Integrand size = 27, antiderivative size = 200 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {13 d^6 x \sqrt {d^2-e^2 x^2}}{128 e^4}+\frac {8 d^3 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^3}-\frac {13 d^2 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{48 e^2}+\frac {2 d x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}+\frac {d^4 (1024 d-1365 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6720 e^5}+\frac {13 d^8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e^5} \]
8/35*d^3*x^2*(-e^2*x^2+d^2)^(3/2)/e^3-13/48*d^2*x^3*(-e^2*x^2+d^2)^(3/2)/e ^2+2/7*d*x^4*(-e^2*x^2+d^2)^(3/2)/e-1/8*x^5*(-e^2*x^2+d^2)^(3/2)+1/6720*d^ 4*(-1365*e*x+1024*d)*(-e^2*x^2+d^2)^(3/2)/e^5+13/128*d^8*arctan(e*x/(-e^2* x^2+d^2)^(1/2))/e^5+13/128*d^6*x*(-e^2*x^2+d^2)^(1/2)/e^4
Time = 0.40 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.68 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (2048 d^7-1365 d^6 e x+1024 d^5 e^2 x^2-910 d^4 e^3 x^3+768 d^3 e^4 x^4+1960 d^2 e^5 x^5-3840 d e^6 x^6+1680 e^7 x^7\right )-2730 d^8 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{13440 e^5} \]
(Sqrt[d^2 - e^2*x^2]*(2048*d^7 - 1365*d^6*e*x + 1024*d^5*e^2*x^2 - 910*d^4 *e^3*x^3 + 768*d^3*e^4*x^4 + 1960*d^2*e^5*x^5 - 3840*d*e^6*x^6 + 1680*e^7* x^7) - 2730*d^8*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(13440*e^ 5)
Time = 0.40 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.23, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {562, 541, 25, 27, 533, 25, 27, 533, 27, 533, 25, 27, 533, 25, 27, 455, 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.
| \(\displaystyle \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 562 |
| \(\displaystyle \int x^4 (d-e x)^2 \sqrt {d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle -\frac {\int -d e^2 x^4 (13 d-16 e x) \sqrt {d^2-e^2 x^2}dx}{8 e^2}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int d e^2 x^4 (13 d-16 e x) \sqrt {d^2-e^2 x^2}dx}{8 e^2}-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} d \int x^4 (13 d-16 e x) \sqrt {d^2-e^2 x^2}dx-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {1}{8} d \left (\frac {\int -d e x^3 (64 d-91 e x) \sqrt {d^2-e^2 x^2}dx}{7 e^2}+\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {\int d e x^3 (64 d-91 e x) \sqrt {d^2-e^2 x^2}dx}{7 e^2}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \int x^3 (64 d-91 e x) \sqrt {d^2-e^2 x^2}dx}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {\int -3 d e x^2 (91 d-128 e x) \sqrt {d^2-e^2 x^2}dx}{6 e^2}+\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \int x^2 (91 d-128 e x) \sqrt {d^2-e^2 x^2}dx}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {\int -d e x (256 d-455 e x) \sqrt {d^2-e^2 x^2}dx}{5 e^2}+\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {\int d e x (256 d-455 e x) \sqrt {d^2-e^2 x^2}dx}{5 e^2}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \int x (256 d-455 e x) \sqrt {d^2-e^2 x^2}dx}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {\int -d e (455 d-1024 e x) \sqrt {d^2-e^2 x^2}dx}{4 e^2}+\frac {455 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}\right )}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {455 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {\int d e (455 d-1024 e x) \sqrt {d^2-e^2 x^2}dx}{4 e^2}\right )}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {455 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \int (455 d-1024 e x) \sqrt {d^2-e^2 x^2}dx}{4 e}\right )}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {455 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (455 d \int \sqrt {d^2-e^2 x^2}dx+\frac {1024 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {455 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (455 d \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1024 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {455 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (455 d \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1024 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{8} d \left (\frac {16 x^4 \left (d^2-e^2 x^2\right )^{3/2}}{7 e}-\frac {d \left (\frac {91 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{6 e}-\frac {d \left (\frac {128 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {455 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (455 d \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {1024 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\right )}{2 e}\right )}{7 e}\right )-\frac {1}{8} x^5 \left (d^2-e^2 x^2\right )^{3/2}\) |
-1/8*(x^5*(d^2 - e^2*x^2)^(3/2)) + (d*((16*x^4*(d^2 - e^2*x^2)^(3/2))/(7*e ) - (d*((91*x^3*(d^2 - e^2*x^2)^(3/2))/(6*e) - (d*((128*x^2*(d^2 - e^2*x^2 )^(3/2))/(5*e) - (d*((455*x*(d^2 - e^2*x^2)^(3/2))/(4*e) - (d*((1024*(d^2 - e^2*x^2)^(3/2))/(3*e) + 455*d*((x*Sqrt[d^2 - e^2*x^2])/2 + (d^2*ArcTan[( e*x)/Sqrt[d^2 - e^2*x^2]])/(2*e))))/(4*e)))/(5*e)))/(2*e)))/(7*e)))/8
3.2.58.3.1 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[(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 = 130, normalized size of antiderivative = 0.65
| method | result | size |
| risch | \(\frac {\left (1680 e^{7} x^{7}-3840 d \,e^{6} x^{6}+1960 d^{2} e^{5} x^{5}+768 d^{3} e^{4} x^{4}-910 d^{4} e^{3} x^{3}+1024 d^{5} e^{2} x^{2}-1365 d^{6} e x +2048 d^{7}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{13440 e^{5}}+\frac {13 d^{8} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 e^{4} \sqrt {e^{2}}}\) | \(130\) |
| default | \(\frac {-\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}}}{e^{2}}+\frac {3 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 )}{e^{4}}+\frac {2 d \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{5}}+\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 {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \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 )}{3 d}\right )}{e^{6}}-\frac {4 d^{3} \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}}\) | \(696\) |
1/13440*(1680*e^7*x^7-3840*d*e^6*x^6+1960*d^2*e^5*x^5+768*d^3*e^4*x^4-910* d^4*e^3*x^3+1024*d^5*e^2*x^2-1365*d^6*e*x+2048*d^7)/e^5*(-e^2*x^2+d^2)^(1/ 2)+13/128*d^8/e^4/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))
Time = 0.36 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.64 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {2730 \, d^{8} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (1680 \, e^{7} x^{7} - 3840 \, d e^{6} x^{6} + 1960 \, d^{2} e^{5} x^{5} + 768 \, d^{3} e^{4} x^{4} - 910 \, d^{4} e^{3} x^{3} + 1024 \, d^{5} e^{2} x^{2} - 1365 \, d^{6} e x + 2048 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{13440 \, e^{5}} \]
-1/13440*(2730*d^8*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - (1680*e^7*x ^7 - 3840*d*e^6*x^6 + 1960*d^2*e^5*x^5 + 768*d^3*e^4*x^4 - 910*d^4*e^3*x^3 + 1024*d^5*e^2*x^2 - 1365*d^6*e*x + 2048*d^7)*sqrt(-e^2*x^2 + d^2))/e^5
Time = 2.03 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.70 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=d^{2} \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \frac {5 d^{8} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{128 e^{6}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {5 d^{6} x}{128 e^{6}} - \frac {5 d^{4} x^{3}}{192 e^{4}} - \frac {d^{2} x^{5}}{48 e^{2}} + \frac {x^{7}}{8}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{7} \sqrt {d^{2}}}{7} & \text {otherwise} \end {cases}\right ) \]
d**2*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)) - 2*d*e*Piecewise((s qrt(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)) + 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))/(12 8*e**6) + sqrt(d**2 - e**2*x**2)*(-5*d**6*x/(128*e**6) - 5*d**4*x**3/(192* e**4) - d**2*x**5/(48*e**2) + x**7/8), Ne(e**2, 0)), (x**7*sqrt(d**2)/7, T rue))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.38 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{4 \, {\left (e^{6} x + d e^{5}\right )}} + \frac {7 i \, d^{8} \arcsin \left (\frac {e x}{d} + 2\right )}{8 \, e^{5}} + \frac {125 \, d^{8} \arcsin \left (\frac {e x}{d}\right )}{128 \, e^{5}} - \frac {7 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6} x}{8 \, e^{4}} + \frac {125 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6} x}{128 \, e^{4}} - \frac {7 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{7}}{4 \, e^{5}} - \frac {67 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4} x}{192 \, e^{4}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{12 \, e^{5}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} x}{48 \, e^{4}} - \frac {4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{5 \, e^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} x}{8 \, e^{4}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{7 \, e^{5}} \]
1/4*(-e^2*x^2 + d^2)^(5/2)*d^4/(e^6*x + d*e^5) + 7/8*I*d^8*arcsin(e*x/d + 2)/e^5 + 125/128*d^8*arcsin(e*x/d)/e^5 - 7/8*sqrt(e^2*x^2 + 4*d*e*x + 3*d^ 2)*d^6*x/e^4 + 125/128*sqrt(-e^2*x^2 + d^2)*d^6*x/e^4 - 7/4*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^7/e^5 - 67/192*(-e^2*x^2 + d^2)^(3/2)*d^4*x/e^4 + 5/12 *(-e^2*x^2 + d^2)^(3/2)*d^5/e^5 + 25/48*(-e^2*x^2 + d^2)^(5/2)*d^2*x/e^4 - 4/5*(-e^2*x^2 + d^2)^(5/2)*d^3/e^5 - 1/8*(-e^2*x^2 + d^2)^(7/2)*x/e^4 + 2 /7*(-e^2*x^2 + d^2)^(7/2)*d/e^5
Time = 0.34 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.56 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {{\left (349440 \, d^{9} e^{9} \arctan \left (\sqrt {\frac {2 \, d}{e x + d} - 1}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + \frac {{\left (1365 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {15}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 61215 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {13}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 20517 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 141159 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 34969 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 34853 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 10465 \, d^{9} e^{9} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 1365 \, d^{9} e^{9} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )}^{8}}{d^{8}}\right )} {\left | e \right |}}{1720320 \, d e^{15}} \]
-1/1720320*(349440*d^9*e^9*arctan(sqrt(2*d/(e*x + d) - 1))*sgn(1/(e*x + d) )*sgn(e) + (1365*d^9*e^9*(2*d/(e*x + d) - 1)^(15/2)*sgn(1/(e*x + d))*sgn(e ) - 61215*d^9*e^9*(2*d/(e*x + d) - 1)^(13/2)*sgn(1/(e*x + d))*sgn(e) + 205 17*d^9*e^9*(2*d/(e*x + d) - 1)^(11/2)*sgn(1/(e*x + d))*sgn(e) - 141159*d^9 *e^9*(2*d/(e*x + d) - 1)^(9/2)*sgn(1/(e*x + d))*sgn(e) - 34969*d^9*e^9*(2* d/(e*x + d) - 1)^(7/2)*sgn(1/(e*x + d))*sgn(e) - 34853*d^9*e^9*(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))*sgn(e) - 10465*d^9*e^9*(2*d/(e*x + d) - 1) ^(3/2)*sgn(1/(e*x + d))*sgn(e) - 1365*d^9*e^9*sqrt(2*d/(e*x + d) - 1)*sgn( 1/(e*x + d))*sgn(e))*(e*x + d)^8/d^8)*abs(e)/(d*e^15)
Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]