Integrand size = 25, antiderivative size = 201 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}-\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d^3 (128 d+315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]
1/64*d^5*x*(-e^2*x^2+d^2)^(3/2)/e^4-4/63*d^2*x^2*(-e^2*x^2+d^2)^(5/2)/e^3- 1/8*d*x^3*(-e^2*x^2+d^2)^(5/2)/e^2-1/9*x^4*(-e^2*x^2+d^2)^(5/2)/e-1/5040*d ^3*(315*e*x+128*d)*(-e^2*x^2+d^2)^(5/2)/e^5+3/128*d^9*arctan(e*x/(-e^2*x^2 +d^2)^(1/2))/e^5+3/128*d^7*x*(-e^2*x^2+d^2)^(1/2)/e^4
Time = 0.43 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (1024 d^8+945 d^7 e x+512 d^6 e^2 x^2+630 d^5 e^3 x^3+384 d^4 e^4 x^4-7560 d^3 e^5 x^5-6400 d^2 e^6 x^6+5040 d e^7 x^7+4480 e^8 x^8\right )+1890 d^9 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{40320 e^5} \]
-1/40320*(Sqrt[d^2 - e^2*x^2]*(1024*d^8 + 945*d^7*e*x + 512*d^6*e^2*x^2 + 630*d^5*e^3*x^3 + 384*d^4*e^4*x^4 - 7560*d^3*e^5*x^5 - 6400*d^2*e^6*x^6 + 5040*d*e^7*x^7 + 4480*e^8*x^8) + 1890*d^9*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d ^2 - e^2*x^2])])/e^5
Time = 0.36 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.22, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {533, 27, 533, 27, 533, 27, 533, 27, 455, 211, 211, 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.
| \(\displaystyle \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {\int d e x^3 (4 d+9 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{9 e^2}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \int x^3 (4 d+9 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {d \left (\frac {\int d e x^2 (27 d+32 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{8 e^2}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \int x^2 (27 d+32 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {\int d e x (64 d+189 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{7 e^2}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \int x (64 d+189 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{7 e}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (\frac {\int 3 d e (63 d+128 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{6 e^2}-\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (\frac {d \int (63 d+128 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{2 e}-\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (\frac {d \left (63 d \int \left (d^2-e^2 x^2\right )^{3/2}dx-\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (\frac {d \left (63 d \left (\frac {3}{4} d^2 \int \sqrt {d^2-e^2 x^2}dx+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (\frac {d \left (63 d \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (\frac {d \left (63 d \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d \left (\frac {d \left (\frac {d \left (\frac {d \left (63 d \left (\frac {3}{4} d^2 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )-\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}-\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}-\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}-\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}-\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
-1/9*(x^4*(d^2 - e^2*x^2)^(5/2))/e + (d*((-9*x^3*(d^2 - e^2*x^2)^(5/2))/(8 *e) + (d*((-32*x^2*(d^2 - e^2*x^2)^(5/2))/(7*e) + (d*((-63*x*(d^2 - e^2*x^ 2)^(5/2))/(2*e) + (d*((-128*(d^2 - e^2*x^2)^(5/2))/(5*e) + 63*d*((x*(d^2 - e^2*x^2)^(3/2))/4 + (3*d^2*((x*Sqrt[d^2 - e^2*x^2])/2 + (d^2*ArcTan[(e*x) /Sqrt[d^2 - e^2*x^2]])/(2*e)))/4)))/(2*e)))/(7*e)))/(8*e)))/(9*e)
3.1.2.3.1 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)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* p + 2)) Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer Q[2*p]
Time = 0.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {\left (4480 e^{8} x^{8}+5040 d \,e^{7} x^{7}-6400 d^{2} e^{6} x^{6}-7560 d^{3} e^{5} x^{5}+384 d^{4} x^{4} e^{4}+630 d^{5} e^{3} x^{3}+512 d^{6} e^{2} x^{2}+945 d^{7} e x +1024 d^{8}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{40320 e^{5}}+\frac {3 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{4} \sqrt {e^{2}}}\) | \(141\) |
| default | \(e \left (-\frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{9 e^{2}}+\frac {4 d^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{7 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{35 e^{4}}\right )}{9 e^{2}}\right )+d \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6 e^{2}}\right )}{8 e^{2}}\right )\) | \(215\) |
-1/40320*(4480*e^8*x^8+5040*d*e^7*x^7-6400*d^2*e^6*x^6-7560*d^3*e^5*x^5+38 4*d^4*e^4*x^4+630*d^5*e^3*x^3+512*d^6*e^2*x^2+945*d^7*e*x+1024*d^8)/e^5*(- e^2*x^2+d^2)^(1/2)+3/128*d^9/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^ 2+d^2)^(1/2))
Time = 0.34 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.69 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {1890 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (4480 \, e^{8} x^{8} + 5040 \, d e^{7} x^{7} - 6400 \, d^{2} e^{6} x^{6} - 7560 \, d^{3} e^{5} x^{5} + 384 \, d^{4} e^{4} x^{4} + 630 \, d^{5} e^{3} x^{3} + 512 \, d^{6} e^{2} x^{2} + 945 \, d^{7} e x + 1024 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{40320 \, e^{5}} \]
-1/40320*(1890*d^9*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (4480*e^8*x ^8 + 5040*d*e^7*x^7 - 6400*d^2*e^6*x^6 - 7560*d^3*e^5*x^5 + 384*d^4*e^4*x^ 4 + 630*d^5*e^3*x^3 + 512*d^6*e^2*x^2 + 945*d^7*e*x + 1024*d^8)*sqrt(-e^2* x^2 + d^2))/e^5
Time = 0.57 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {3 d^{9} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{8}}{315 e^{5}} - \frac {3 d^{7} x}{128 e^{4}} - \frac {4 d^{6} x^{2}}{315 e^{3}} - \frac {d^{5} x^{3}}{64 e^{2}} - \frac {d^{4} x^{4}}{105 e} + \frac {3 d^{3} x^{5}}{16} + \frac {10 d^{2} e x^{6}}{63} - \frac {d e^{2} x^{7}}{8} - \frac {e^{3} x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d x^{5}}{5} + \frac {e x^{6}}{6}\right ) \left (d^{2}\right )^{\frac {3}{2}} & \text {otherwise} \end {cases} \]
Piecewise((3*d**9*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e** 2*x**2))/sqrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/(12 8*e**4) + sqrt(d**2 - e**2*x**2)*(-8*d**8/(315*e**5) - 3*d**7*x/(128*e**4) - 4*d**6*x**2/(315*e**3) - d**5*x**3/(64*e**2) - d**4*x**4/(105*e) + 3*d* *3*x**5/16 + 10*d**2*e*x**6/63 - d*e**2*x**7/8 - e**3*x**8/9), Ne(e**2, 0) ), ((d*x**5/5 + e*x**6/6)*(d**2)**(3/2), True))
Time = 0.28 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.94 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} x^{4}}{9 \, e} + \frac {3 \, d^{9} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{128 \, \sqrt {e^{2}} e^{4}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x}{128 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x^{3}}{8 \, e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x}{64 \, e^{4}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x^{2}}{63 \, e^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x}{16 \, e^{4}} - \frac {8 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{315 \, e^{5}} \]
-1/9*(-e^2*x^2 + d^2)^(5/2)*x^4/e + 3/128*d^9*arcsin(e^2*x/(d*sqrt(e^2)))/ (sqrt(e^2)*e^4) + 3/128*sqrt(-e^2*x^2 + d^2)*d^7*x/e^4 - 1/8*(-e^2*x^2 + d ^2)^(5/2)*d*x^3/e^2 + 1/64*(-e^2*x^2 + d^2)^(3/2)*d^5*x/e^4 - 4/63*(-e^2*x ^2 + d^2)^(5/2)*d^2*x^2/e^3 - 1/16*(-e^2*x^2 + d^2)^(5/2)*d^3*x/e^4 - 8/31 5*(-e^2*x^2 + d^2)^(5/2)*d^4/e^5
Time = 0.31 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65 \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\frac {3 \, d^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, e^{4} {\left | e \right |}} - \frac {1}{40320} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {1024 \, d^{8}}{e^{5}} + {\left (\frac {945 \, d^{7}}{e^{4}} + 2 \, {\left (\frac {256 \, d^{6}}{e^{3}} + {\left (\frac {315 \, d^{5}}{e^{2}} + 4 \, {\left (\frac {48 \, d^{4}}{e} - 5 \, {\left (189 \, d^{3} + 2 \, {\left (80 \, d^{2} e - 7 \, {\left (8 \, e^{3} x + 9 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
3/128*d^9*arcsin(e*x/d)*sgn(d)*sgn(e)/(e^4*abs(e)) - 1/40320*sqrt(-e^2*x^2 + d^2)*(1024*d^8/e^5 + (945*d^7/e^4 + 2*(256*d^6/e^3 + (315*d^5/e^2 + 4*( 48*d^4/e - 5*(189*d^3 + 2*(80*d^2*e - 7*(8*e^3*x + 9*d*e^2)*x)*x)*x)*x)*x) *x)*x)
Timed out. \[ \int x^4 (d+e x) \left (d^2-e^2 x^2\right )^{3/2} \, dx=\int x^4\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right ) \,d x \]