Integrand size = 27, antiderivative size = 201 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 d^7 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {d^5 x \left (d^2-e^2 x^2\right )^{3/2}}{64 e^4}+\frac {4 d^2 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{63 e^3}-\frac {d x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}+\frac {d^3 (128 d-315 e x) \left (d^2-e^2 x^2\right )^{5/2}}{5040 e^5}+\frac {3 d^9 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]
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 = 0.42 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.73 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (1024 d^8-945 d^7 e x+512 d^6 e^2 x^2-630 d^5 e^3 x^3+384 d^4 e^4 x^4+7560 d^3 e^5 x^5-6400 d^2 e^6 x^6-5040 d e^7 x^7+4480 e^8 x^8\right )-1890 d^9 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{40320 e^5} \]
(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])])/(40320*e^5)
Time = 0.39 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.22, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {562, 533, 25, 27, 533, 25, 27, 533, 25, 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.
| \(\displaystyle \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 562 |
| \(\displaystyle \int x^4 (d-e x) \left (d^2-e^2 x^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {\int -d e x^3 (4 d-9 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{9 e^2}+\frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {\int d e x^3 (4 d-9 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{9 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \int x^3 (4 d-9 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{9 e}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {\int -d e x^2 (27 d-32 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{8 e^2}+\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {\int d e x^2 (27 d-32 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{8 e^2}\right )}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \int x^2 (27 d-32 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {\int -d e x (64 d-189 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{7 e^2}+\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {\int d e x (64 d-189 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{7 e^2}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {d \int x (64 d-189 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{7 e}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {d \left (\frac {\int -3 d e (63 d-128 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{6 e^2}+\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}\right )}{7 e}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {d \left (\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}-\frac {d \int (63 d-128 e x) \left (d^2-e^2 x^2\right )^{3/2}dx}{2 e}\right )}{7 e}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {d \left (\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}-\frac {d \left (63 d \int \left (d^2-e^2 x^2\right )^{3/2}dx+\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}\right )}{7 e}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {d \left (\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}-\frac {d \left (63 d \left (\frac {3}{4} d^2 \int \sqrt {d^2-e^2 x^2}dx+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}\right )}{7 e}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {d \left (\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}-\frac {d \left (63 d \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}\right )}{7 e}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {d \left (\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}-\frac {d \left (63 d \left (\frac {3}{4} d^2 \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}\right )}{7 e}\right )}{8 e}\right )}{9 e}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{9 e}-\frac {d \left (\frac {9 x^3 \left (d^2-e^2 x^2\right )^{5/2}}{8 e}-\frac {d \left (\frac {32 x^2 \left (d^2-e^2 x^2\right )^{5/2}}{7 e}-\frac {d \left (\frac {63 x \left (d^2-e^2 x^2\right )^{5/2}}{2 e}-\frac {d \left (63 d \left (\frac {3}{4} d^2 \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1}{4} x \left (d^2-e^2 x^2\right )^{3/2}\right )+\frac {128 \left (d^2-e^2 x^2\right )^{5/2}}{5 e}\right )}{2 e}\right )}{7 e}\right )}{8 e}\right )}{9 e}\) |
(x^4*(d^2 - e^2*x^2)^(5/2))/(9*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)
3.2.3.3.1 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[c^(2*n)/a^n Int[x^m*((a + b*x^2)^(n + p)/(c - d*x)^n), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[m, 0] && ILtQ[n, 0] && IGtQ[n + p + 1/2, 0]
Time = 0.40 (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 | \(\frac {-\frac {x^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 e^{2}}-\frac {2 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 e^{4}}}{e}-\frac {d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{5}}-\frac {d^{3} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{e^{4}}-\frac {d \left (-\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 e^{2}}+\frac {d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 d^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{6}\right )}{8 e^{2}}\right )}{e^{2}}+\frac {d^{4} \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{e^{5}}\) | \(502\) |
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+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)/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 = 0.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.69 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {1890 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (4480 \, e^{8} x^{8} - 5040 \, d e^{7} x^{7} - 6400 \, d^{2} e^{6} x^{6} + 7560 \, d^{3} e^{5} x^{5} + 384 \, d^{4} e^{4} x^{4} - 630 \, d^{5} e^{3} x^{3} + 512 \, d^{6} e^{2} x^{2} - 945 \, d^{7} e x + 1024 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{40320 \, e^{5}} \]
-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 = 1.47 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.12 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=d^{3} \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) - d^{2} e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {5 d^{8} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{6}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {5 d^{6} x}{128 e^{6}} - \frac {5 d^{4} x^{3}}{192 e^{4}} - \frac {d^{2} x^{5}}{48 e^{2}} + \frac {x^{7}}{8}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{7} \sqrt {d^{2}}}{7} & \text {otherwise} \end {cases}\right ) + e^{3} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {16 d^{8}}{315 e^{8}} - \frac {8 d^{6} x^{2}}{315 e^{6}} - \frac {2 d^{4} x^{4}}{105 e^{4}} - \frac {d^{2} x^{6}}{63 e^{2}} + \frac {x^{8}}{9}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{8} \sqrt {d^{2}}}{8} & \text {otherwise} \end {cases}\right ) \]
d**3*Piecewise((d**6*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))/ (16*e**4) + sqrt(d**2 - e**2*x**2)*(-d**4*x/(16*e**4) - d**2*x**3/(24*e**2 ) + x**5/6), Ne(e**2, 0)), (x**5*sqrt(d**2)/5, True)) - d**2*e*Piecewise(( sqrt(d**2 - e**2*x**2)*(-8*d**6/(105*e**6) - 4*d**4*x**2/(105*e**4) - d**2 *x**4/(35*e**2) + x**6/7), Ne(e**2, 0)), (x**6*sqrt(d**2)/6, True)) - d*e* *2*Piecewise((5*d**8*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))/ (128*e**6) + sqrt(d**2 - e**2*x**2)*(-5*d**6*x/(128*e**6) - 5*d**4*x**3/(1 92*e**4) - d**2*x**5/(48*e**2) + x**7/8), Ne(e**2, 0)), (x**7*sqrt(d**2)/7 , True)) + e**3*Piecewise((sqrt(d**2 - e**2*x**2)*(-16*d**8/(315*e**8) - 8 *d**6*x**2/(315*e**6) - 2*d**4*x**4/(105*e**4) - d**2*x**6/(63*e**2) + x** 8/9), Ne(e**2, 0)), (x**8*sqrt(d**2)/8, True))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.22 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=-\frac {3 i \, d^{9} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e^{5}} - \frac {45 \, d^{9} \arcsin \left (\frac {e x}{d}\right )}{128 \, e^{5}} + \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7} x}{8 \, e^{4}} - \frac {45 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x}{128 \, e^{4}} + \frac {3 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{8}}{4 \, e^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x}{64 \, e^{4}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x}{16 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x^{2}}{9 \, e^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{5 \, e^{5}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x}{8 \, e^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e^{5}} \]
-3/8*I*d^9*arcsin(e*x/d + 2)/e^5 - 45/128*d^9*arcsin(e*x/d)/e^5 + 3/8*sqrt (e^2*x^2 + 4*d*e*x + 3*d^2)*d^7*x/e^4 - 45/128*sqrt(-e^2*x^2 + d^2)*d^7*x/ e^4 + 3/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^8/e^5 + 1/64*(-e^2*x^2 + d^2)^ (3/2)*d^5*x/e^4 - 3/16*(-e^2*x^2 + d^2)^(5/2)*d^3*x/e^4 - 1/9*(-e^2*x^2 + d^2)^(7/2)*x^2/e^3 + 1/5*(-e^2*x^2 + d^2)^(5/2)*d^4/e^5 + 1/8*(-e^2*x^2 + d^2)^(7/2)*d*x/e^4 - 11/63*(-e^2*x^2 + d^2)^(7/2)*d^2/e^5
Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.66 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\frac {3 \, d^{9} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{128 \, e^{4} {\left | e \right |}} + \frac {1}{40320} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left (\frac {1024 \, d^{8}}{e^{5}} - {\left (\frac {945 \, d^{7}}{e^{4}} - 2 \, {\left (\frac {256 \, d^{6}}{e^{3}} - {\left (\frac {315 \, d^{5}}{e^{2}} - 4 \, {\left (\frac {48 \, d^{4}}{e} + 5 \, {\left (189 \, d^{3} - 2 \, {\left (80 \, d^{2} e - 7 \, {\left (8 \, e^{3} x - 9 \, d e^{2}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{d+e\,x} \,d x \]