Integrand size = 25, antiderivative size = 211 \[ \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {64 x}{2145 d^5 e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{143 d e^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^2 e^2 (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {32}{1287 d^3 e^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {256 x}{6435 d^7 e \left (d^2-e^2 x^2\right )^{3/2}}+\frac {512 x}{6435 d^9 e \sqrt {d^2-e^2 x^2}} \]
64/2145*x/d^5/e/(-e^2*x^2+d^2)^(5/2)+1/13/e^2/(e*x+d)^4/(-e^2*x^2+d^2)^(5/ 2)-4/143/d/e^2/(e*x+d)^3/(-e^2*x^2+d^2)^(5/2)-32/1287/d^2/e^2/(e*x+d)^2/(- e^2*x^2+d^2)^(5/2)-32/1287/d^3/e^2/(e*x+d)/(-e^2*x^2+d^2)^(5/2)+256/6435*x /d^7/e/(-e^2*x^2+d^2)^(3/2)+512/6435*x/d^9/e/(-e^2*x^2+d^2)^(1/2)
Time = 0.59 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.65 \[ \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-5 d^9-20 d^8 e x+3200 d^7 e^2 x^2+4320 d^6 e^3 x^3-1280 d^5 e^4 x^4-6208 d^4 e^5 x^5-3072 d^3 e^6 x^6+1792 d^2 e^7 x^7+2048 d e^8 x^8+512 e^9 x^9\right )}{6435 d^9 e^2 (d-e x)^3 (d+e x)^7} \]
(Sqrt[d^2 - e^2*x^2]*(-5*d^9 - 20*d^8*e*x + 3200*d^7*e^2*x^2 + 4320*d^6*e^ 3*x^3 - 1280*d^5*e^4*x^4 - 6208*d^4*e^5*x^5 - 3072*d^3*e^6*x^6 + 1792*d^2* e^7*x^7 + 2048*d*e^8*x^8 + 512*e^9*x^9))/(6435*d^9*e^2*(d - e*x)^3*(d + e* x)^7)
Time = 0.32 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.15, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {571, 461, 461, 470, 209, 209, 208}
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.
| \(\displaystyle \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 571 |
| \(\displaystyle \frac {4 \int \frac {1}{(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}dx}{13 e}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {4 \left (\frac {8 \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}dx}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 e}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 461 |
| \(\displaystyle \frac {4 \left (\frac {8 \left (\frac {7 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}}dx}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 e}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 470 |
| \(\displaystyle \frac {4 \left (\frac {8 \left (\frac {7 \left (\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}}dx}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 e}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {4 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 e}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {4 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}\right )}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 e}+\frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {1}{13 e^2 (d+e x)^4 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \left (\frac {8 \left (\frac {7 \left (\frac {6 \left (\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \left (\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^4 \sqrt {d^2-e^2 x^2}}\right )}{5 d^2}\right )}{7 d}-\frac {1}{7 d e (d+e x) \left (d^2-e^2 x^2\right )^{5/2}}\right )}{9 d}-\frac {1}{9 d e (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{13 e}\) |
1/(13*e^2*(d + e*x)^4*(d^2 - e^2*x^2)^(5/2)) + (4*(-1/11*1/(d*e*(d + e*x)^ 3*(d^2 - e^2*x^2)^(5/2)) + (8*(-1/9*1/(d*e*(d + e*x)^2*(d^2 - e^2*x^2)^(5/ 2)) + (7*(-1/7*1/(d*e*(d + e*x)*(d^2 - e^2*x^2)^(5/2)) + (6*(x/(5*d^2*(d^2 - e^2*x^2)^(5/2)) + (4*(x/(3*d^2*(d^2 - e^2*x^2)^(3/2)) + (2*x)/(3*d^4*Sq rt[d^2 - e^2*x^2])))/(5*d^2)))/(7*d)))/(9*d)))/(11*d)))/(13*e)
3.3.14.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl ify[n + 2*p + 2]/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[(n + 2*p + 2)/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[n, 0] && NeQ[n + p + 1, 0] && IntegerQ[2*p]
Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*(n + p + 1))), x] + Simp[n/(2*d* (n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ((LtQ[n, -1] && !IGtQ[n + p + 1, 0]) || (LtQ[n, 0] && LtQ[p, -1]) || EqQ[n + 2*p + 2, 0]) && NeQ[n + p + 1, 0]
Time = 0.40 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.63
| method | result | size |
| gosper | \(-\frac {\left (-e x +d \right ) \left (-512 e^{9} x^{9}-2048 d \,e^{8} x^{8}-1792 d^{2} e^{7} x^{7}+3072 d^{3} e^{6} x^{6}+6208 d^{4} e^{5} x^{5}+1280 d^{5} e^{4} x^{4}-4320 d^{6} e^{3} x^{3}-3200 x^{2} d^{7} e^{2}+20 x \,d^{8} e +5 d^{9}\right )}{6435 \left (e x +d \right )^{3} d^{9} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(132\) |
| trager | \(-\frac {\left (-512 e^{9} x^{9}-2048 d \,e^{8} x^{8}-1792 d^{2} e^{7} x^{7}+3072 d^{3} e^{6} x^{6}+6208 d^{4} e^{5} x^{5}+1280 d^{5} e^{4} x^{4}-4320 d^{6} e^{3} x^{3}-3200 x^{2} d^{7} e^{2}+20 x \,d^{8} e +5 d^{9}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{6435 d^{9} \left (e x +d \right )^{7} \left (-e x +d \right )^{3} e^{2}}\) | \(134\) |
| default | \(\frac {-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{3} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {8 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{10 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right )}{15 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right )}{15 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{e^{4}}-\frac {d \left (-\frac {1}{13 d e \left (x +\frac {d}{e}\right )^{4} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {9 e \left (-\frac {1}{11 d e \left (x +\frac {d}{e}\right )^{3} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {8 e \left (-\frac {1}{9 d e \left (x +\frac {d}{e}\right )^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {7 e \left (-\frac {1}{7 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{10 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right )}{15 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right )}{15 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}}{d^{2}}\right )}{7 d}\right )}{9 d}\right )}{11 d}\right )}{13 d}\right )}{e^{5}}\) | \(708\) |
-1/6435*(-e*x+d)*(-512*e^9*x^9-2048*d*e^8*x^8-1792*d^2*e^7*x^7+3072*d^3*e^ 6*x^6+6208*d^4*e^5*x^5+1280*d^5*e^4*x^4-4320*d^6*e^3*x^3-3200*d^7*e^2*x^2+ 20*d^8*e*x+5*d^9)/(e*x+d)^3/d^9/e^2/(-e^2*x^2+d^2)^(7/2)
Time = 0.68 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.50 \[ \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {5 \, e^{10} x^{10} + 20 \, d e^{9} x^{9} + 15 \, d^{2} e^{8} x^{8} - 40 \, d^{3} e^{7} x^{7} - 70 \, d^{4} e^{6} x^{6} + 70 \, d^{6} e^{4} x^{4} + 40 \, d^{7} e^{3} x^{3} - 15 \, d^{8} e^{2} x^{2} - 20 \, d^{9} e x - 5 \, d^{10} + {\left (512 \, e^{9} x^{9} + 2048 \, d e^{8} x^{8} + 1792 \, d^{2} e^{7} x^{7} - 3072 \, d^{3} e^{6} x^{6} - 6208 \, d^{4} e^{5} x^{5} - 1280 \, d^{5} e^{4} x^{4} + 4320 \, d^{6} e^{3} x^{3} + 3200 \, d^{7} e^{2} x^{2} - 20 \, d^{8} e x - 5 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6435 \, {\left (d^{9} e^{12} x^{10} + 4 \, d^{10} e^{11} x^{9} + 3 \, d^{11} e^{10} x^{8} - 8 \, d^{12} e^{9} x^{7} - 14 \, d^{13} e^{8} x^{6} + 14 \, d^{15} e^{6} x^{4} + 8 \, d^{16} e^{5} x^{3} - 3 \, d^{17} e^{4} x^{2} - 4 \, d^{18} e^{3} x - d^{19} e^{2}\right )}} \]
-1/6435*(5*e^10*x^10 + 20*d*e^9*x^9 + 15*d^2*e^8*x^8 - 40*d^3*e^7*x^7 - 70 *d^4*e^6*x^6 + 70*d^6*e^4*x^4 + 40*d^7*e^3*x^3 - 15*d^8*e^2*x^2 - 20*d^9*e *x - 5*d^10 + (512*e^9*x^9 + 2048*d*e^8*x^8 + 1792*d^2*e^7*x^7 - 3072*d^3* e^6*x^6 - 6208*d^4*e^5*x^5 - 1280*d^5*e^4*x^4 + 4320*d^6*e^3*x^3 + 3200*d^ 7*e^2*x^2 - 20*d^8*e*x - 5*d^9)*sqrt(-e^2*x^2 + d^2))/(d^9*e^12*x^10 + 4*d ^10*e^11*x^9 + 3*d^11*e^10*x^8 - 8*d^12*e^9*x^7 - 14*d^13*e^8*x^6 + 14*d^1 5*e^6*x^4 + 8*d^16*e^5*x^3 - 3*d^17*e^4*x^2 - 4*d^18*e^3*x - d^19*e^2)
\[ \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )^{4}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (183) = 366\).
Time = 0.22 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.92 \[ \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {1}{13 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6} x^{4} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{3} + 6 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 4 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {4}{143 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{5} x^{3} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {32}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2} e^{4} x^{2} + 2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} - \frac {32}{1287 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} e^{3} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} e^{2}\right )}} + \frac {64 \, x}{2145 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} e} + \frac {256 \, x}{6435 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7} e} + \frac {512 \, x}{6435 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{9} e} \]
1/13/((-e^2*x^2 + d^2)^(5/2)*e^6*x^4 + 4*(-e^2*x^2 + d^2)^(5/2)*d*e^5*x^3 + 6*(-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x^2 + 4*(-e^2*x^2 + d^2)^(5/2)*d^3*e^3* x + (-e^2*x^2 + d^2)^(5/2)*d^4*e^2) - 4/143/((-e^2*x^2 + d^2)^(5/2)*d*e^5* x^3 + 3*(-e^2*x^2 + d^2)^(5/2)*d^2*e^4*x^2 + 3*(-e^2*x^2 + d^2)^(5/2)*d^3* e^3*x + (-e^2*x^2 + d^2)^(5/2)*d^4*e^2) - 32/1287/((-e^2*x^2 + d^2)^(5/2)* d^2*e^4*x^2 + 2*(-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x + (-e^2*x^2 + d^2)^(5/2)* d^4*e^2) - 32/1287/((-e^2*x^2 + d^2)^(5/2)*d^3*e^3*x + (-e^2*x^2 + d^2)^(5 /2)*d^4*e^2) + 64/2145*x/((-e^2*x^2 + d^2)^(5/2)*d^5*e) + 256/6435*x/((-e^ 2*x^2 + d^2)^(3/2)*d^7*e) + 512/6435*x/(sqrt(-e^2*x^2 + d^2)*d^9*e)
\[ \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {x}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}^{4}} \,d x } \]
Time = 11.92 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.19 \[ \int \frac {x}{(d+e x)^4 \left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {41}{41184\,d^6\,e^2}+\frac {256\,x}{6435\,d^7\,e}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {47}{1716\,d^4\,e^2}-\frac {1369\,x}{34320\,d^5\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}}{104\,d^3\,e^2\,{\left (d+e\,x\right )}^7}+\frac {25\,\sqrt {d^2-e^2\,x^2}}{2288\,d^4\,e^2\,{\left (d+e\,x\right )}^6}+\frac {125\,\sqrt {d^2-e^2\,x^2}}{20592\,d^5\,e^2\,{\left (d+e\,x\right )}^5}-\frac {41\,\sqrt {d^2-e^2\,x^2}}{41184\,d^6\,e^2\,{\left (d+e\,x\right )}^4}+\frac {512\,x\,\sqrt {d^2-e^2\,x^2}}{6435\,d^9\,e\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
((d^2 - e^2*x^2)^(1/2)*(41/(41184*d^6*e^2) + (256*x)/(6435*d^7*e)))/((d + e*x)^2*(d - e*x)^2) - ((d^2 - e^2*x^2)^(1/2)*(47/(1716*d^4*e^2) - (1369*x) /(34320*d^5*e)))/((d + e*x)^3*(d - e*x)^3) + (d^2 - e^2*x^2)^(1/2)/(104*d^ 3*e^2*(d + e*x)^7) + (25*(d^2 - e^2*x^2)^(1/2))/(2288*d^4*e^2*(d + e*x)^6) + (125*(d^2 - e^2*x^2)^(1/2))/(20592*d^5*e^2*(d + e*x)^5) - (41*(d^2 - e^ 2*x^2)^(1/2))/(41184*d^6*e^2*(d + e*x)^4) + (512*x*(d^2 - e^2*x^2)^(1/2))/ (6435*d^9*e*(d + e*x)*(d - e*x))