Integrand size = 24, antiderivative size = 210 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {a x}{d \left (d+e x^2\right )^{11/2}}+\frac {(b d+10 a e) x^3}{3 d^2 \left (d+e x^2\right )^{11/2}}+\frac {\left (3 c d^2+8 e (b d+10 a e)\right ) x^5}{15 d^3 \left (d+e x^2\right )^{11/2}}+\frac {2 e \left (3 c d^2+8 e (b d+10 a e)\right ) x^7}{35 d^4 \left (d+e x^2\right )^{11/2}}+\frac {8 e^2 \left (3 c d^2+8 e (b d+10 a e)\right ) x^9}{315 d^5 \left (d+e x^2\right )^{11/2}}+\frac {16 e^3 \left (3 c d^2+8 e (b d+10 a e)\right ) x^{11}}{3465 d^6 \left (d+e x^2\right )^{11/2}} \]
a*x/d/(e*x^2+d)^(11/2)+1/3*(10*a*e+b*d)*x^3/d^2/(e*x^2+d)^(11/2)+1/15*(3*c *d^2+8*e*(10*a*e+b*d))*x^5/d^3/(e*x^2+d)^(11/2)+2/35*e*(3*c*d^2+8*e*(10*a* e+b*d))*x^7/d^4/(e*x^2+d)^(11/2)+8/315*e^2*(3*c*d^2+8*e*(10*a*e+b*d))*x^9/ d^5/(e*x^2+d)^(11/2)+16/3465*e^3*(3*c*d^2+8*e*(10*a*e+b*d))*x^11/d^6/(e*x^ 2+d)^(11/2)
Time = 0.46 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.80 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {5 a \left (693 d^5 x+2310 d^4 e x^3+3696 d^3 e^2 x^5+3168 d^2 e^3 x^7+1408 d e^4 x^9+256 e^5 x^{11}\right )+d x^3 \left (3 c d x^2 \left (231 d^3+198 d^2 e x^2+88 d e^2 x^4+16 e^3 x^6\right )+b \left (1155 d^4+1848 d^3 e x^2+1584 d^2 e^2 x^4+704 d e^3 x^6+128 e^4 x^8\right )\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}} \]
(5*a*(693*d^5*x + 2310*d^4*e*x^3 + 3696*d^3*e^2*x^5 + 3168*d^2*e^3*x^7 + 1 408*d*e^4*x^9 + 256*e^5*x^11) + d*x^3*(3*c*d*x^2*(231*d^3 + 198*d^2*e*x^2 + 88*d*e^2*x^4 + 16*e^3*x^6) + b*(1155*d^4 + 1848*d^3*e*x^2 + 1584*d^2*e^2 *x^4 + 704*d*e^3*x^6 + 128*e^4*x^8)))/(3465*d^6*(d + e*x^2)^(11/2))
Time = 0.36 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1469, 2075, 362, 245, 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.
| \(\displaystyle \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx\) |
| \(\Big \downarrow \) 1469 |
| \(\displaystyle \frac {\int \frac {x^2 \left (10 a e+d \left (c x^2+b\right )\right )}{\left (e x^2+d\right )^{13/2}}dx}{d}+\frac {a x}{d \left (d+e x^2\right )^{11/2}}\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \frac {\int \frac {x^2 \left (c d x^2+b d+10 a e\right )}{\left (e x^2+d\right )^{13/2}}dx}{d}+\frac {a x}{d \left (d+e x^2\right )^{11/2}}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {80 a e}{d}+8 b+\frac {3 c d}{e}\right ) \int \frac {x^2}{\left (e x^2+d\right )^{11/2}}dx+\frac {x^3 \left (10 a e+b d-\frac {c d^2}{e}\right )}{11 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{11/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {80 a e}{d}+8 b+\frac {3 c d}{e}\right ) \left (\frac {2 e \int \frac {x^4}{\left (e x^2+d\right )^{11/2}}dx}{d}+\frac {x^3}{3 d \left (d+e x^2\right )^{9/2}}\right )+\frac {x^3 \left (10 a e+b d-\frac {c d^2}{e}\right )}{11 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{11/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {80 a e}{d}+8 b+\frac {3 c d}{e}\right ) \left (\frac {2 e \left (\frac {4 e \int \frac {x^6}{\left (e x^2+d\right )^{11/2}}dx}{5 d}+\frac {x^5}{5 d \left (d+e x^2\right )^{9/2}}\right )}{d}+\frac {x^3}{3 d \left (d+e x^2\right )^{9/2}}\right )+\frac {x^3 \left (10 a e+b d-\frac {c d^2}{e}\right )}{11 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{11/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {80 a e}{d}+8 b+\frac {3 c d}{e}\right ) \left (\frac {2 e \left (\frac {4 e \left (\frac {2 e \int \frac {x^8}{\left (e x^2+d\right )^{11/2}}dx}{7 d}+\frac {x^7}{7 d \left (d+e x^2\right )^{9/2}}\right )}{5 d}+\frac {x^5}{5 d \left (d+e x^2\right )^{9/2}}\right )}{d}+\frac {x^3}{3 d \left (d+e x^2\right )^{9/2}}\right )+\frac {x^3 \left (10 a e+b d-\frac {c d^2}{e}\right )}{11 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{11/2}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {\frac {x^3 \left (10 a e+b d-\frac {c d^2}{e}\right )}{11 d \left (d+e x^2\right )^{11/2}}+\frac {1}{11} \left (\frac {2 e \left (\frac {4 e \left (\frac {2 e x^9}{63 d^2 \left (d+e x^2\right )^{9/2}}+\frac {x^7}{7 d \left (d+e x^2\right )^{9/2}}\right )}{5 d}+\frac {x^5}{5 d \left (d+e x^2\right )^{9/2}}\right )}{d}+\frac {x^3}{3 d \left (d+e x^2\right )^{9/2}}\right ) \left (\frac {80 a e}{d}+8 b+\frac {3 c d}{e}\right )}{d}+\frac {a x}{d \left (d+e x^2\right )^{11/2}}\) |
(a*x)/(d*(d + e*x^2)^(11/2)) + (((b*d - (c*d^2)/e + 10*a*e)*x^3)/(11*d*(d + e*x^2)^(11/2)) + ((8*b + (3*c*d)/e + (80*a*e)/d)*(x^3/(3*d*(d + e*x^2)^( 9/2)) + (2*e*(x^5/(5*d*(d + e*x^2)^(9/2)) + (4*e*(x^7/(7*d*(d + e*x^2)^(9/ 2)) + (2*e*x^9)/(63*d^2*(d + e*x^2)^(9/2))))/(5*d)))/d))/11)/d
3.3.85.3.1 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 = 133, normalized size of antiderivative = 0.63
| method | result | size |
| pseudoelliptic | \(\frac {\left (\left (\frac {1}{5} c \,x^{4}+\frac {1}{3} b \,x^{2}+a \right ) d^{5}+\frac {10 e \left (\frac {9}{175} c \,x^{4}+\frac {4}{25} b \,x^{2}+a \right ) x^{2} d^{4}}{3}+\frac {16 e^{2} \left (\frac {1}{70} c \,x^{4}+\frac {3}{35} b \,x^{2}+a \right ) x^{4} d^{3}}{3}+\frac {32 e^{3} x^{6} \left (\frac {1}{330} c \,x^{4}+\frac {2}{45} b \,x^{2}+a \right ) d^{2}}{7}+\frac {128 e^{4} x^{8} \left (\frac {b \,x^{2}}{55}+a \right ) d}{63}+\frac {256 a \,e^{5} x^{10}}{693}\right ) x}{\left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\) | \(133\) |
| gosper | \(\frac {x \left (1280 a \,e^{5} x^{10}+128 b d \,e^{4} x^{10}+48 c \,d^{2} e^{3} x^{10}+7040 a d \,e^{4} x^{8}+704 b \,d^{2} e^{3} x^{8}+264 c \,d^{3} e^{2} x^{8}+15840 a \,d^{2} e^{3} x^{6}+1584 b \,d^{3} e^{2} x^{6}+594 c \,d^{4} e \,x^{6}+18480 a \,d^{3} e^{2} x^{4}+1848 b \,d^{4} e \,x^{4}+693 c \,d^{5} x^{4}+11550 a \,d^{4} e \,x^{2}+1155 b \,d^{5} x^{2}+3465 a \,d^{5}\right )}{3465 \left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\) | \(172\) |
| trager | \(\frac {x \left (1280 a \,e^{5} x^{10}+128 b d \,e^{4} x^{10}+48 c \,d^{2} e^{3} x^{10}+7040 a d \,e^{4} x^{8}+704 b \,d^{2} e^{3} x^{8}+264 c \,d^{3} e^{2} x^{8}+15840 a \,d^{2} e^{3} x^{6}+1584 b \,d^{3} e^{2} x^{6}+594 c \,d^{4} e \,x^{6}+18480 a \,d^{3} e^{2} x^{4}+1848 b \,d^{4} e \,x^{4}+693 c \,d^{5} x^{4}+11550 a \,d^{4} e \,x^{2}+1155 b \,d^{5} x^{2}+3465 a \,d^{5}\right )}{3465 \left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\) | \(172\) |
| default | \(a \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )+c \left (-\frac {x^{3}}{8 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {3 d \left (-\frac {x}{10 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {d \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )}{10 e}\right )}{8 e}\right )+b \left (-\frac {x}{10 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {d \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )}{10 e}\right )\) | \(421\) |
((1/5*c*x^4+1/3*b*x^2+a)*d^5+10/3*e*(9/175*c*x^4+4/25*b*x^2+a)*x^2*d^4+16/ 3*e^2*(1/70*c*x^4+3/35*b*x^2+a)*x^4*d^3+32/7*e^3*x^6*(1/330*c*x^4+2/45*b*x ^2+a)*d^2+128/63*e^4*x^8*(1/55*b*x^2+a)*d+256/693*a*e^5*x^10)/(e*x^2+d)^(1 1/2)*x/d^6
Time = 0.44 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.07 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {{\left (16 \, {\left (3 \, c d^{2} e^{3} + 8 \, b d e^{4} + 80 \, a e^{5}\right )} x^{11} + 88 \, {\left (3 \, c d^{3} e^{2} + 8 \, b d^{2} e^{3} + 80 \, a d e^{4}\right )} x^{9} + 198 \, {\left (3 \, c d^{4} e + 8 \, b d^{3} e^{2} + 80 \, a d^{2} e^{3}\right )} x^{7} + 3465 \, a d^{5} x + 231 \, {\left (3 \, c d^{5} + 8 \, b d^{4} e + 80 \, a d^{3} e^{2}\right )} x^{5} + 1155 \, {\left (b d^{5} + 10 \, a d^{4} e\right )} x^{3}\right )} \sqrt {e x^{2} + d}}{3465 \, {\left (d^{6} e^{6} x^{12} + 6 \, d^{7} e^{5} x^{10} + 15 \, d^{8} e^{4} x^{8} + 20 \, d^{9} e^{3} x^{6} + 15 \, d^{10} e^{2} x^{4} + 6 \, d^{11} e x^{2} + d^{12}\right )}} \]
1/3465*(16*(3*c*d^2*e^3 + 8*b*d*e^4 + 80*a*e^5)*x^11 + 88*(3*c*d^3*e^2 + 8 *b*d^2*e^3 + 80*a*d*e^4)*x^9 + 198*(3*c*d^4*e + 8*b*d^3*e^2 + 80*a*d^2*e^3 )*x^7 + 3465*a*d^5*x + 231*(3*c*d^5 + 8*b*d^4*e + 80*a*d^3*e^2)*x^5 + 1155 *(b*d^5 + 10*a*d^4*e)*x^3)*sqrt(e*x^2 + d)/(d^6*e^6*x^12 + 6*d^7*e^5*x^10 + 15*d^8*e^4*x^8 + 20*d^9*e^3*x^6 + 15*d^10*e^2*x^4 + 6*d^11*e*x^2 + d^12)
Leaf count of result is larger than twice the leaf count of optimal. 11602 vs. \(2 (206) = 412\).
Time = 154.15 (sec) , antiderivative size = 11602, normalized size of antiderivative = 55.25 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\text {Too large to display} \]
a*(693*d**55*x/(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x** 2*sqrt(1 + e*x**2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315 315*d**(117/2)*e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8* sqrt(1 + e*x**2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 34 68465*d**(111/2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x **14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d**(105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e **10*x**20*sqrt(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x **2/d) + 315315*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2) *e**13*x**26*sqrt(1 + e*x**2/d) + 10395*d**(95/2)*e**14*x**28*sqrt(1 + e*x **2/d) + 693*d**(93/2)*e**15*x**30*sqrt(1 + e*x**2/d)) + 9240*d**54*e*x**3 /(693*d**(123/2)*sqrt(1 + e*x**2/d) + 10395*d**(121/2)*e*x**2*sqrt(1 + e*x **2/d) + 72765*d**(119/2)*e**2*x**4*sqrt(1 + e*x**2/d) + 315315*d**(117/2) *e**3*x**6*sqrt(1 + e*x**2/d) + 945945*d**(115/2)*e**4*x**8*sqrt(1 + e*x** 2/d) + 2081079*d**(113/2)*e**5*x**10*sqrt(1 + e*x**2/d) + 3468465*d**(111/ 2)*e**6*x**12*sqrt(1 + e*x**2/d) + 4459455*d**(109/2)*e**7*x**14*sqrt(1 + e*x**2/d) + 4459455*d**(107/2)*e**8*x**16*sqrt(1 + e*x**2/d) + 3468465*d** (105/2)*e**9*x**18*sqrt(1 + e*x**2/d) + 2081079*d**(103/2)*e**10*x**20*sqr t(1 + e*x**2/d) + 945945*d**(101/2)*e**11*x**22*sqrt(1 + e*x**2/d) + 31531 5*d**(99/2)*e**12*x**24*sqrt(1 + e*x**2/d) + 72765*d**(97/2)*e**13*x**2...
Time = 0.21 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.60 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=-\frac {c x^{3}}{8 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e} + \frac {256 \, a x}{693 \, \sqrt {e x^{2} + d} d^{6}} + \frac {128 \, a x}{693 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{5}} + \frac {32 \, a x}{231 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{4}} + \frac {80 \, a x}{693 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{3}} + \frac {10 \, a x}{99 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d^{2}} + \frac {a x}{11 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} d} + \frac {c x}{264 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} + \frac {16 \, c x}{1155 \, \sqrt {e x^{2} + d} d^{4} e^{2}} + \frac {8 \, c x}{1155 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} e^{2}} + \frac {2 \, c x}{385 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} e^{2}} + \frac {c x}{231 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d e^{2}} - \frac {3 \, c d x}{88 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e^{2}} - \frac {b x}{11 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e} + \frac {128 \, b x}{3465 \, \sqrt {e x^{2} + d} d^{5} e} + \frac {64 \, b x}{3465 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{4} e} + \frac {16 \, b x}{1155 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{3} e} + \frac {8 \, b x}{693 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{2} e} + \frac {b x}{99 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d e} \]
-1/8*c*x^3/((e*x^2 + d)^(11/2)*e) + 256/693*a*x/(sqrt(e*x^2 + d)*d^6) + 12 8/693*a*x/((e*x^2 + d)^(3/2)*d^5) + 32/231*a*x/((e*x^2 + d)^(5/2)*d^4) + 8 0/693*a*x/((e*x^2 + d)^(7/2)*d^3) + 10/99*a*x/((e*x^2 + d)^(9/2)*d^2) + 1/ 11*a*x/((e*x^2 + d)^(11/2)*d) + 1/264*c*x/((e*x^2 + d)^(9/2)*e^2) + 16/115 5*c*x/(sqrt(e*x^2 + d)*d^4*e^2) + 8/1155*c*x/((e*x^2 + d)^(3/2)*d^3*e^2) + 2/385*c*x/((e*x^2 + d)^(5/2)*d^2*e^2) + 1/231*c*x/((e*x^2 + d)^(7/2)*d*e^ 2) - 3/88*c*d*x/((e*x^2 + d)^(11/2)*e^2) - 1/11*b*x/((e*x^2 + d)^(11/2)*e) + 128/3465*b*x/(sqrt(e*x^2 + d)*d^5*e) + 64/3465*b*x/((e*x^2 + d)^(3/2)*d ^4*e) + 16/1155*b*x/((e*x^2 + d)^(5/2)*d^3*e) + 8/693*b*x/((e*x^2 + d)^(7/ 2)*d^2*e) + 1/99*b*x/((e*x^2 + d)^(9/2)*d*e)
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.99 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {{\left ({\left ({\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, {\left (3 \, c d^{2} e^{8} + 8 \, b d e^{9} + 80 \, a e^{10}\right )} x^{2}}{d^{6} e^{5}} + \frac {11 \, {\left (3 \, c d^{3} e^{7} + 8 \, b d^{2} e^{8} + 80 \, a d e^{9}\right )}}{d^{6} e^{5}}\right )} + \frac {99 \, {\left (3 \, c d^{4} e^{6} + 8 \, b d^{3} e^{7} + 80 \, a d^{2} e^{8}\right )}}{d^{6} e^{5}}\right )} x^{2} + \frac {231 \, {\left (3 \, c d^{5} e^{5} + 8 \, b d^{4} e^{6} + 80 \, a d^{3} e^{7}\right )}}{d^{6} e^{5}}\right )} x^{2} + \frac {1155 \, {\left (b d^{5} e^{5} + 10 \, a d^{4} e^{6}\right )}}{d^{6} e^{5}}\right )} x^{2} + \frac {3465 \, a}{d}\right )} x}{3465 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}}} \]
1/3465*(((2*(4*x^2*(2*(3*c*d^2*e^8 + 8*b*d*e^9 + 80*a*e^10)*x^2/(d^6*e^5) + 11*(3*c*d^3*e^7 + 8*b*d^2*e^8 + 80*a*d*e^9)/(d^6*e^5)) + 99*(3*c*d^4*e^6 + 8*b*d^3*e^7 + 80*a*d^2*e^8)/(d^6*e^5))*x^2 + 231*(3*c*d^5*e^5 + 8*b*d^4 *e^6 + 80*a*d^3*e^7)/(d^6*e^5))*x^2 + 1155*(b*d^5*e^5 + 10*a*d^4*e^6)/(d^6 *e^5))*x^2 + 3465*a/d)*x/(e*x^2 + d)^(11/2)
Time = 7.91 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {x\,\left (\frac {a}{11\,d}-\frac {d\,\left (\frac {b}{11\,d}-\frac {c}{11\,e}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{11/2}}-\frac {x\,\left (\frac {c}{9\,e^2}-\frac {-c\,d^2+b\,d\,e+10\,a\,e^2}{99\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{9/2}}+\frac {x\,\left (3\,c\,d^2+8\,b\,d\,e+80\,a\,e^2\right )}{693\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (6\,c\,d^2+16\,b\,d\,e+160\,a\,e^2\right )}{1155\,d^4\,e^2\,{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (24\,c\,d^2+64\,b\,d\,e+640\,a\,e^2\right )}{3465\,d^5\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (48\,c\,d^2+128\,b\,d\,e+1280\,a\,e^2\right )}{3465\,d^6\,e^2\,\sqrt {e\,x^2+d}} \]
(x*(a/(11*d) - (d*(b/(11*d) - c/(11*e)))/e))/(d + e*x^2)^(11/2) - (x*(c/(9 *e^2) - (10*a*e^2 - c*d^2 + b*d*e)/(99*d^2*e^2)))/(d + e*x^2)^(9/2) + (x*( 80*a*e^2 + 3*c*d^2 + 8*b*d*e))/(693*d^3*e^2*(d + e*x^2)^(7/2)) + (x*(160*a *e^2 + 6*c*d^2 + 16*b*d*e))/(1155*d^4*e^2*(d + e*x^2)^(5/2)) + (x*(640*a*e ^2 + 24*c*d^2 + 64*b*d*e))/(3465*d^5*e^2*(d + e*x^2)^(3/2)) + (x*(1280*a*e ^2 + 48*c*d^2 + 128*b*d*e))/(3465*d^6*e^2*(d + e*x^2)^(1/2))