Integrand size = 25, antiderivative size = 161 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x^6 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^4 (6 d+7 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x^2 (24 d+35 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {(32 d+35 e x) \sqrt {d^2-e^2 x^2}}{10 e^8}-\frac {7 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^8} \]
1/5*x^6*(e*x+d)/e^2/(-e^2*x^2+d^2)^(5/2)-1/15*x^4*(7*e*x+6*d)/e^4/(-e^2*x^ 2+d^2)^(3/2)-7/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^8+1/15*x^2*(35*e*x +24*d)/e^6/(-e^2*x^2+d^2)^(1/2)+1/10*(35*e*x+32*d)*(-e^2*x^2+d^2)^(1/2)/e^ 8
Time = 0.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.87 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (96 d^6+9 d^5 e x-249 d^4 e^2 x^2+4 d^3 e^3 x^3+176 d^2 e^4 x^4-15 d e^5 x^5-15 e^6 x^6\right )}{(d-e x)^3 (d+e x)^2}+210 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{30 e^8} \]
((Sqrt[d^2 - e^2*x^2]*(96*d^6 + 9*d^5*e*x - 249*d^4*e^2*x^2 + 4*d^3*e^3*x^ 3 + 176*d^2*e^4*x^4 - 15*d*e^5*x^5 - 15*e^6*x^6))/((d - e*x)^3*(d + e*x)^2 ) + 210*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(30*e^8)
Time = 0.60 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.27, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {529, 2345, 2345, 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.
| \(\displaystyle \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {\frac {d^7}{e^7}+\frac {5 x d^6}{e^6}+\frac {5 x^2 d^5}{e^5}+\frac {5 x^3 d^4}{e^4}+\frac {5 x^4 d^3}{e^3}+\frac {5 x^5 d^2}{e^2}+\frac {5 x^6 d}{e}}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {\frac {13 d^7}{e^7}+\frac {30 x d^6}{e^6}+\frac {30 x^2 d^5}{e^5}+\frac {15 x^3 d^4}{e^4}+\frac {15 x^4 d^3}{e^3}}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^5 (45 d+58 e x)}{e^8 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 \left (\frac {3 d^7}{e^7}+\frac {x d^6}{e^6}+\frac {x^2 d^5}{e^5}\right )}{\sqrt {d^2-e^2 x^2}}dx}{d^2}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^5 (45 d+58 e x)}{e^8 \sqrt {d^2-e^2 x^2}}-\frac {15 \int \frac {\frac {3 d^7}{e^7}+\frac {x d^6}{e^6}+\frac {x^2 d^5}{e^5}}{\sqrt {d^2-e^2 x^2}}dx}{d^2}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^5 (45 d+58 e x)}{e^8 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (-\frac {\int -\frac {d^6 (7 d+2 e x)}{e^5 \sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{2 e^7}\right )}{d^2}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^5 (45 d+58 e x)}{e^8 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {\int \frac {d^6 (7 d+2 e x)}{e^5 \sqrt {d^2-e^2 x^2}}dx}{2 e^2}-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{2 e^7}\right )}{d^2}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^5 (45 d+58 e x)}{e^8 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^6 \int \frac {7 d+2 e x}{\sqrt {d^2-e^2 x^2}}dx}{2 e^7}-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{2 e^7}\right )}{d^2}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^5 (45 d+58 e x)}{e^8 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^6 \left (7 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {2 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^7}-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{2 e^7}\right )}{d^2}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^5 (45 d+58 e x)}{e^8 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^6 \left (7 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {2 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^7}-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{2 e^7}\right )}{d^2}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d^6 (d+e x)}{5 e^8 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^5 (15 d+16 e x)}{3 e^8 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^5 (45 d+58 e x)}{e^8 \sqrt {d^2-e^2 x^2}}-\frac {15 \left (\frac {d^6 \left (\frac {7 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {2 \sqrt {d^2-e^2 x^2}}{e}\right )}{2 e^7}-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{2 e^7}\right )}{d^2}}{3 d^2}}{5 d}\) |
(d^6*(d + e*x))/(5*e^8*(d^2 - e^2*x^2)^(5/2)) - ((d^5*(15*d + 16*e*x))/(3* e^8*(d^2 - e^2*x^2)^(3/2)) - ((d^5*(45*d + 58*e*x))/(e^8*Sqrt[d^2 - e^2*x^ 2]) - (15*(-1/2*(d^5*x*Sqrt[d^2 - e^2*x^2])/e^7 + (d^6*((-2*Sqrt[d^2 - e^2 *x^2])/e + (7*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/(2*e^7)))/d^2)/(3*d ^2))/(5*d)
3.1.19.3.1 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_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[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)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
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 = 0.44 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(e \left (-\frac {x^{7}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 d^{2} \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )}{2 e^{2}}\right )+d \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )\) | \(251\) |
| risch | \(\frac {\left (e x +2 d \right ) \sqrt {-e^{2} x^{2}+d^{2}}}{2 e^{8}}-\frac {7 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 e^{7} \sqrt {e^{2}}}-\frac {7 d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{15 e^{10} \left (x -\frac {d}{e}\right )^{2}}-\frac {773 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{240 e^{9} \left (x -\frac {d}{e}\right )}+\frac {d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{24 e^{10} \left (x +\frac {d}{e}\right )^{2}}-\frac {31 d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{48 e^{9} \left (x +\frac {d}{e}\right )}-\frac {d^{4} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{20 e^{11} \left (x -\frac {d}{e}\right )^{3}}\) | \(297\) |
e*(-1/2*x^7/e^2/(-e^2*x^2+d^2)^(5/2)+7/2*d^2/e^2*(1/5*x^5/e^2/(-e^2*x^2+d^ 2)^(5/2)-1/e^2*(1/3*x^3/e^2/(-e^2*x^2+d^2)^(3/2)-1/e^2*(x/e^2/(-e^2*x^2+d^ 2)^(1/2)-1/e^2/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))))))+ d*(-x^6/e^2/(-e^2*x^2+d^2)^(5/2)+6*d^2/e^2*(x^4/e^2/(-e^2*x^2+d^2)^(5/2)-4 *d^2/e^2*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(-e^2*x^2+d^2)^(5/ 2))))
Time = 0.30 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.73 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {96 \, d^{2} e^{5} x^{5} - 96 \, d^{3} e^{4} x^{4} - 192 \, d^{4} e^{3} x^{3} + 192 \, d^{5} e^{2} x^{2} + 96 \, d^{6} e x - 96 \, d^{7} + 210 \, {\left (d^{2} e^{5} x^{5} - d^{3} e^{4} x^{4} - 2 \, d^{4} e^{3} x^{3} + 2 \, d^{5} e^{2} x^{2} + d^{6} e x - d^{7}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{6} x^{6} + 15 \, d e^{5} x^{5} - 176 \, d^{2} e^{4} x^{4} - 4 \, d^{3} e^{3} x^{3} + 249 \, d^{4} e^{2} x^{2} - 9 \, d^{5} e x - 96 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{13} x^{5} - d e^{12} x^{4} - 2 \, d^{2} e^{11} x^{3} + 2 \, d^{3} e^{10} x^{2} + d^{4} e^{9} x - d^{5} e^{8}\right )}} \]
1/30*(96*d^2*e^5*x^5 - 96*d^3*e^4*x^4 - 192*d^4*e^3*x^3 + 192*d^5*e^2*x^2 + 96*d^6*e*x - 96*d^7 + 210*(d^2*e^5*x^5 - d^3*e^4*x^4 - 2*d^4*e^3*x^3 + 2 *d^5*e^2*x^2 + d^6*e*x - d^7)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (15*e^6*x^6 + 15*d*e^5*x^5 - 176*d^2*e^4*x^4 - 4*d^3*e^3*x^3 + 249*d^4*e^2 *x^2 - 9*d^5*e*x - 96*d^6)*sqrt(-e^2*x^2 + d^2))/(e^13*x^5 - d*e^12*x^4 - 2*d^2*e^11*x^3 + 2*d^3*e^10*x^2 + d^4*e^9*x - d^5*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (144) = 288\).
Time = 13.69 (sec) , antiderivative size = 2004, normalized size of antiderivative = 12.45 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
d*Piecewise((16*d**6/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x **2*sqrt(d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) - 40*d** 4*e**2*x**2/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt( d**2 - e**2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) + 30*d**2*e**4*x* *4/(5*d**4*e**8*sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e* *2*x**2) + 5*e**12*x**4*sqrt(d**2 - e**2*x**2)) - 5*e**6*x**6/(5*d**4*e**8 *sqrt(d**2 - e**2*x**2) - 10*d**2*e**10*x**2*sqrt(d**2 - e**2*x**2) + 5*e* *12*x**4*sqrt(d**2 - e**2*x**2)), Ne(e, 0)), (x**8/(8*(d**2)**(7/2)), True )) + e*Piecewise((210*I*d**7*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(60*d* *5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x** 2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) - 105*pi*d**7*sqrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**1 1*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d** 2)) - 210*I*d**6*e*x/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e* *11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d **2)) - 420*I*d**5*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(60*d* *5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120*d**3*e**11*x**2*sqrt(-1 + e**2*x** 2/d**2) + 60*d*e**13*x**4*sqrt(-1 + e**2*x**2/d**2)) + 210*pi*d**5*e**2*x* *2*sqrt(-1 + e**2*x**2/d**2)/(60*d**5*e**9*sqrt(-1 + e**2*x**2/d**2) - 120 *d**3*e**11*x**2*sqrt(-1 + e**2*x**2/d**2) + 60*d*e**13*x**4*sqrt(-1 + ...
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (141) = 282\).
Time = 0.28 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.01 \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {x^{7}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {7 \, d^{2} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )}}{30 \, e} - \frac {d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {7 \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{6 \, e^{3}} + \frac {6 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} - \frac {8 \, d^{5} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} + \frac {16 \, d^{7}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{8}} + \frac {14 \, d^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{7}} - \frac {49 \, d^{2} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{7}} - \frac {7 \, d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}} e^{7}} \]
-1/2*x^7/((-e^2*x^2 + d^2)^(5/2)*e) + 7/30*d^2*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))/e - d*x^6/((-e^2*x^2 + d^2)^(5/2)*e^2) - 7/6*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^3 + 6*d^3*x^4/((-e^2*x^2 + d^2)^(5/2)*e^4) - 8*d^5*x^2/((-e^2*x^2 + d^2)^(5/2 )*e^6) + 16/5*d^7/((-e^2*x^2 + d^2)^(5/2)*e^8) + 14/15*d^4*x/((-e^2*x^2 + d^2)^(3/2)*e^7) - 49/30*d^2*x/(sqrt(-e^2*x^2 + d^2)*e^7) - 7/2*d^2*arcsin( e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^7)
\[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} x^{7}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^7 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x^7\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]