Integrand size = 27, antiderivative size = 171 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {d^5 x \sqrt {d^2-e^2 x^2}}{8 e^3}-\frac {11 d^2 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{35 e^2}+\frac {d x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}-\frac {d^3 (88 d-105 e x) \left (d^2-e^2 x^2\right )^{3/2}}{420 e^4}-\frac {d^7 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{8 e^4} \]
-11/35*d^2*x^2*(-e^2*x^2+d^2)^(3/2)/e^2+1/3*d*x^3*(-e^2*x^2+d^2)^(3/2)/e-1 /7*x^4*(-e^2*x^2+d^2)^(3/2)-1/420*d^3*(-105*e*x+88*d)*(-e^2*x^2+d^2)^(3/2) /e^4-1/8*d^7*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^4-1/8*d^5*x*(-e^2*x^2+d^2) ^(1/2)/e^3
Time = 0.25 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {e \sqrt {d^2-e^2 x^2} \left (-176 d^6+105 d^5 e x-88 d^4 e^2 x^2+70 d^3 e^3 x^3+144 d^2 e^4 x^4-280 d e^5 x^5+120 e^6 x^6\right )-105 d^7 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{840 e^5} \]
(e*Sqrt[d^2 - e^2*x^2]*(-176*d^6 + 105*d^5*e*x - 88*d^4*e^2*x^2 + 70*d^3*e ^3*x^3 + 144*d^2*e^4*x^4 - 280*d*e^5*x^5 + 120*e^6*x^6) - 105*d^7*Sqrt[-e^ 2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/(840*e^5)
Time = 0.37 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.22, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {562, 541, 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.
| \(\displaystyle \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 562 |
| \(\displaystyle \int x^3 (d-e x)^2 \sqrt {d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 541 |
| \(\displaystyle -\frac {\int -d e^2 x^3 (11 d-14 e x) \sqrt {d^2-e^2 x^2}dx}{7 e^2}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int d e^2 x^3 (11 d-14 e x) \sqrt {d^2-e^2 x^2}dx}{7 e^2}-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} d \int x^3 (11 d-14 e x) \sqrt {d^2-e^2 x^2}dx-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {1}{7} d \left (\frac {\int -6 d e x^2 (7 d-11 e x) \sqrt {d^2-e^2 x^2}dx}{6 e^2}+\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \int x^2 (7 d-11 e x) \sqrt {d^2-e^2 x^2}dx}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {\int -d e x (22 d-35 e x) \sqrt {d^2-e^2 x^2}dx}{5 e^2}+\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {\int d e x (22 d-35 e x) \sqrt {d^2-e^2 x^2}dx}{5 e^2}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \int x (22 d-35 e x) \sqrt {d^2-e^2 x^2}dx}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 533 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {\int -d e (35 d-88 e x) \sqrt {d^2-e^2 x^2}dx}{4 e^2}+\frac {35 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}\right )}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {35 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {\int d e (35 d-88 e x) \sqrt {d^2-e^2 x^2}dx}{4 e^2}\right )}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {35 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \int (35 d-88 e x) \sqrt {d^2-e^2 x^2}dx}{4 e}\right )}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {35 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (35 d \int \sqrt {d^2-e^2 x^2}dx+\frac {88 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {35 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (35 d \left (\frac {1}{2} d^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {88 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {35 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (35 d \left (\frac {1}{2} d^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {88 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{7} d \left (\frac {7 x^3 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}-\frac {d \left (\frac {11 x^2 \left (d^2-e^2 x^2\right )^{3/2}}{5 e}-\frac {d \left (\frac {35 x \left (d^2-e^2 x^2\right )^{3/2}}{4 e}-\frac {d \left (35 d \left (\frac {d^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e}+\frac {1}{2} x \sqrt {d^2-e^2 x^2}\right )+\frac {88 \left (d^2-e^2 x^2\right )^{3/2}}{3 e}\right )}{4 e}\right )}{5 e}\right )}{e}\right )-\frac {1}{7} x^4 \left (d^2-e^2 x^2\right )^{3/2}\) |
-1/7*(x^4*(d^2 - e^2*x^2)^(3/2)) + (d*((7*x^3*(d^2 - e^2*x^2)^(3/2))/(3*e) - (d*((11*x^2*(d^2 - e^2*x^2)^(3/2))/(5*e) - (d*((35*x*(d^2 - e^2*x^2)^(3 /2))/(4*e) - (d*((88*(d^2 - e^2*x^2)^(3/2))/(3*e) + 35*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) ))/e))/7
3.2.59.3.1 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 = 0.39 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {\left (-120 e^{6} x^{6}+280 d \,e^{5} x^{5}-144 d^{2} e^{4} x^{4}-70 d^{3} x^{3} e^{3}+88 d^{4} e^{2} x^{2}-105 d^{5} e x +176 d^{6}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{840 e^{4}}-\frac {d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{8 e^{3} \sqrt {e^{2}}}\) | \(119\) |
| default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{4}}-\frac {2 d \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^{3}}-\frac {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 {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^{5}}+\frac {3 d^{2} \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^{4}}\) | \(565\) |
-1/840*(-120*e^6*x^6+280*d*e^5*x^5-144*d^2*e^4*x^4-70*d^3*e^3*x^3+88*d^4*e ^2*x^2-105*d^5*e*x+176*d^6)/e^4*(-e^2*x^2+d^2)^(1/2)-1/8*d^7/e^3/(e^2)^(1/ 2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))
Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.68 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {210 \, d^{7} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (120 \, e^{6} x^{6} - 280 \, d e^{5} x^{5} + 144 \, d^{2} e^{4} x^{4} + 70 \, d^{3} e^{3} x^{3} - 88 \, d^{4} e^{2} x^{2} + 105 \, d^{5} e x - 176 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{840 \, e^{4}} \]
1/840*(210*d^7*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (120*e^6*x^6 - 280*d*e^5*x^5 + 144*d^2*e^4*x^4 + 70*d^3*e^3*x^3 - 88*d^4*e^2*x^2 + 105*d^ 5*e*x - 176*d^6)*sqrt(-e^2*x^2 + d^2))/e^4
Time = 1.96 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.49 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=d^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {2 d^{4}}{15 e^{4}} - \frac {d^{2} x^{2}}{15 e^{2}} + \frac {x^{4}}{5}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{4} \sqrt {d^{2}}}{4} & \text {otherwise} \end {cases}\right ) - 2 d e \left (\begin {cases} \frac {d^{6} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{16 e^{4}} + \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {d^{4} x}{16 e^{4}} - \frac {d^{2} x^{3}}{24 e^{2}} + \frac {x^{5}}{6}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{5} \sqrt {d^{2}}}{5} & \text {otherwise} \end {cases}\right ) + e^{2} \left (\begin {cases} \sqrt {d^{2} - e^{2} x^{2}} \left (- \frac {8 d^{6}}{105 e^{6}} - \frac {4 d^{4} x^{2}}{105 e^{4}} - \frac {d^{2} x^{4}}{35 e^{2}} + \frac {x^{6}}{7}\right ) & \text {for}\: e^{2} \neq 0 \\\frac {x^{6} \sqrt {d^{2}}}{6} & \text {otherwise} \end {cases}\right ) \]
d**2*Piecewise((sqrt(d**2 - e**2*x**2)*(-2*d**4/(15*e**4) - d**2*x**2/(15* e**2) + x**4/5), Ne(e**2, 0)), (x**4*sqrt(d**2)/4, True)) - 2*d*e*Piecewis e((d**6*Piecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/s qrt(-e**2), Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/(16*e**4) + s qrt(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)) + e**2*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))
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{4 \, {\left (e^{5} x + d e^{4}\right )}} - \frac {i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{4}} - \frac {5 \, d^{7} \arcsin \left (\frac {e x}{d}\right )}{8 \, e^{4}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x}{2 \, e^{3}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} x}{8 \, e^{3}} + \frac {\sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{e^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x}{3 \, e^{3}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{12 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x}{3 \, e^{3}} + \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{5 \, e^{4}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e^{4}} \]
-1/4*(-e^2*x^2 + d^2)^(5/2)*d^3/(e^5*x + d*e^4) - 1/2*I*d^7*arcsin(e*x/d + 2)/e^4 - 5/8*d^7*arcsin(e*x/d)/e^4 + 1/2*sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)* d^5*x/e^3 - 5/8*sqrt(-e^2*x^2 + d^2)*d^5*x/e^3 + sqrt(e^2*x^2 + 4*d*e*x + 3*d^2)*d^6/e^4 + 1/3*(-e^2*x^2 + d^2)^(3/2)*d^3*x/e^3 - 5/12*(-e^2*x^2 + d ^2)^(3/2)*d^4/e^4 - 1/3*(-e^2*x^2 + d^2)^(5/2)*d*x/e^3 + 3/5*(-e^2*x^2 + d ^2)^(5/2)*d^2/e^4 - 1/7*(-e^2*x^2 + d^2)^(7/2)/e^4
Time = 0.33 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.64 \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\frac {{\left (13440 \, d^{8} e^{8} \arctan \left (\sqrt {\frac {2 \, d}{e x + d} - 1}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + \frac {{\left (105 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {13}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 3780 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {11}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) + 189 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 4992 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 1981 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 700 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - 105 \, d^{8} e^{8} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} {\left (e x + d\right )}^{7}}{d^{7}}\right )} {\left | e \right |}}{53760 \, d e^{13}} \]
1/53760*(13440*d^8*e^8*arctan(sqrt(2*d/(e*x + d) - 1))*sgn(1/(e*x + d))*sg n(e) + (105*d^8*e^8*(2*d/(e*x + d) - 1)^(13/2)*sgn(1/(e*x + d))*sgn(e) - 3 780*d^8*e^8*(2*d/(e*x + d) - 1)^(11/2)*sgn(1/(e*x + d))*sgn(e) + 189*d^8*e ^8*(2*d/(e*x + d) - 1)^(9/2)*sgn(1/(e*x + d))*sgn(e) - 4992*d^8*e^8*(2*d/( e*x + d) - 1)^(7/2)*sgn(1/(e*x + d))*sgn(e) - 1981*d^8*e^8*(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))*sgn(e) - 700*d^8*e^8*(2*d/(e*x + d) - 1)^(3/2) *sgn(1/(e*x + d))*sgn(e) - 105*d^8*e^8*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))*sgn(e))*(e*x + d)^7/d^7)*abs(e)/(d*e^13)
Timed out. \[ \int \frac {x^3 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx=\int \frac {x^3\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \]