Integrand size = 25, antiderivative size = 230 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^{10} \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2} \]
11/128*d^6*x*(-e^2*x^2+d^2)^(3/2)/e+11/160*d^4*x*(-e^2*x^2+d^2)^(5/2)/e-33 /560*d^3*(-e^2*x^2+d^2)^(7/2)/e^2-11/240*d^2*(e*x+d)*(-e^2*x^2+d^2)^(7/2)/ e^2-1/30*d*(e*x+d)^2*(-e^2*x^2+d^2)^(7/2)/e^2-1/10*(e*x+d)^3*(-e^2*x^2+d^2 )^(7/2)/e^2+33/256*d^10*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^2+33/256*d^8*x* (-e^2*x^2+d^2)^(1/2)/e
Time = 0.58 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.69 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-6400 d^9-3465 d^8 e x+10240 d^7 e^2 x^2+24570 d^6 e^3 x^3+7680 d^5 e^4 x^4-23352 d^4 e^5 x^5-20480 d^3 e^6 x^6+3024 d^2 e^7 x^7+8960 d e^8 x^8+2688 e^9 x^9\right )-6930 d^{10} \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{26880 e^2} \]
(Sqrt[d^2 - e^2*x^2]*(-6400*d^9 - 3465*d^8*e*x + 10240*d^7*e^2*x^2 + 24570 *d^6*e^3*x^3 + 7680*d^5*e^4*x^4 - 23352*d^4*e^5*x^5 - 20480*d^3*e^6*x^6 + 3024*d^2*e^7*x^7 + 8960*d*e^8*x^8 + 2688*e^9*x^9) - 6930*d^10*ArcTan[(e*x) /(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(26880*e^2)
Time = 0.43 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {541, 25, 2340, 27, 533, 27, 455, 211, 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.
| \(\displaystyle \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle -\frac {\int -x \left (d^2-e^2 x^2\right )^{5/2} \left (30 d x^2 e^4+33 d^2 x e^3+10 d^3 e^2\right )dx}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int x \left (d^2-e^2 x^2\right )^{5/2} \left (30 d x^2 e^4+33 d^2 x e^3+10 d^3 e^2\right )dx}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle \frac {-\frac {\int -3 d^2 e^4 x (50 d+99 e x) \left (d^2-e^2 x^2\right )^{5/2}dx}{9 e^2}-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \int x (50 d+99 e x) \left (d^2-e^2 x^2\right )^{5/2}dx-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {\int d e (99 d+400 e x) \left (d^2-e^2 x^2\right )^{5/2}dx}{8 e^2}-\frac {99 x \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {d \int (99 d+400 e x) \left (d^2-e^2 x^2\right )^{5/2}dx}{8 e}-\frac {99 x \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {d \left (99 d \int \left (d^2-e^2 x^2\right )^{5/2}dx-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )}{8 e}-\frac {99 x \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {d \left (99 d \left (\frac {5}{6} d^2 \int \left (d^2-e^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )}{8 e}-\frac {99 x \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {d \left (99 d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \int \sqrt {d^2-e^2 x^2}dx+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )}{8 e}-\frac {99 x \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {d \left (99 d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )}{8 e}-\frac {99 x \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {d \left (99 d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )}{8 e}-\frac {99 x \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{3} d^2 e^2 \left (\frac {d \left (99 d \left (\frac {5}{6} d^2 \left (\frac {3}{4} d^2 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (d^2-e^2 x^2\right )^{5/2}\right )-\frac {400 \left (d^2-e^2 x^2\right )^{7/2}}{7 e}\right )}{8 e}-\frac {99 x \left (d^2-e^2 x^2\right )^{7/2}}{8 e}\right )-\frac {10}{3} d e^2 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}-\frac {1}{10} e x^3 \left (d^2-e^2 x^2\right )^{7/2}\) |
-1/10*(e*x^3*(d^2 - e^2*x^2)^(7/2)) + ((-10*d*e^2*x^2*(d^2 - e^2*x^2)^(7/2 ))/3 + (d^2*e^2*((-99*x*(d^2 - e^2*x^2)^(7/2))/(8*e) + (d*((-400*(d^2 - e^ 2*x^2)^(7/2))/(7*e) + 99*d*((x*(d^2 - e^2*x^2)^(5/2))/6 + (5*d^2*((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))/6)))/(8*e)))/3)/(10*e^2)
3.1.69.3.1 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]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x ] + Simp[1/(b*(m + n + 2*p + 1)) Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) *x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
Time = 0.43 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(-\frac {\left (-2688 e^{9} x^{9}-8960 d \,e^{8} x^{8}-3024 d^{2} e^{7} x^{7}+20480 d^{3} e^{6} x^{6}+23352 d^{4} e^{5} x^{5}-7680 d^{5} e^{4} x^{4}-24570 d^{6} e^{3} x^{3}-10240 x^{2} d^{7} e^{2}+3465 x \,d^{8} e +6400 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{26880 e^{2}}+\frac {33 d^{10} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 e \sqrt {e^{2}}}\) | \(152\) |
| default | \(e^{3} \left (-\frac {x^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{10 e^{2}}+\frac {3 d^{2} \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{10 e^{2}}\right )-\frac {d^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{2}}+3 d \,e^{2} \left (-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}\right )+3 d^{2} e \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )\) | \(366\) |
-1/26880*(-2688*e^9*x^9-8960*d*e^8*x^8-3024*d^2*e^7*x^7+20480*d^3*e^6*x^6+ 23352*d^4*e^5*x^5-7680*d^5*e^4*x^4-24570*d^6*e^3*x^3-10240*d^7*e^2*x^2+346 5*d^8*e*x+6400*d^9)/e^2*(-e^2*x^2+d^2)^(1/2)+33/256*d^10/e/(e^2)^(1/2)*arc tan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))
Time = 0.37 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.65 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=-\frac {6930 \, d^{10} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (2688 \, e^{9} x^{9} + 8960 \, d e^{8} x^{8} + 3024 \, d^{2} e^{7} x^{7} - 20480 \, d^{3} e^{6} x^{6} - 23352 \, d^{4} e^{5} x^{5} + 7680 \, d^{5} e^{4} x^{4} + 24570 \, d^{6} e^{3} x^{3} + 10240 \, d^{7} e^{2} x^{2} - 3465 \, d^{8} e x - 6400 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{26880 \, e^{2}} \]
-1/26880*(6930*d^10*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (2688*e^9* x^9 + 8960*d*e^8*x^8 + 3024*d^2*e^7*x^7 - 20480*d^3*e^6*x^6 - 23352*d^4*e^ 5*x^5 + 7680*d^5*e^4*x^4 + 24570*d^6*e^3*x^3 + 10240*d^7*e^2*x^2 - 3465*d^ 8*e*x - 6400*d^9)*sqrt(-e^2*x^2 + d^2))/e^2
Time = 0.62 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.05 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\begin {cases} \frac {33 d^{10} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{256 e} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {5 d^{9}}{21 e^{2}} - \frac {33 d^{8} x}{256 e} + \frac {8 d^{7} x^{2}}{21} + \frac {117 d^{6} e x^{3}}{128} + \frac {2 d^{5} e^{2} x^{4}}{7} - \frac {139 d^{4} e^{3} x^{5}}{160} - \frac {16 d^{3} e^{4} x^{6}}{21} + \frac {9 d^{2} e^{5} x^{7}}{80} + \frac {d e^{6} x^{8}}{3} + \frac {e^{7} x^{9}}{10}\right ) & \text {for}\: e^{2} \neq 0 \\\left (\frac {d^{3} x^{2}}{2} + d^{2} e x^{3} + \frac {3 d e^{2} x^{4}}{4} + \frac {e^{3} x^{5}}{5}\right ) \left (d^{2}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]
Piecewise((33*d**10*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))/( 256*e) + sqrt(d**2 - e**2*x**2)*(-5*d**9/(21*e**2) - 33*d**8*x/(256*e) + 8 *d**7*x**2/21 + 117*d**6*e*x**3/128 + 2*d**5*e**2*x**4/7 - 139*d**4*e**3*x **5/160 - 16*d**3*e**4*x**6/21 + 9*d**2*e**5*x**7/80 + d*e**6*x**8/3 + e** 7*x**9/10), Ne(e**2, 0)), ((d**3*x**2/2 + d**2*e*x**3 + 3*d*e**2*x**4/4 + e**3*x**5/5)*(d**2)**(5/2), True))
Time = 0.29 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.79 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {33 \, d^{10} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{256 \, \sqrt {e^{2}} e} + \frac {33 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8} x}{256 \, e} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} x}{128 \, e} - \frac {1}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{3} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x}{160 \, e} - \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{2} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x}{80 \, e} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3}}{21 \, e^{2}} \]
33/256*d^10*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e) + 33/256*sqrt(-e^2*x ^2 + d^2)*d^8*x/e + 11/128*(-e^2*x^2 + d^2)^(3/2)*d^6*x/e - 1/10*(-e^2*x^2 + d^2)^(7/2)*e*x^3 + 11/160*(-e^2*x^2 + d^2)^(5/2)*d^4*x/e - 1/3*(-e^2*x^ 2 + d^2)^(7/2)*d*x^2 - 33/80*(-e^2*x^2 + d^2)^(7/2)*d^2*x/e - 5/21*(-e^2*x ^2 + d^2)^(7/2)*d^3/e^2
Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.62 \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\frac {33 \, d^{10} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{256 \, e {\left | e \right |}} - \frac {1}{26880} \, {\left (\frac {6400 \, d^{9}}{e^{2}} + {\left (\frac {3465 \, d^{8}}{e} - 2 \, {\left (5120 \, d^{7} + {\left (12285 \, d^{6} e + 4 \, {\left (960 \, d^{5} e^{2} - {\left (2919 \, d^{4} e^{3} + 2 \, {\left (1280 \, d^{3} e^{4} - 7 \, {\left (27 \, d^{2} e^{5} + 8 \, {\left (3 \, e^{7} x + 10 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-e^{2} x^{2} + d^{2}} \]
33/256*d^10*arcsin(e*x/d)*sgn(d)*sgn(e)/(e*abs(e)) - 1/26880*(6400*d^9/e^2 + (3465*d^8/e - 2*(5120*d^7 + (12285*d^6*e + 4*(960*d^5*e^2 - (2919*d^4*e ^3 + 2*(1280*d^3*e^4 - 7*(27*d^2*e^5 + 8*(3*e^7*x + 10*d*e^6)*x)*x)*x)*x)* x)*x)*x)*x)*sqrt(-e^2*x^2 + d^2)
Timed out. \[ \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx=\int x\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]