac {d}{e}\right )}+\frac {17 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{15 d^{5} \left (x -\frac {d}{e}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{5 d^{4} e \left (x -\frac {d}{e}\right )^{3}}\) | \(214\) |
| default | \(e^{3} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )+d^{3} \left (-\frac {1}{2 d^{2} x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )}{2 d^{2}}\right )+3 d \,e^{2} \left (\frac {1}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}}{d^{2}}\right )+3 d^{2} e \left (-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 e^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{d^{2}}\right )\) | \(436\) |
-1/2*(-e^2*x^2+d^2)^(1/2)*(6*e*x+d)/d^6/x^2-13/2/d^5*e^2/(d^2)^(1/2)*ln((2 *d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-107/15/d^6*e/(x-d/e)*(-(x-d/e) ^2*e^2-2*d*e*(x-d/e))^(1/2)+17/15/d^5/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d /e))^(1/2)-1/5/d^4/e/(x-d/e)^3*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)
Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.13 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {254 \, e^{5} x^{5} - 762 \, d e^{4} x^{4} + 762 \, d^{2} e^{3} x^{3} - 254 \, d^{3} e^{2} x^{2} + 195 \, {\left (e^{5} x^{5} - 3 \, d e^{4} x^{4} + 3 \, d^{2} e^{3} x^{3} - d^{3} e^{2} x^{2}\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (304 \, e^{4} x^{4} - 717 \, d e^{3} x^{3} + 479 \, d^{2} e^{2} x^{2} - 45 \, d^{3} e x - 15 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (d^{6} e^{3} x^{5} - 3 \, d^{7} e^{2} x^{4} + 3 \, d^{8} e x^{3} - d^{9} x^{2}\right )}} \]
1/30*(254*e^5*x^5 - 762*d*e^4*x^4 + 762*d^2*e^3*x^3 - 254*d^3*e^2*x^2 + 19 5*(e^5*x^5 - 3*d*e^4*x^4 + 3*d^2*e^3*x^3 - d^3*e^2*x^2)*log(-(d - sqrt(-e^ 2*x^2 + d^2))/x) - (304*e^4*x^4 - 717*d*e^3*x^3 + 479*d^2*e^2*x^2 - 45*d^3 *e*x - 15*d^4)*sqrt(-e^2*x^2 + d^2))/(d^6*e^3*x^5 - 3*d^7*e^2*x^4 + 3*d^8* e*x^3 - d^9*x^2)
\[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{3}}{x^{3} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Time = 0.19 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {19 \, e^{3} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}} + \frac {13 \, e^{2}}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d} + \frac {76 \, e^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}} + \frac {13 \, e^{2}}{6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3}} - \frac {3 \, e}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x} + \frac {152 \, e^{3} x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}} - \frac {13 \, e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{2 \, d^{6}} + \frac {13 \, e^{2}}{2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5}} - \frac {d}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{2}} \]
19/5*e^3*x/((-e^2*x^2 + d^2)^(5/2)*d^2) + 13/10*e^2/((-e^2*x^2 + d^2)^(5/2 )*d) + 76/15*e^3*x/((-e^2*x^2 + d^2)^(3/2)*d^4) + 13/6*e^2/((-e^2*x^2 + d^ 2)^(3/2)*d^3) - 3*e/((-e^2*x^2 + d^2)^(5/2)*x) + 152/15*e^3*x/(sqrt(-e^2*x ^2 + d^2)*d^6) - 13/2*e^2*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs( x))/d^6 + 13/2*e^2/(sqrt(-e^2*x^2 + d^2)*d^5) - 1/2*d/((-e^2*x^2 + d^2)^(5 /2)*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (160) = 320\).
Time = 0.31 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.09 \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {13 \, e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d^{6} {\left | e \right |}} - \frac {{\left (15 \, e^{3} + \frac {105 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e}{x} - \frac {2782 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e x^{2}} + \frac {9410 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{3} x^{3}} - \frac {13645 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{5} x^{4}} + \frac {9285 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{7} x^{5}} - \frac {2580 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{9} x^{6}}\right )} e^{4} x^{2}}{120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{6} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} - \frac {\frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{6} e {\left | e \right |}}{x} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{6} {\left | e \right |}}{e x^{2}}}{8 \, d^{12} e^{2}} \]
-13/2*e^3*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)) )/(d^6*abs(e)) - 1/120*(15*e^3 + 105*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e /x - 2782*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2/(e*x^2) + 9410*(d*e + sqrt (-e^2*x^2 + d^2)*abs(e))^3/(e^3*x^3) - 13645*(d*e + sqrt(-e^2*x^2 + d^2)*a bs(e))^4/(e^5*x^4) + 9285*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e^7*x^5) - 2580*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6/(e^9*x^6))*e^4*x^2/((d*e + sq rt(-e^2*x^2 + d^2)*abs(e))^2*d^6*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2 *x) - 1)^5*abs(e)) - 1/8*(12*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^6*e*abs (e)/x + (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^6*abs(e)/(e*x^2))/(d^12*e^ 2)
Timed out. \[ \int \frac {(d+e x)^3}{x^3 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^3}{x^3\,{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]