Integrand size = 19, antiderivative size = 158 \[ \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {\left (a+\frac {c d^2}{e^2}\right ) x}{9 d \left (d+e x^2\right )^{9/2}}+\frac {2 \left (\frac {4 a}{d^2}-\frac {5 c}{e^2}\right ) x}{63 \left (d+e x^2\right )^{7/2}}+\frac {\left (\frac {16 a}{d^2}+\frac {c}{e^2}\right ) x}{105 d \left (d+e x^2\right )^{5/2}}+\frac {4 \left (c d^2+16 a e^2\right ) x}{315 d^4 e^2 \left (d+e x^2\right )^{3/2}}+\frac {8 \left (c d^2+16 a e^2\right ) x}{315 d^5 e^2 \sqrt {d+e x^2}} \] Output:
1/9*(a+c*d^2/e^2)*x/d/(e*x^2+d)^(9/2)+2/63*(4*a/d^2-5*c/e^2)*x/(e*x^2+d)^( 7/2)+1/105*(16*a/d^2+c/e^2)*x/d/(e*x^2+d)^(5/2)+4/315*(16*a*e^2+c*d^2)*x/d ^4/e^2/(e*x^2+d)^(3/2)+8/315*(16*a*e^2+c*d^2)*x/d^5/e^2/(e*x^2+d)^(1/2)
Time = 0.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {c d^2 x^5 \left (63 d^2+36 d e x^2+8 e^2 x^4\right )+a \left (315 d^4 x+840 d^3 e x^3+1008 d^2 e^2 x^5+576 d e^3 x^7+128 e^4 x^9\right )}{315 d^5 \left (d+e x^2\right )^{9/2}} \] Input:
Integrate[(a + c*x^4)/(d + e*x^2)^(11/2),x]
Output:
(c*d^2*x^5*(63*d^2 + 36*d*e*x^2 + 8*e^2*x^4) + 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))/(315*d^5*(d + e*x^2)^ (9/2))
Time = 0.43 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1470, 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.
| \(\displaystyle \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 1470 |
| \(\displaystyle \frac {\int \frac {x^2 \left (c d x^2+8 a e\right )}{\left (e x^2+d\right )^{11/2}}dx}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 362 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {16 a e}{d}+\frac {c d}{e}\right ) \int \frac {x^2}{\left (e x^2+d\right )^{9/2}}dx-\frac {x^3 \left (\frac {c d}{e}-\frac {8 a e}{d}\right )}{9 \left (d+e x^2\right )^{9/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {16 a e}{d}+\frac {c d}{e}\right ) \left (\frac {4 e \int \frac {x^4}{\left (e x^2+d\right )^{9/2}}dx}{3 d}+\frac {x^3}{3 d \left (d+e x^2\right )^{7/2}}\right )-\frac {x^3 \left (\frac {c d}{e}-\frac {8 a e}{d}\right )}{9 \left (d+e x^2\right )^{9/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {16 a e}{d}+\frac {c d}{e}\right ) \left (\frac {4 e \left (\frac {2 e \int \frac {x^6}{\left (e x^2+d\right )^{9/2}}dx}{5 d}+\frac {x^5}{5 d \left (d+e x^2\right )^{7/2}}\right )}{3 d}+\frac {x^3}{3 d \left (d+e x^2\right )^{7/2}}\right )-\frac {x^3 \left (\frac {c d}{e}-\frac {8 a e}{d}\right )}{9 \left (d+e x^2\right )^{9/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {4 e \left (\frac {2 e x^7}{35 d^2 \left (d+e x^2\right )^{7/2}}+\frac {x^5}{5 d \left (d+e x^2\right )^{7/2}}\right )}{3 d}+\frac {x^3}{3 d \left (d+e x^2\right )^{7/2}}\right ) \left (\frac {16 a e}{d}+\frac {c d}{e}\right )-\frac {x^3 \left (\frac {c d}{e}-\frac {8 a e}{d}\right )}{9 \left (d+e x^2\right )^{9/2}}}{d}+\frac {a x}{d \left (d+e x^2\right )^{9/2}}\) |
Input:
Int[(a + c*x^4)/(d + e*x^2)^(11/2),x]
Output:
(a*x)/(d*(d + e*x^2)^(9/2)) + (-1/9*(((c*d)/e - (8*a*e)/d)*x^3)/(d + e*x^2 )^(9/2) + (((c*d)/e + (16*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
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_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si mp[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 + c*x^4)^p - a^p, x^2, x] - e*a^p*(2*q + 3)), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] && ILtQ[q + 1/2, 0] && LtQ[4*p + 2*q + 1, 0]
Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.53
| method | result | size |
| pseudoelliptic | \(\frac {x \left (\left (\frac {c \,x^{4}}{5}+a \right ) d^{4}+\frac {8 \left (\frac {3 c \,x^{4}}{70}+a \right ) x^{2} e \,d^{3}}{3}+\frac {16 x^{4} \left (\frac {c \,x^{4}}{126}+a \right ) e^{2} d^{2}}{5}+\frac {64 a d \,e^{3} x^{6}}{35}+\frac {128 a \,e^{4} x^{8}}{315}\right )}{\left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\) | \(83\) |
| gosper | \(\frac {x \left (128 a \,e^{4} x^{8}+8 c \,d^{2} e^{2} x^{8}+576 a d \,e^{3} x^{6}+36 c \,d^{3} e \,x^{6}+1008 a \,d^{2} e^{2} x^{4}+63 c \,d^{4} x^{4}+840 a \,d^{3} e \,x^{2}+315 d^{4} a \right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\) | \(95\) |
| trager | \(\frac {x \left (128 a \,e^{4} x^{8}+8 c \,d^{2} e^{2} x^{8}+576 a d \,e^{3} x^{6}+36 c \,d^{3} e \,x^{6}+1008 a \,d^{2} e^{2} x^{4}+63 c \,d^{4} x^{4}+840 a \,d^{3} e \,x^{2}+315 d^{4} a \right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\) | \(95\) |
| orering | \(\frac {x \left (128 a \,e^{4} x^{8}+8 c \,d^{2} e^{2} x^{8}+576 a d \,e^{3} x^{6}+36 c \,d^{3} e \,x^{6}+1008 a \,d^{2} e^{2} x^{4}+63 c \,d^{4} x^{4}+840 a \,d^{3} e \,x^{2}+315 d^{4} a \right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\) | \(95\) |
| default | \(a \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )+c \left (-\frac {x^{3}}{6 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (-\frac {x}{8 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )}{8 e}\right )}{2 e}\right )\) | \(240\) |
Input:
int((c*x^4+a)/(e*x^2+d)^(11/2),x,method=_RETURNVERBOSE)
Output:
x/(e*x^2+d)^(9/2)*((1/5*c*x^4+a)*d^4+8/3*(3/70*c*x^4+a)*x^2*e*d^3+16/5*x^4 *(1/126*c*x^4+a)*e^2*d^2+64/35*a*d*e^3*x^6+128/315*a*e^4*x^8)/d^5
Time = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.92 \[ \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {{\left (8 \, {\left (c d^{2} e^{2} + 16 \, a e^{4}\right )} x^{9} + 840 \, a d^{3} e x^{3} + 36 \, {\left (c d^{3} e + 16 \, a d e^{3}\right )} x^{7} + 315 \, a d^{4} x + 63 \, {\left (c d^{4} + 16 \, a d^{2} e^{2}\right )} x^{5}\right )} \sqrt {e x^{2} + d}}{315 \, {\left (d^{5} e^{5} x^{10} + 5 \, d^{6} e^{4} x^{8} + 10 \, d^{7} e^{3} x^{6} + 10 \, d^{8} e^{2} x^{4} + 5 \, d^{9} e x^{2} + d^{10}\right )}} \] Input:
integrate((c*x^4+a)/(e*x^2+d)^(11/2),x, algorithm="fricas")
Output:
1/315*(8*(c*d^2*e^2 + 16*a*e^4)*x^9 + 840*a*d^3*e*x^3 + 36*(c*d^3*e + 16*a *d*e^3)*x^7 + 315*a*d^4*x + 63*(c*d^4 + 16*a*d^2*e^2)*x^5)*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. 3738 vs. \(2 (155) = 310\).
Time = 47.79 (sec) , antiderivative size = 3738, normalized size of antiderivative = 23.66 \[ \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\text {Too large to display} \] Input:
integrate((c*x**4+a)/(e*x**2+d)**(11/2),x)
Output:
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 = 0.03 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.20 \[ \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=-\frac {c x^{3}}{6 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} + \frac {128 \, a x}{315 \, \sqrt {e x^{2} + d} d^{5}} + \frac {64 \, a x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{4}} + \frac {16 \, a x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{3}} + \frac {8 \, a x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{2}} + \frac {a x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d} + \frac {c x}{126 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{2}} + \frac {8 \, c x}{315 \, \sqrt {e x^{2} + d} d^{3} e^{2}} + \frac {4 \, c x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {c x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d e^{2}} - \frac {c d x}{18 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} \] Input:
integrate((c*x^4+a)/(e*x^2+d)^(11/2),x, algorithm="maxima")
Output:
-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)
Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.75 \[ \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {{\left ({\left ({\left (4 \, x^{2} {\left (\frac {2 \, {\left (c d^{2} e^{6} + 16 \, a e^{8}\right )} x^{2}}{d^{5} e^{4}} + \frac {9 \, {\left (c d^{3} e^{5} + 16 \, a d e^{7}\right )}}{d^{5} e^{4}}\right )} + \frac {63 \, {\left (c d^{4} e^{4} + 16 \, a d^{2} e^{6}\right )}}{d^{5} e^{4}}\right )} x^{2} + \frac {840 \, a e}{d^{2}}\right )} x^{2} + \frac {315 \, a}{d}\right )} x}{315 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}}} \] Input:
integrate((c*x^4+a)/(e*x^2+d)^(11/2),x, algorithm="giac")
Output:
1/315*(((4*x^2*(2*(c*d^2*e^6 + 16*a*e^8)*x^2/(d^5*e^4) + 9*(c*d^3*e^5 + 16 *a*d*e^7)/(d^5*e^4)) + 63*(c*d^4*e^4 + 16*a*d^2*e^6)/(d^5*e^4))*x^2 + 840* a*e/d^2)*x^2 + 315*a/d)*x/(e*x^2 + d)^(9/2)
Time = 17.51 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00 \[ \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {x\,\left (\frac {a}{9\,d}+\frac {c\,d}{9\,e^2}\right )}{{\left (e\,x^2+d\right )}^{9/2}}-\frac {x\,\left (\frac {c}{7\,e^2}-\frac {8\,a\,e^2-c\,d^2}{63\,d^2\,e^2}\right )}{{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (c\,d^2+16\,a\,e^2\right )}{105\,d^3\,e^2\,{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (4\,c\,d^2+64\,a\,e^2\right )}{315\,d^4\,e^2\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (8\,c\,d^2+128\,a\,e^2\right )}{315\,d^5\,e^2\,\sqrt {e\,x^2+d}} \] Input:
int((a + c*x^4)/(d + e*x^2)^(11/2),x)
Output:
(x*(a/(9*d) + (c*d)/(9*e^2)))/(d + e*x^2)^(9/2) - (x*(c/(7*e^2) - (8*a*e^2 - c*d^2)/(63*d^2*e^2)))/(d + e*x^2)^(7/2) + (x*(16*a*e^2 + c*d^2))/(105*d ^3*e^2*(d + e*x^2)^(5/2)) + (x*(64*a*e^2 + 4*c*d^2))/(315*d^4*e^2*(d + e*x ^2)^(3/2)) + (x*(128*a*e^2 + 8*c*d^2))/(315*d^5*e^2*(d + e*x^2)^(1/2))
Time = 0.20 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.32 \[ \int \frac {a+c x^4}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {315 \sqrt {e \,x^{2}+d}\, a \,d^{4} e^{3} x +840 \sqrt {e \,x^{2}+d}\, a \,d^{3} e^{4} x^{3}+1008 \sqrt {e \,x^{2}+d}\, a \,d^{2} e^{5} x^{5}+576 \sqrt {e \,x^{2}+d}\, a d \,e^{6} x^{7}+128 \sqrt {e \,x^{2}+d}\, a \,e^{7} x^{9}+63 \sqrt {e \,x^{2}+d}\, c \,d^{4} e^{3} x^{5}+36 \sqrt {e \,x^{2}+d}\, c \,d^{3} e^{4} x^{7}+8 \sqrt {e \,x^{2}+d}\, c \,d^{2} e^{5} x^{9}-128 \sqrt {e}\, a \,d^{5} e^{2}-640 \sqrt {e}\, a \,d^{4} e^{3} x^{2}-1280 \sqrt {e}\, a \,d^{3} e^{4} x^{4}-1280 \sqrt {e}\, a \,d^{2} e^{5} x^{6}-640 \sqrt {e}\, a d \,e^{6} x^{8}-128 \sqrt {e}\, a \,e^{7} x^{10}-8 \sqrt {e}\, c \,d^{7}-40 \sqrt {e}\, c \,d^{6} e \,x^{2}-80 \sqrt {e}\, c \,d^{5} e^{2} x^{4}-80 \sqrt {e}\, c \,d^{4} e^{3} x^{6}-40 \sqrt {e}\, c \,d^{3} e^{4} x^{8}-8 \sqrt {e}\, c \,d^{2} e^{5} x^{10}}{315 d^{5} e^{3} \left (e^{5} x^{10}+5 d \,e^{4} x^{8}+10 d^{2} e^{3} x^{6}+10 d^{3} e^{2} x^{4}+5 d^{4} e \,x^{2}+d^{5}\right )} \] Input:
int((c*x^4+a)/(e*x^2+d)^(11/2),x)
Output:
(315*sqrt(d + e*x**2)*a*d**4*e**3*x + 840*sqrt(d + e*x**2)*a*d**3*e**4*x** 3 + 1008*sqrt(d + e*x**2)*a*d**2*e**5*x**5 + 576*sqrt(d + e*x**2)*a*d*e**6 *x**7 + 128*sqrt(d + e*x**2)*a*e**7*x**9 + 63*sqrt(d + e*x**2)*c*d**4*e**3 *x**5 + 36*sqrt(d + e*x**2)*c*d**3*e**4*x**7 + 8*sqrt(d + e*x**2)*c*d**2*e **5*x**9 - 128*sqrt(e)*a*d**5*e**2 - 640*sqrt(e)*a*d**4*e**3*x**2 - 1280*s qrt(e)*a*d**3*e**4*x**4 - 1280*sqrt(e)*a*d**2*e**5*x**6 - 640*sqrt(e)*a*d* e**6*x**8 - 128*sqrt(e)*a*e**7*x**10 - 8*sqrt(e)*c*d**7 - 40*sqrt(e)*c*d** 6*e*x**2 - 80*sqrt(e)*c*d**5*e**2*x**4 - 80*sqrt(e)*c*d**4*e**3*x**6 - 40* sqrt(e)*c*d**3*e**4*x**8 - 8*sqrt(e)*c*d**2*e**5*x**10)/(315*d**5*e**3*(d* *5 + 5*d**4*e*x**2 + 10*d**3*e**2*x**4 + 10*d**2*e**3*x**6 + 5*d*e**4*x**8 + e**5*x**10))