Integrand size = 24, antiderivative size = 165 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {a x}{d \left (d+e x^2\right )^{9/2}}+\frac {(b d+8 a e) x^3}{3 d^2 \left (d+e x^2\right )^{9/2}}+\frac {\left (c d^2+2 e (b d+8 a e)\right ) x^5}{5 d^3 \left (d+e x^2\right )^{9/2}}+\frac {4 e \left (c d^2+2 e (b d+8 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{9/2}}+\frac {8 e^2 \left (c d^2+2 e (b d+8 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{9/2}} \]
a*x/d/(e*x^2+d)^(9/2)+1/3*(8*a*e+b*d)*x^3/d^2/(e*x^2+d)^(9/2)+1/5*(c*d^2+2 *e*(8*a*e+b*d))*x^5/d^3/(e*x^2+d)^(9/2)+4/35*e*(c*d^2+2*e*(8*a*e+b*d))*x^7 /d^4/(e*x^2+d)^(9/2)+8/315*e^2*(c*d^2+2*e*(8*a*e+b*d))*x^9/d^5/(e*x^2+d)^( 9/2)
Time = 0.29 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.80 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {a \left (315 d^4 x+840 d^3 e x^3+1008 d^2 e^2 x^5+576 d e^3 x^7+128 e^4 x^9\right )+d x^3 \left (c d x^2 \left (63 d^2+36 d e x^2+8 e^2 x^4\right )+b \left (105 d^3+126 d^2 e x^2+72 d e^2 x^4+16 e^3 x^6\right )\right )}{315 d^5 \left (d+e x^2\right )^{9/2}} \]
(a*(315*d^4*x + 840*d^3*e*x^3 + 1008*d^2*e^2*x^5 + 576*d*e^3*x^7 + 128*e^4 *x^9) + d*x^3*(c*d*x^2*(63*d^2 + 36*d*e*x^2 + 8*e^2*x^4) + b*(105*d^3 + 12 6*d^2*e*x^2 + 72*d*e^2*x^4 + 16*e^3*x^6)))/(315*d^5*(d + e*x^2)^(9/2))
Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1469, 2075, 362, 245, 245, 242}
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.
| \(\displaystyle \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1469 |
| \(\displaystyle \frac {\int \frac {x^2 \left (8 a e+d \left (c x^2+b\right )\right )}{\left (e x^2+d\right )^{11/2}}dx}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \frac {\int \frac {x^2 \left (c d x^2+b d+8 a e\right )}{\left (e x^2+d\right )^{11/2}}dx}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 (8 a e+b d)}{d}+\frac {c d}{e}\right ) \int \frac {x^2}{\left (e x^2+d\right )^{9/2}}dx+\frac {x^3 \left (8 a e+b d-\frac {c d^2}{e}\right )}{9 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 (8 a e+b d)}{d}+\frac {c d}{e}\right ) \left (\frac {4 e \int \frac {x^4}{\left (e x^2+d\right )^{9/2}}dx}{3 d}+\frac {x^3}{3 d \left (d+e x^2\right )^{7/2}}\right )+\frac {x^3 \left (8 a e+b d-\frac {c d^2}{e}\right )}{9 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {2 (8 a e+b d)}{d}+\frac {c d}{e}\right ) \left (\frac {4 e \left (\frac {2 e \int \frac {x^6}{\left (e x^2+d\right )^{9/2}}dx}{5 d}+\frac {x^5}{5 d \left (d+e x^2\right )^{7/2}}\right )}{3 d}+\frac {x^3}{3 d \left (d+e x^2\right )^{7/2}}\right )+\frac {x^3 \left (8 a e+b d-\frac {c d^2}{e}\right )}{9 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {\frac {x^3 \left (8 a e+b d-\frac {c d^2}{e}\right )}{9 d \left (d+e x^2\right )^{9/2}}+\frac {1}{3} \left (\frac {4 e \left (\frac {2 e x^7}{35 d^2 \left (d+e x^2\right )^{7/2}}+\frac {x^5}{5 d \left (d+e x^2\right )^{7/2}}\right )}{3 d}+\frac {x^3}{3 d \left (d+e x^2\right )^{7/2}}\right ) \left (\frac {2 (8 a e+b d)}{d}+\frac {c d}{e}\right )}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
(a*x)/(d*(d + e*x^2)^(9/2)) + (((b*d - (c*d^2)/e + 8*a*e)*x^3)/(9*d*(d + e *x^2)^(9/2)) + (((c*d)/e + (2*(b*d + 8*a*e))/d)*(x^3/(3*d*(d + e*x^2)^(7/2 )) + (4*e*(x^5/(5*d*(d + e*x^2)^(7/2)) + (2*e*x^7)/(35*d^2*(d + e*x^2)^(7/ 2))))/(3*d)))/3)/d
3.3.84.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*b*e *(p + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(2*a*b*(p + 1)) I nt[(e*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && N eQ[b*c - a*d, 0] && LtQ[p, -1] && (( !IntegerQ[p + 1/2] && NeQ[p, -5/4]) || !RationalQ[m] || (ILtQ[p + 1/2, 0] && LeQ[-1, m, -2*(p + 1)]))
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[a^p*x*((d + e*x^2)^(q + 1)/d), x] + Simp[1/d Int[x^2*(d + e*x^2)^q*(d*PolynomialQuotient[(a + b*x^2 + c*x^4)^p - a^p, x^2, x] - e* a^p*(2*q + 3)), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4 *p + 2*q + 1, 0]
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && ! BinomialMatchQ[{u, v}, x]
Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.65
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (\frac {1}{5} c \,x^{4}+\frac {1}{3} b \,x^{2}+a \right ) d^{4}+\frac {8 e \left (\frac {3}{70} c \,x^{4}+\frac {3}{20} b \,x^{2}+a \right ) x^{2} d^{3}}{3}+\frac {16 e^{2} \left (\frac {1}{126} c \,x^{4}+\frac {1}{14} b \,x^{2}+a \right ) x^{4} d^{2}}{5}+\frac {64 \left (\frac {b \,x^{2}}{36}+a \right ) e^{3} x^{6} d}{35}+\frac {128 a \,e^{4} x^{8}}{315}\right ) x}{\left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\) | \(108\) |
| gosper | \(\frac {x \left (128 a \,e^{4} x^{8}+16 b d \,e^{3} x^{8}+8 c \,d^{2} e^{2} x^{8}+576 a d \,e^{3} x^{6}+72 b \,d^{2} e^{2} x^{6}+36 c \,d^{3} e \,x^{6}+1008 a \,d^{2} e^{2} x^{4}+126 b \,d^{3} e \,x^{4}+63 c \,d^{4} x^{4}+840 a \,d^{3} e \,x^{2}+105 b \,d^{4} x^{2}+315 d^{4} a \right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\) | \(136\) |
| trager | \(\frac {x \left (128 a \,e^{4} x^{8}+16 b d \,e^{3} x^{8}+8 c \,d^{2} e^{2} x^{8}+576 a d \,e^{3} x^{6}+72 b \,d^{2} e^{2} x^{6}+36 c \,d^{3} e \,x^{6}+1008 a \,d^{2} e^{2} x^{4}+126 b \,d^{3} e \,x^{4}+63 c \,d^{4} x^{4}+840 a \,d^{3} e \,x^{2}+105 b \,d^{4} x^{2}+315 d^{4} a \right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\) | \(136\) |
| default | \(a \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )+c \left (-\frac {x^{3}}{6 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (-\frac {x}{8 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )}{8 e}\right )}{2 e}\right )+b \left (-\frac {x}{8 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )}{8 e}\right )\) | \(358\) |
((1/5*c*x^4+1/3*b*x^2+a)*d^4+8/3*e*(3/70*c*x^4+3/20*b*x^2+a)*x^2*d^3+16/5* e^2*(1/126*c*x^4+1/14*b*x^2+a)*x^4*d^2+64/35*(1/36*b*x^2+a)*e^3*x^6*d+128/ 315*a*e^4*x^8)/(e*x^2+d)^(9/2)*x/d^5
Time = 0.40 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.07 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {{\left (8 \, {\left (c d^{2} e^{2} + 2 \, b d e^{3} + 16 \, a e^{4}\right )} x^{9} + 36 \, {\left (c d^{3} e + 2 \, b d^{2} e^{2} + 16 \, a d e^{3}\right )} x^{7} + 315 \, a d^{4} x + 63 \, {\left (c d^{4} + 2 \, b d^{3} e + 16 \, a d^{2} e^{2}\right )} x^{5} + 105 \, {\left (b d^{4} + 8 \, a d^{3} e\right )} x^{3}\right )} \sqrt {e x^{2} + d}}{315 \, {\left (d^{5} e^{5} x^{10} + 5 \, d^{6} e^{4} x^{8} + 10 \, d^{7} e^{3} x^{6} + 10 \, d^{8} e^{2} x^{4} + 5 \, d^{9} e x^{2} + d^{10}\right )}} \]
1/315*(8*(c*d^2*e^2 + 2*b*d*e^3 + 16*a*e^4)*x^9 + 36*(c*d^3*e + 2*b*d^2*e^ 2 + 16*a*d*e^3)*x^7 + 315*a*d^4*x + 63*(c*d^4 + 2*b*d^3*e + 16*a*d^2*e^2)* x^5 + 105*(b*d^4 + 8*a*d^3*e)*x^3)*sqrt(e*x^2 + d)/(d^5*e^5*x^10 + 5*d^6*e ^4*x^8 + 10*d^7*e^3*x^6 + 10*d^8*e^2*x^4 + 5*d^9*e*x^2 + d^10)
Leaf count of result is larger than twice the leaf count of optimal. 5187 vs. \(2 (160) = 320\).
Time = 80.52 (sec) , antiderivative size = 5187, normalized size of antiderivative = 31.44 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\text {Too large to display} \]
a*(315*d**30*x/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e*x**2*s qrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d **(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150*d**(59/ 2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt(1 + e*x **2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(53/2)*e* *9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e*x**2/d) ) + 2730*d**29*e*x**3/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150*d**(69/2)*e *x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)*e**4*x**8*s qrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 66150* d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7*x**14*sqrt( 1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3150*d**(5 3/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 315*d**(51/2)*e**10*x**20*sqrt(1 + e* x**2/d)) + 10773*d**28*e**2*x**5/(315*d**(71/2)*sqrt(1 + e*x**2/d) + 3150* d**(69/2)*e*x**2*sqrt(1 + e*x**2/d) + 14175*d**(67/2)*e**2*x**4*sqrt(1 + e *x**2/d) + 37800*d**(65/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 66150*d**(63/2)* e**4*x**8*sqrt(1 + e*x**2/d) + 79380*d**(61/2)*e**5*x**10*sqrt(1 + e*x**2/ d) + 66150*d**(59/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 37800*d**(57/2)*e**7* x**14*sqrt(1 + e*x**2/d) + 14175*d**(55/2)*e**8*x**16*sqrt(1 + e*x**2/d...
Time = 0.19 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.70 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=-\frac {c x^{3}}{6 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} + \frac {128 \, a x}{315 \, \sqrt {e x^{2} + d} d^{5}} + \frac {64 \, a x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{4}} + \frac {16 \, a x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{3}} + \frac {8 \, a x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{2}} + \frac {a x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d} + \frac {c x}{126 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{2}} + \frac {8 \, c x}{315 \, \sqrt {e x^{2} + d} d^{3} e^{2}} + \frac {4 \, c x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {c x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d e^{2}} - \frac {c d x}{18 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} - \frac {b x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} + \frac {16 \, b x}{315 \, \sqrt {e x^{2} + d} d^{4} e} + \frac {8 \, b x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} e} + \frac {2 \, b x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} e} + \frac {b x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d e} \]
-1/6*c*x^3/((e*x^2 + d)^(9/2)*e) + 128/315*a*x/(sqrt(e*x^2 + d)*d^5) + 64/ 315*a*x/((e*x^2 + d)^(3/2)*d^4) + 16/105*a*x/((e*x^2 + d)^(5/2)*d^3) + 8/6 3*a*x/((e*x^2 + d)^(7/2)*d^2) + 1/9*a*x/((e*x^2 + d)^(9/2)*d) + 1/126*c*x/ ((e*x^2 + d)^(7/2)*e^2) + 8/315*c*x/(sqrt(e*x^2 + d)*d^3*e^2) + 4/315*c*x/ ((e*x^2 + d)^(3/2)*d^2*e^2) + 1/105*c*x/((e*x^2 + d)^(5/2)*d*e^2) - 1/18*c *d*x/((e*x^2 + d)^(9/2)*e^2) - 1/9*b*x/((e*x^2 + d)^(9/2)*e) + 16/315*b*x/ (sqrt(e*x^2 + d)*d^4*e) + 8/315*b*x/((e*x^2 + d)^(3/2)*d^3*e) + 2/105*b*x/ ((e*x^2 + d)^(5/2)*d^2*e) + 1/63*b*x/((e*x^2 + d)^(7/2)*d*e)
Time = 0.30 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.98 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {{\left ({\left ({\left (4 \, x^{2} {\left (\frac {2 \, {\left (c d^{2} e^{6} + 2 \, b d e^{7} + 16 \, a e^{8}\right )} x^{2}}{d^{5} e^{4}} + \frac {9 \, {\left (c d^{3} e^{5} + 2 \, b d^{2} e^{6} + 16 \, a d e^{7}\right )}}{d^{5} e^{4}}\right )} + \frac {63 \, {\left (c d^{4} e^{4} + 2 \, b d^{3} e^{5} + 16 \, a d^{2} e^{6}\right )}}{d^{5} e^{4}}\right )} x^{2} + \frac {105 \, {\left (b d^{4} e^{4} + 8 \, a d^{3} e^{5}\right )}}{d^{5} e^{4}}\right )} x^{2} + \frac {315 \, a}{d}\right )} x}{315 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}}} \]
1/315*(((4*x^2*(2*(c*d^2*e^6 + 2*b*d*e^7 + 16*a*e^8)*x^2/(d^5*e^4) + 9*(c* d^3*e^5 + 2*b*d^2*e^6 + 16*a*d*e^7)/(d^5*e^4)) + 63*(c*d^4*e^4 + 2*b*d^3*e ^5 + 16*a*d^2*e^6)/(d^5*e^4))*x^2 + 105*(b*d^4*e^4 + 8*a*d^3*e^5)/(d^5*e^4 ))*x^2 + 315*a/d)*x/(e*x^2 + d)^(9/2)
Time = 7.87 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.15 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {x\,\left (\frac {a}{9\,d}-\frac {d\,\left (\frac {b}{9\,d}-\frac {c}{9\,e}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{9/2}}-\frac {x\,\left (\frac {c}{7\,e^2}-\frac {-c\,d^2+b\,d\,e+8\,a\,e^2}{63\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (c\,d^2+2\,b\,d\,e+16\,a\,e^2\right )}{105\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (4\,c\,d^2+8\,b\,d\,e+64\,a\,e^2\right )}{315\,d^4\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (8\,c\,d^2+16\,b\,d\,e+128\,a\,e^2\right )}{315\,d^5\,e^2\,\sqrt {e\,x^2+d}} \]
(x*(a/(9*d) - (d*(b/(9*d) - c/(9*e)))/e))/(d + e*x^2)^(9/2) - (x*(c/(7*e^2 ) - (8*a*e^2 - c*d^2 + b*d*e)/(63*d^2*e^2)))/(d + e*x^2)^(7/2) + (x*(16*a* e^2 + c*d^2 + 2*b*d*e))/(105*d^3*e^2*(d + e*x^2)^(5/2)) + (x*(64*a*e^2 + 4 *c*d^2 + 8*b*d*e))/(315*d^4*e^2*(d + e*x^2)^(3/2)) + (x*(128*a*e^2 + 8*c*d ^2 + 16*b*d*e))/(315*d^5*e^2*(d + e*x^2)^(1/2))