Integrand size = 24, antiderivative size = 173 \[ \int \frac {(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 (d+e x)^5}{5 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {42 (d+e x)^3}{5 e \sqrt {d^2-e^2 x^2}}+\frac {63 d \sqrt {d^2-e^2 x^2}}{2 e}+\frac {21 (d+e x) \sqrt {d^2-e^2 x^2}}{2 e}-\frac {63 d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e} \]
2/5*(e*x+d)^7/e/(-e^2*x^2+d^2)^(5/2)-6/5*(e*x+d)^5/e/(-e^2*x^2+d^2)^(3/2)- 63/2*d^2*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e+42/5*(e*x+d)^3/e/(-e^2*x^2+d^2 )^(1/2)+63/2*d*(-e^2*x^2+d^2)^(1/2)/e+21/2*(e*x+d)*(-e^2*x^2+d^2)^(1/2)/e
Time = 0.66 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (496 d^4-1163 d^3 e x+801 d^2 e^2 x^2-65 d e^3 x^3-5 e^4 x^4\right )}{(d-e x)^3}+630 d^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{10 e} \]
((Sqrt[d^2 - e^2*x^2]*(496*d^4 - 1163*d^3*e*x + 801*d^2*e^2*x^2 - 65*d*e^3 *x^3 - 5*e^4*x^4))/(d - e*x)^3 + 630*d^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^ 2 - e^2*x^2])])/(10*e)
Time = 0.35 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {468, 468, 462, 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.
| \(\displaystyle \int \frac {(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 468 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \int \frac {(d+e x)^6}{\left (d^2-e^2 x^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 468 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \int \frac {(d+e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}}dx\right )\) |
| \(\Big \downarrow \) 462 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}-\int \frac {7 d^2+4 e x d+e^2 x^2}{\sqrt {d^2-e^2 x^2}}dx\right )\right )\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (\frac {\int -\frac {d e^2 (15 d+8 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {\int \frac {d e^2 (15 d+8 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {1}{2} d \int \frac {15 d+8 e x}{\sqrt {d^2-e^2 x^2}}dx+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {1}{2} d \left (15 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {1}{2} d \left (15 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 (d+e x)^7}{5 e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {9}{5} \left (\frac {2 (d+e x)^5}{3 e \left (d^2-e^2 x^2\right )^{3/2}}-\frac {7}{3} \left (-\frac {1}{2} d \left (\frac {15 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {8 \sqrt {d^2-e^2 x^2}}{e}\right )+\frac {8 d^2 (d+e x)}{e \sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )\right )\) |
(2*(d + e*x)^7)/(5*e*(d^2 - e^2*x^2)^(5/2)) - (9*((2*(d + e*x)^5)/(3*e*(d^ 2 - e^2*x^2)^(3/2)) - (7*((8*d^2*(d + e*x))/(e*Sqrt[d^2 - e^2*x^2]) + (x*S qrt[d^2 - e^2*x^2])/2 - (d*((-8*Sqrt[d^2 - e^2*x^2])/e + (15*d*ArcTan[(e*x )/Sqrt[d^2 - e^2*x^2]])/e))/2))/3))/5
3.9.47.3.1 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[((c_) + (d_.)*(x_))^(n_)/((a_) + (b_.)*(x_)^2)^(3/2), x_Symbol] :> Simp [(-2^(n - 1))*d*c^(n - 2)*((c + d*x)/(b*Sqrt[a + b*x^2])), x] + Simp[d^2/b Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(n - 1)*c^(n - 1) - (c + d*x)^(n - 1))/(c - d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[n, 2]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((n + p)/(b*(p + 1))) Int[(c + d*x)^(n - 2)*(a + b*x^2)^(p + 1), x], x] /; Free Q[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[p, -1] && GtQ[n, 1] && I ntegerQ[2*p]
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 = 2.50 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.18
| method | result | size |
| risch | \(\frac {\left (e x +16 d \right ) \sqrt {-x^{2} e^{2}+d^{2}}}{2 e}-\frac {63 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {32 d^{4} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{4} \left (x -\frac {d}{e}\right )^{3}}-\frac {112 d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{3} \left (x -\frac {d}{e}\right )^{2}}-\frac {288 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 \left (x -\frac {d}{e}\right ) d e}}{5 e^{2} \left (x -\frac {d}{e}\right )}\) | \(204\) |
| default | \(d^{8} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )+e^{8} \left (-\frac {x^{7}}{2 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {7 d^{2} \left (\frac {x^{5}}{5 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-x^{2} e^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )}{2 e^{2}}\right )+8 d \,e^{7} \left (-\frac {x^{6}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )+\frac {8 d^{7}}{5 e \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+28 d^{2} e^{6} \left (\frac {x^{5}}{5 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-x^{2} e^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-x^{2} e^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )+56 d^{3} e^{5} \left (\frac {x^{4}}{e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+70 d^{4} e^{4} \left (\frac {x^{3}}{2 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+56 d^{5} e^{3} \left (\frac {x^{2}}{3 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+28 d^{6} e^{2} \left (\frac {x}{4 e^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-x^{2} e^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-x^{2} e^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )\) | \(841\) |
1/2*(e*x+16*d)/e*(-e^2*x^2+d^2)^(1/2)-63/2*d^2/(e^2)^(1/2)*arctan((e^2)^(1 /2)*x/(-e^2*x^2+d^2)^(1/2))-32/5*d^4/e^4/(x-d/e)^3*(-(x-d/e)^2*e^2-2*(x-d/ e)*d*e)^(1/2)-112/5*d^3/e^3/(x-d/e)^2*(-(x-d/e)^2*e^2-2*(x-d/e)*d*e)^(1/2) -288/5*d^2/e^2/(x-d/e)*(-(x-d/e)^2*e^2-2*(x-d/e)*d*e)^(1/2)
Time = 0.31 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {496 \, d^{2} e^{3} x^{3} - 1488 \, d^{3} e^{2} x^{2} + 1488 \, d^{4} e x - 496 \, d^{5} + 630 \, {\left (d^{2} e^{3} x^{3} - 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x - d^{5}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (5 \, e^{4} x^{4} + 65 \, d e^{3} x^{3} - 801 \, d^{2} e^{2} x^{2} + 1163 \, d^{3} e x - 496 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{10 \, {\left (e^{4} x^{3} - 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x - d^{3} e\right )}} \]
1/10*(496*d^2*e^3*x^3 - 1488*d^3*e^2*x^2 + 1488*d^4*e*x - 496*d^5 + 630*(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)) + (5*e^4*x^4 + 65*d*e^3*x^3 - 801*d^2*e^2*x^2 + 1163*d^3*e*x - 496*d^4)*sqrt(-e^2*x^2 + d^2))/(e^4*x^3 - 3*d*e^3*x^2 + 3*d^2*e^2*x - d ^3*e)
\[ \int \frac {(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {\left (d + e x\right )^{8}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (149) = 298\).
Time = 0.33 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.07 \[ \int \frac {(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {e^{6} x^{7}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {21}{10} \, d^{2} e^{6} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {8 \, d e^{5} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {21}{2} \, d^{2} e^{4} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {104 \, d^{3} e^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {35 \, d^{4} e^{2} x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {120 \, d^{5} e x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {76 \, d^{6} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {248 \, d^{7}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {69 \, d^{4} x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}}} - \frac {39 \, d^{2} x}{10 \, \sqrt {-e^{2} x^{2} + d^{2}}} - \frac {63 \, d^{2} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} \]
-1/2*e^6*x^7/(-e^2*x^2 + d^2)^(5/2) + 21/10*d^2*e^6*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)) - 8*d*e^5*x^6/(-e^2*x^2 + d^2)^(5/2) - 21/2*d^2*e^4 *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 )) + 104*d^3*e^3*x^4/(-e^2*x^2 + d^2)^(5/2) + 35*d^4*e^2*x^3/(-e^2*x^2 + d ^2)^(5/2) - 120*d^5*e*x^2/(-e^2*x^2 + d^2)^(5/2) - 76/5*d^6*x/(-e^2*x^2 + d^2)^(5/2) + 248/5*d^7/((-e^2*x^2 + d^2)^(5/2)*e) + 69/5*d^4*x/(-e^2*x^2 + d^2)^(3/2) - 39/10*d^2*x/sqrt(-e^2*x^2 + d^2) - 63/2*d^2*arcsin(e^2*x/(d* sqrt(e^2)))/sqrt(e^2)
Time = 0.33 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {63 \, d^{2} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} + \frac {1}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (x + \frac {16 \, d}{e}\right )} + \frac {32 \, {\left (13 \, d^{2} - \frac {55 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2}}{e^{2} x} + \frac {85 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2}}{e^{4} x^{2}} - \frac {45 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{2}}{e^{6} x^{3}} + \frac {10 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{2}}{e^{8} x^{4}}\right )}}{5 \, {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]
-63/2*d^2*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 1/2*sqrt(-e^2*x^2 + d^2)*(x + 16*d/e) + 32/5*(13*d^2 - 55*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2/(e^ 2*x) + 85*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^2/(e^4*x^2) - 45*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^2/(e^6*x^3) + 10*(d*e + sqrt(-e^2*x^2 + d ^2)*abs(e))^4*d^2/(e^8*x^4))/(((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*abs(e))
Timed out. \[ \int \frac {(d+e x)^8}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^8}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]