Integrand size = 27, antiderivative size = 174 \[ \int \frac {x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {127 d^2 (d+e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {3 d \sqrt {d^2-e^2 x^2}}{e^6}+\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}-\frac {13 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \]
1/5*d^4*(e*x+d)^3/e^6/(-e^2*x^2+d^2)^(5/2)-23/15*d^3*(e*x+d)^2/e^6/(-e^2*x ^2+d^2)^(3/2)-13/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^6+127/15*d^2*(e* x+d)/e^6/(-e^2*x^2+d^2)^(1/2)+3*d*(-e^2*x^2+d^2)^(1/2)/e^6+1/2*x*(-e^2*x^2 +d^2)^(1/2)/e^5
Time = 0.55 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.64 \[ \int \frac {x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (304 d^4-717 d^3 e x+479 d^2 e^2 x^2-45 d e^3 x^3-15 e^4 x^4\right )}{(d-e x)^3}+390 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{30 e^6} \]
((Sqrt[d^2 - e^2*x^2]*(304*d^4 - 717*d^3*e*x + 479*d^2*e^2*x^2 - 45*d*e^3* x^3 - 15*e^4*x^4))/(d - e*x)^3 + 390*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^ 2 - e^2*x^2])])/(30*e^6)
Time = 0.68 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {529, 2166, 2166, 27, 2346, 25, 27, 455, 224, 216}
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.
| \(\displaystyle \int \frac {x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (\frac {3 d^5}{e^5}+\frac {5 x d^4}{e^4}+\frac {5 x^2 d^3}{e^3}+\frac {5 x^3 d^2}{e^2}+\frac {5 x^4 d}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {(d+e x) \left (\frac {37 d^5}{e^5}+\frac {45 x d^4}{e^4}+\frac {30 x^2 d^3}{e^3}+\frac {15 x^3 d^2}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {127 d^4 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 \left (\frac {6 d^5}{e^5}+\frac {3 x d^4}{e^4}+\frac {x^2 d^3}{e^3}\right )}{\sqrt {d^2-e^2 x^2}}dx}{d}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {127 d^4 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \int \frac {\frac {6 d^5}{e^5}+\frac {3 x d^4}{e^4}+\frac {x^2 d^3}{e^3}}{\sqrt {d^2-e^2 x^2}}dx}{d}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {127 d^4 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (-\frac {\int -\frac {d^4 (13 d+6 e x)}{e^3 \sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {127 d^4 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {\int \frac {d^4 (13 d+6 e x)}{e^3 \sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {127 d^4 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^4 \int \frac {13 d+6 e x}{\sqrt {d^2-e^2 x^2}}dx}{2 e^5}-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {127 d^4 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^4 \left (13 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {6 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^5}-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {127 d^4 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^4 \left (13 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {6 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^5}-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {23 d^4 (d+e x)^2}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {127 d^4 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^4 \left (\frac {13 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {6 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^5}-\frac {d^3 x \sqrt {d^2-e^2 x^2}}{2 e^5}\right )}{d}}{3 d}}{5 d}\) |
(d^4*(d + e*x)^3)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - ((23*d^4*(d + e*x)^2)/(3 *e^6*(d^2 - e^2*x^2)^(3/2)) - ((127*d^4*(d + e*x))/(e^6*Sqrt[d^2 - e^2*x^2 ]) - (15*(-1/2*(d^3*x*Sqrt[d^2 - e^2*x^2])/e^5 + (d^4*((-6*Sqrt[d^2 - e^2* x^2])/e + (13*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/(2*e^5)))/d)/(3*d)) /(5*d)
3.1.83.3.1 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_)^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[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[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 0.46 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(\frac {\left (e x +6 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{6}}-\frac {13 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{5} \sqrt {e^{2}}}-\frac {d^{4} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{5 e^{9} \left (x -\frac {d}{e}\right )^{3}}-\frac {23 d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{15 e^{8} \left (x -\frac {d}{e}\right )^{2}}-\frac {127 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{15 e^{7} \left (x -\frac {d}{e}\right )}\) | \(207\) |
| default | \(e^{3} \left (-\frac {x^{7}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 d^{2} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )}{2 e^{2}}\right )+d^{3} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+3 d \,e^{2} \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )+3 d^{2} e \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )\) | \(450\) |
1/2*(e*x+6*d)/e^6*(-e^2*x^2+d^2)^(1/2)-13/2*d^2/e^5/(e^2)^(1/2)*arctan((e^ 2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/5*d^4/e^9/(x-d/e)^3*(-(x-d/e)^2*e^2-2*d *e*(x-d/e))^(1/2)-23/15*d^3/e^8/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^( 1/2)-127/15*d^2/e^7/(x-d/e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)
Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10 \[ \int \frac {x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {304 \, d^{2} e^{3} x^{3} - 912 \, d^{3} e^{2} x^{2} + 912 \, d^{4} e x - 304 \, d^{5} + 390 \, {\left (d^{2} e^{3} x^{3} - 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x - d^{5}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{4} x^{4} + 45 \, d e^{3} x^{3} - 479 \, d^{2} e^{2} x^{2} + 717 \, d^{3} e x - 304 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{9} x^{3} - 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x - d^{3} e^{6}\right )}} \]
1/30*(304*d^2*e^3*x^3 - 912*d^3*e^2*x^2 + 912*d^4*e*x - 304*d^5 + 390*(d^2 *e^3*x^3 - 3*d^3*e^2*x^2 + 3*d^4*e*x - d^5)*arctan(-(d - sqrt(-e^2*x^2 + d ^2))/(e*x)) + (15*e^4*x^4 + 45*d*e^3*x^3 - 479*d^2*e^2*x^2 + 717*d^3*e*x - 304*d^4)*sqrt(-e^2*x^2 + d^2))/(e^9*x^3 - 3*d*e^8*x^2 + 3*d^2*e^7*x - d^3 *e^6)
\[ \int \frac {x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x^{5} \left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (152) = 304\).
Time = 0.28 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.82 \[ \int \frac {x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {e x^{7}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {13}{30} \, d^{2} e x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {3 \, d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {13 \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{6 \, e} + \frac {19 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {76 \, d^{5} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {152 \, d^{7}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} + \frac {26 \, d^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}} - \frac {91 \, d^{2} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{5}} - \frac {13 \, d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}} e^{5}} \]
-1/2*e*x^7/(-e^2*x^2 + d^2)^(5/2) + 13/30*d^2*e*x*(15*x^4/((-e^2*x^2 + d^2 )^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - 3*d*x^6/(-e^2*x^2 + d^2)^(5/2) - 13/6*d^2*x*(3*x^2/(( -e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^2 + d^2)^(3/2)*e^4))/e + 19*d^ 3*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 76/3*d^5*x^2/((-e^2*x^2 + d^2)^(5/2)* e^4) + 152/15*d^7/((-e^2*x^2 + d^2)^(5/2)*e^6) + 26/15*d^4*x/((-e^2*x^2 + d^2)^(3/2)*e^5) - 91/30*d^2*x/(sqrt(-e^2*x^2 + d^2)*e^5) - 13/2*d^2*arcsin (e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^5)
Time = 0.30 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.34 \[ \int \frac {x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {x}{e^{5}} + \frac {6 \, d}{e^{6}}\right )} - \frac {13 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, e^{5} {\left | e \right |}} + \frac {2 \, {\left (107 \, d^{2} - \frac {445 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2}}{e^{2} x} + \frac {665 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2}}{e^{4} x^{2}} - \frac {405 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2}}{e^{6} x^{3}} + \frac {90 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{2}}{e^{8} x^{4}}\right )}}{15 \, e^{5} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]
1/2*sqrt(-e^2*x^2 + d^2)*(x/e^5 + 6*d/e^6) - 13/2*d^2*arcsin(e*x/d)*sgn(d) *sgn(e)/(e^5*abs(e)) + 2/15*(107*d^2 - 445*(d*e + sqrt(-e^2*x^2 + d^2)*abs (e))*d^2/(e^2*x) + 665*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2/(e^4*x^2) - 405*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^2/(e^6*x^3) + 90*(d*e + sqr t(-e^2*x^2 + d^2)*abs(e))^4*d^2/(e^8*x^4))/(e^5*((d*e + sqrt(-e^2*x^2 + d^ 2)*abs(e))/(e^2*x) - 1)^5*abs(e))
Timed out. \[ \int \frac {x^5 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x^5\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]