\(\int \frac {(a+c x^4)^2}{(d+e x^2)^{13/2}} \, dx\) [381]

Optimal result

Integrand size = 21, antiderivative size = 278 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {\left (c d^2+a e^2\right )^2 x}{11 d e^4 \left (d+e x^2\right )^{11/2}}-\frac {2 \left (17 c d^2-5 a e^2\right ) \left (c d^2+a e^2\right ) x}{99 d^2 e^4 \left (d+e x^2\right )^{9/2}}+\frac {2 \left (161 c^2 d^4+3 a c d^2 e^2+40 a^2 e^4\right ) x}{693 d^3 e^4 \left (d+e x^2\right )^{7/2}}-\frac {4 \left (70 c^2 d^4-3 a c d^2 e^2-40 a^2 e^4\right ) x}{1155 d^4 e^4 \left (d+e x^2\right )^{5/2}}+\frac {\left (35 c^2 d^4+48 a c d^2 e^2+640 a^2 e^4\right ) x}{3465 d^5 e^4 \left (d+e x^2\right )^{3/2}}+\frac {2 \left (35 c^2 d^4+48 a c d^2 e^2+640 a^2 e^4\right ) x}{3465 d^6 e^4 \sqrt {d+e x^2}} \] Output:

1/11*(a*e^2+c*d^2)^2*x/d/e^4/(e*x^2+d)^(11/2)-2/99*(-5*a*e^2+17*c*d^2)*(a* 
e^2+c*d^2)*x/d^2/e^4/(e*x^2+d)^(9/2)+2/693*(40*a^2*e^4+3*a*c*d^2*e^2+161*c 
^2*d^4)*x/d^3/e^4/(e*x^2+d)^(7/2)-4/1155*(-40*a^2*e^4-3*a*c*d^2*e^2+70*c^2 
*d^4)*x/d^4/e^4/(e*x^2+d)^(5/2)+1/3465*(640*a^2*e^4+48*a*c*d^2*e^2+35*c^2* 
d^4)*x/d^5/e^4/(e*x^2+d)^(3/2)+2/3465*(640*a^2*e^4+48*a*c*d^2*e^2+35*c^2*d 
^4)*x/d^6/e^4/(e*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.51 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {35 c^2 d^4 x^9 \left (11 d+2 e x^2\right )+6 a c d^2 x^5 \left (231 d^3+198 d^2 e x^2+88 d e^2 x^4+16 e^3 x^6\right )+5 a^2 \left (693 d^5 x+2310 d^4 e x^3+3696 d^3 e^2 x^5+3168 d^2 e^3 x^7+1408 d e^4 x^9+256 e^5 x^{11}\right )}{3465 d^6 \left (d+e x^2\right )^{11/2}} \] Input:

Integrate[(a + c*x^4)^2/(d + e*x^2)^(13/2),x]
 

Output:

(35*c^2*d^4*x^9*(11*d + 2*e*x^2) + 6*a*c*d^2*x^5*(231*d^3 + 198*d^2*e*x^2 
+ 88*d*e^2*x^4 + 16*e^3*x^6) + 5*a^2*(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))/(3465*d^6*( 
d + e*x^2)^(11/2))
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {1470, 2334, 2090, 1587, 9, 27, 25, 362, 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.

\(\displaystyle \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx\)

\(\Big \downarrow \) 1470

\(\displaystyle \frac {\int \frac {x^2 \left (10 e a^2+d \left (c^2 x^6+2 a c x^2\right )\right )}{\left (e x^2+d\right )^{13/2}}dx}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 2334

\(\displaystyle \frac {\frac {\int \frac {x^4 \left (3 c \left (c x^4+2 a\right ) d^2+80 a^2 e^2\right )}{\left (e x^2+d\right )^{13/2}}dx}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 2090

\(\displaystyle \frac {\frac {\int \frac {x^4 \left (3 c^2 d^2 x^4+2 a \left (3 c d^2+40 a e^2\right )\right )}{\left (e x^2+d\right )^{13/2}}dx}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 1587

\(\displaystyle \frac {\frac {\frac {x^5 \left (80 a^2 e^2+6 a c d^2+\frac {3 c^2 d^4}{e^2}\right )}{11 d \left (d+e x^2\right )^{11/2}}-\frac {\int -\frac {3 x^3 \left (\frac {11 c^2 d^3 x^3}{e}+\left (-\frac {5 c^2 d^4}{e^2}+12 a c d^2+160 a^2 e^2\right ) x\right )}{\left (e x^2+d\right )^{11/2}}dx}{11 d}}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 9

\(\displaystyle \frac {\frac {\frac {x^5 \left (80 a^2 e^2+6 a c d^2+\frac {3 c^2 d^4}{e^2}\right )}{11 d \left (d+e x^2\right )^{11/2}}-\frac {\int -\frac {3 x^4 \left (11 c^2 x^2 d^3+e \left (-\frac {5 c^2 d^4}{e^2}+12 a c d^2+160 a^2 e^2\right )\right )}{e \left (e x^2+d\right )^{11/2}}dx}{11 d}}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int -\frac {x^4 \left (\frac {5 c^2 d^4}{e}-11 c^2 x^2 d^3-12 a c e d^2-160 a^2 e^3\right )}{\left (e x^2+d\right )^{11/2}}dx}{11 d e}+\frac {x^5 \left (80 a^2 e^2+6 a c d^2+\frac {3 c^2 d^4}{e^2}\right )}{11 d \left (d+e x^2\right )^{11/2}}}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {x^5 \left (80 a^2 e^2+6 a c d^2+\frac {3 c^2 d^4}{e^2}\right )}{11 d \left (d+e x^2\right )^{11/2}}-\frac {3 \int \frac {x^4 \left (\frac {5 c^2 d^4}{e}-11 c^2 x^2 d^3-12 a c e d^2-160 a^2 e^3\right )}{\left (e x^2+d\right )^{11/2}}dx}{11 d e}}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 362

\(\displaystyle \frac {\frac {\frac {x^5 \left (80 a^2 e^2+6 a c d^2+\frac {3 c^2 d^4}{e^2}\right )}{11 d \left (d+e x^2\right )^{11/2}}-\frac {3 \left (\frac {4 x^5 \left (-40 a^2 e^4-3 a c d^2 e^2+4 c^2 d^4\right )}{9 d e \left (d+e x^2\right )^{9/2}}-\frac {\left (640 a^2 e^4+48 a c d^2 e^2+35 c^2 d^4\right ) \int \frac {x^4}{\left (e x^2+d\right )^{9/2}}dx}{9 d e}\right )}{11 d e}}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 245

\(\displaystyle \frac {\frac {\frac {x^5 \left (80 a^2 e^2+6 a c d^2+\frac {3 c^2 d^4}{e^2}\right )}{11 d \left (d+e x^2\right )^{11/2}}-\frac {3 \left (\frac {4 x^5 \left (-40 a^2 e^4-3 a c d^2 e^2+4 c^2 d^4\right )}{9 d e \left (d+e x^2\right )^{9/2}}-\frac {\left (640 a^2 e^4+48 a c d^2 e^2+35 c^2 d^4\right ) \left (\frac {2 e \int \frac {x^6}{\left (e x^2+d\right )^{9/2}}dx}{5 d}+\frac {x^5}{5 d \left (d+e x^2\right )^{7/2}}\right )}{9 d e}\right )}{11 d e}}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\frac {\frac {x^5 \left (80 a^2 e^2+6 a c d^2+\frac {3 c^2 d^4}{e^2}\right )}{11 d \left (d+e x^2\right )^{11/2}}-\frac {3 \left (\frac {4 x^5 \left (-40 a^2 e^4-3 a c d^2 e^2+4 c^2 d^4\right )}{9 d e \left (d+e x^2\right )^{9/2}}-\frac {\left (\frac {2 e x^7}{35 d^2 \left (d+e x^2\right )^{7/2}}+\frac {x^5}{5 d \left (d+e x^2\right )^{7/2}}\right ) \left (640 a^2 e^4+48 a c d^2 e^2+35 c^2 d^4\right )}{9 d e}\right )}{11 d e}}{3 d}+\frac {10 a^2 e x^3}{3 d \left (d+e x^2\right )^{11/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{11/2}}\)

Input:

Int[(a + c*x^4)^2/(d + e*x^2)^(13/2),x]
 

Output:

(a^2*x)/(d*(d + e*x^2)^(11/2)) + ((10*a^2*e*x^3)/(3*d*(d + e*x^2)^(11/2)) 
+ (((6*a*c*d^2 + (3*c^2*d^4)/e^2 + 80*a^2*e^2)*x^5)/(11*d*(d + e*x^2)^(11/ 
2)) - (3*((4*(4*c^2*d^4 - 3*a*c*d^2*e^2 - 40*a^2*e^4)*x^5)/(9*d*e*(d + e*x 
^2)^(9/2)) - ((35*c^2*d^4 + 48*a*c*d^2*e^2 + 640*a^2*e^4)*(x^5/(5*d*(d + e 
*x^2)^(7/2)) + (2*e*x^7)/(35*d^2*(d + e*x^2)^(7/2))))/(9*d*e)))/(11*d*e))/ 
(3*d))/d
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 242

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]
 

rule 245

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]
 

rule 362

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)]))
 

rule 1470

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]
 

rule 1587

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_ 
.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + c*x^4)^p, d + e*x^2, x] 
, R = Coeff[PolynomialRemainder[(a + c*x^4)^p, d + e*x^2, x], x, 0]}, Simp[ 
(-R)*(f*x)^(m + 1)*((d + e*x^2)^(q + 1)/(2*d*f*(q + 1))), x] + Simp[f/(2*d* 
(q + 1))   Int[(f*x)^(m - 1)*(d + e*x^2)^(q + 1)*ExpandToSum[2*d*(q + 1)*x* 
Qx + R*(m + 2*q + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, f}, x] && IGtQ[p, 
 0] && LtQ[q, -1] && GtQ[m, 0]
 

rule 2090

Int[(u_)^(p_.)*((f_.)*(x_))^(m_.)*(z_)^(q_.), x_Symbol] :> Int[(f*x)^m*Expa 
ndToSum[z, x]^q*ExpandToSum[u, x]^p, x] /; FreeQ[{f, m, p, q}, x] && Binomi 
alQ[z, x] && BinomialQ[u, x] &&  !(BinomialMatchQ[z, x] && BinomialMatchQ[u 
, x])
 

rule 2334

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{A = Coef 
f[Pq, x, 0], Q = PolynomialQuotient[Pq - Coeff[Pq, x, 0], x^2, x]}, Simp[A* 
x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1))   Int[ 
x^(m + 2)*(a + b*x^2)^p*(a*(m + 1)*Q - A*b*(m + 2*(p + 1) + 1)), x], x]] /; 
 FreeQ[{a, b}, x] && PolyQ[Pq, x^2] && IntegerQ[m/2] && ILtQ[(m + 1)/2 + p, 
 0] && LtQ[m + Expon[Pq, x] + 2*p + 1, 0]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.47

method	result	size
pseudoelliptic	\(\frac {x \left (\left (\frac {1}{9} c^{2} x^{8}+\frac {2}{5} a c \,x^{4}+a^{2}\right ) d^{5}+\frac {10 x^{2} \left (\frac {1}{165} c^{2} x^{8}+\frac {18}{175} a c \,x^{4}+a^{2}\right ) e \,d^{4}}{3}+\frac {16 x^{4} \left (\frac {c \,x^{4}}{35}+a \right ) a \,e^{2} d^{3}}{3}+\frac {32 x^{6} a \left (\frac {c \,x^{4}}{165}+a \right ) e^{3} d^{2}}{7}+\frac {128 a^{2} d \,e^{4} x^{8}}{63}+\frac {256 a^{2} e^{5} x^{10}}{693}\right )}{\left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\)	\(130\)
gosper	\(\frac {x \left (1280 a^{2} e^{5} x^{10}+96 a c \,d^{2} e^{3} x^{10}+70 c^{2} d^{4} e \,x^{10}+7040 a^{2} d \,e^{4} x^{8}+528 a c \,d^{3} e^{2} x^{8}+385 c^{2} d^{5} x^{8}+15840 a^{2} d^{2} e^{3} x^{6}+1188 a c \,d^{4} e \,x^{6}+18480 a^{2} d^{3} e^{2} x^{4}+1386 a c \,d^{5} x^{4}+11550 a^{2} e \,x^{2} d^{4}+3465 a^{2} d^{5}\right )}{3465 \left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\)	\(158\)
trager	\(\frac {x \left (1280 a^{2} e^{5} x^{10}+96 a c \,d^{2} e^{3} x^{10}+70 c^{2} d^{4} e \,x^{10}+7040 a^{2} d \,e^{4} x^{8}+528 a c \,d^{3} e^{2} x^{8}+385 c^{2} d^{5} x^{8}+15840 a^{2} d^{2} e^{3} x^{6}+1188 a c \,d^{4} e \,x^{6}+18480 a^{2} d^{3} e^{2} x^{4}+1386 a c \,d^{5} x^{4}+11550 a^{2} e \,x^{2} d^{4}+3465 a^{2} d^{5}\right )}{3465 \left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\)	\(158\)
orering	\(\frac {x \left (1280 a^{2} e^{5} x^{10}+96 a c \,d^{2} e^{3} x^{10}+70 c^{2} d^{4} e \,x^{10}+7040 a^{2} d \,e^{4} x^{8}+528 a c \,d^{3} e^{2} x^{8}+385 c^{2} d^{5} x^{8}+15840 a^{2} d^{2} e^{3} x^{6}+1188 a c \,d^{4} e \,x^{6}+18480 a^{2} d^{3} e^{2} x^{4}+1386 a c \,d^{5} x^{4}+11550 a^{2} e \,x^{2} d^{4}+3465 a^{2} d^{5}\right )}{3465 \left (e \,x^{2}+d \right )^{\frac {11}{2}} d^{6}}\)	\(158\)
default	\(a^{2} \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )+c^{2} \left (-\frac {x^{7}}{4 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {7 d \left (-\frac {x^{5}}{6 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {5 d \left (-\frac {x^{3}}{8 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {3 d \left (-\frac {x}{10 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {d \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )}{10 e}\right )}{8 e}\right )}{6 e}\right )}{4 e}\right )+2 a c \left (-\frac {x^{3}}{8 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {3 d \left (-\frac {x}{10 e \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {d \left (\frac {x}{11 d \left (e \,x^{2}+d \right )^{\frac {11}{2}}}+\frac {\frac {10 x}{99 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {10 \left (\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}\right )}{11 d}}{d}\right )}{10 e}\right )}{8 e}\right )\)	\(499\)

Input:

int((c*x^4+a)^2/(e*x^2+d)^(13/2),x,method=_RETURNVERBOSE)
 

Output:

x/(e*x^2+d)^(11/2)*((1/9*c^2*x^8+2/5*a*c*x^4+a^2)*d^5+10/3*x^2*(1/165*c^2* 
x^8+18/175*a*c*x^4+a^2)*e*d^4+16/3*x^4*(1/35*c*x^4+a)*a*e^2*d^3+32/7*x^6*a 
*(1/165*c*x^4+a)*e^3*d^2+128/63*a^2*d*e^4*x^8+256/693*a^2*e^5*x^10)/d^6
 

Fricas [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {{\left (2 \, {\left (35 \, c^{2} d^{4} e + 48 \, a c d^{2} e^{3} + 640 \, a^{2} e^{5}\right )} x^{11} + 11550 \, a^{2} d^{4} e x^{3} + 11 \, {\left (35 \, c^{2} d^{5} + 48 \, a c d^{3} e^{2} + 640 \, a^{2} d e^{4}\right )} x^{9} + 3465 \, a^{2} d^{5} x + 396 \, {\left (3 \, a c d^{4} e + 40 \, a^{2} d^{2} e^{3}\right )} x^{7} + 462 \, {\left (3 \, a c d^{5} + 40 \, a^{2} d^{3} e^{2}\right )} x^{5}\right )} \sqrt {e x^{2} + d}}{3465 \, {\left (d^{6} e^{6} x^{12} + 6 \, d^{7} e^{5} x^{10} + 15 \, d^{8} e^{4} x^{8} + 20 \, d^{9} e^{3} x^{6} + 15 \, d^{10} e^{2} x^{4} + 6 \, d^{11} e x^{2} + d^{12}\right )}} \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(13/2),x, algorithm="fricas")
 

Output:

1/3465*(2*(35*c^2*d^4*e + 48*a*c*d^2*e^3 + 640*a^2*e^5)*x^11 + 11550*a^2*d 
^4*e*x^3 + 11*(35*c^2*d^5 + 48*a*c*d^3*e^2 + 640*a^2*d*e^4)*x^9 + 3465*a^2 
*d^5*x + 396*(3*a*c*d^4*e + 40*a^2*d^2*e^3)*x^7 + 462*(3*a*c*d^5 + 40*a^2* 
d^3*e^2)*x^5)*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)
 

Sympy [F]

\[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx=\int \frac {\left (a + c x^{4}\right )^{2}}{\left (d + e x^{2}\right )^{\frac {13}{2}}}\, dx \] Input:

integrate((c*x**4+a)**2/(e*x**2+d)**(13/2),x)
 

Output:

Integral((a + c*x**4)**2/(d + e*x**2)**(13/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx=-\frac {c^{2} x^{7}}{4 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e} - \frac {7 \, c^{2} d x^{5}}{24 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e^{2}} - \frac {35 \, c^{2} d^{2} x^{3}}{192 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e^{3}} - \frac {a c x^{3}}{4 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e} + \frac {256 \, a^{2} x}{693 \, \sqrt {e x^{2} + d} d^{6}} + \frac {128 \, a^{2} x}{693 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{5}} + \frac {32 \, a^{2} x}{231 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{4}} + \frac {80 \, a^{2} x}{693 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{3}} + \frac {10 \, a^{2} x}{99 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d^{2}} + \frac {a^{2} x}{11 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} d} + \frac {c^{2} x}{132 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} e^{4}} + \frac {2 \, c^{2} x}{99 \, \sqrt {e x^{2} + d} d^{2} e^{4}} + \frac {c^{2} x}{99 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d e^{4}} + \frac {5 \, c^{2} d x}{792 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{4}} + \frac {35 \, c^{2} d^{2} x}{6336 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{4}} - \frac {35 \, c^{2} d^{3} x}{704 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e^{4}} + \frac {a c x}{132 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} + \frac {32 \, a c x}{1155 \, \sqrt {e x^{2} + d} d^{4} e^{2}} + \frac {16 \, a c x}{1155 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{3} e^{2}} + \frac {4 \, a c x}{385 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{2} e^{2}} + \frac {2 \, a c x}{231 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d e^{2}} - \frac {3 \, a c d x}{44 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}} e^{2}} \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(13/2),x, algorithm="maxima")
 

Output:

-1/4*c^2*x^7/((e*x^2 + d)^(11/2)*e) - 7/24*c^2*d*x^5/((e*x^2 + d)^(11/2)*e 
^2) - 35/192*c^2*d^2*x^3/((e*x^2 + d)^(11/2)*e^3) - 1/4*a*c*x^3/((e*x^2 + 
d)^(11/2)*e) + 256/693*a^2*x/(sqrt(e*x^2 + d)*d^6) + 128/693*a^2*x/((e*x^2 
 + d)^(3/2)*d^5) + 32/231*a^2*x/((e*x^2 + d)^(5/2)*d^4) + 80/693*a^2*x/((e 
*x^2 + d)^(7/2)*d^3) + 10/99*a^2*x/((e*x^2 + d)^(9/2)*d^2) + 1/11*a^2*x/(( 
e*x^2 + d)^(11/2)*d) + 1/132*c^2*x/((e*x^2 + d)^(5/2)*e^4) + 2/99*c^2*x/(s 
qrt(e*x^2 + d)*d^2*e^4) + 1/99*c^2*x/((e*x^2 + d)^(3/2)*d*e^4) + 5/792*c^2 
*d*x/((e*x^2 + d)^(7/2)*e^4) + 35/6336*c^2*d^2*x/((e*x^2 + d)^(9/2)*e^4) - 
 35/704*c^2*d^3*x/((e*x^2 + d)^(11/2)*e^4) + 1/132*a*c*x/((e*x^2 + d)^(9/2 
)*e^2) + 32/1155*a*c*x/(sqrt(e*x^2 + d)*d^4*e^2) + 16/1155*a*c*x/((e*x^2 + 
 d)^(3/2)*d^3*e^2) + 4/385*a*c*x/((e*x^2 + d)^(5/2)*d^2*e^2) + 2/231*a*c*x 
/((e*x^2 + d)^(7/2)*d*e^2) - 3/44*a*c*d*x/((e*x^2 + d)^(11/2)*e^2)
 

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.68 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {{\left ({\left ({\left ({\left (x^{2} {\left (\frac {2 \, {\left (35 \, c^{2} d^{4} e^{6} + 48 \, a c d^{2} e^{8} + 640 \, a^{2} e^{10}\right )} x^{2}}{d^{6} e^{5}} + \frac {11 \, {\left (35 \, c^{2} d^{5} e^{5} + 48 \, a c d^{3} e^{7} + 640 \, a^{2} d e^{9}\right )}}{d^{6} e^{5}}\right )} + \frac {396 \, {\left (3 \, a c d^{4} e^{6} + 40 \, a^{2} d^{2} e^{8}\right )}}{d^{6} e^{5}}\right )} x^{2} + \frac {462 \, {\left (3 \, a c d^{5} e^{5} + 40 \, a^{2} d^{3} e^{7}\right )}}{d^{6} e^{5}}\right )} x^{2} + \frac {11550 \, a^{2} e}{d^{2}}\right )} x^{2} + \frac {3465 \, a^{2}}{d}\right )} x}{3465 \, {\left (e x^{2} + d\right )}^{\frac {11}{2}}} \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(13/2),x, algorithm="giac")
 

Output:

1/3465*((((x^2*(2*(35*c^2*d^4*e^6 + 48*a*c*d^2*e^8 + 640*a^2*e^10)*x^2/(d^ 
6*e^5) + 11*(35*c^2*d^5*e^5 + 48*a*c*d^3*e^7 + 640*a^2*d*e^9)/(d^6*e^5)) + 
 396*(3*a*c*d^4*e^6 + 40*a^2*d^2*e^8)/(d^6*e^5))*x^2 + 462*(3*a*c*d^5*e^5 
+ 40*a^2*d^3*e^7)/(d^6*e^5))*x^2 + 11550*a^2*e/d^2)*x^2 + 3465*a^2/d)*x/(e 
*x^2 + d)^(11/2)
 

Mupad [B] (verification not implemented)

Time = 17.70 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {x\,\left (\frac {a^2}{11\,d}+\frac {d^2\,\left (\frac {c^2\,d}{11\,e^2}+\frac {2\,a\,c}{11\,d}\right )}{e^2}\right )}{{\left (e\,x^2+d\right )}^{11/2}}-\frac {x\,\left (\frac {-10\,a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}{99\,d^2\,e^4}+\frac {d\,\left (\frac {2\,c^2\,d}{9\,e^3}+\frac {c\,\left (c\,d^2+2\,a\,e^2\right )}{9\,d\,e^3}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{9/2}}-\frac {x\,\left (\frac {c^2}{5\,e^4}-\frac {160\,a^2\,e^4+12\,a\,c\,d^2\,e^2-49\,c^2\,d^4}{1155\,d^4\,e^4}\right )}{{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (\frac {3\,c^2\,d}{7\,e^4}+\frac {80\,a^2\,e^4+6\,a\,c\,d^2\,e^2+25\,c^2\,d^4}{693\,d^3\,e^4}\right )}{{\left (e\,x^2+d\right )}^{7/2}}+\frac {x\,\left (640\,a^2\,e^4+48\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{3465\,d^5\,e^4\,{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (1280\,a^2\,e^4+96\,a\,c\,d^2\,e^2+70\,c^2\,d^4\right )}{3465\,d^6\,e^4\,\sqrt {e\,x^2+d}} \] Input:

int((a + c*x^4)^2/(d + e*x^2)^(13/2),x)
 

Output:

(x*(a^2/(11*d) + (d^2*((c^2*d)/(11*e^2) + (2*a*c)/(11*d)))/e^2))/(d + e*x^ 
2)^(11/2) - (x*((c^2*d^4 - 10*a^2*e^4 + 2*a*c*d^2*e^2)/(99*d^2*e^4) + (d*( 
(2*c^2*d)/(9*e^3) + (c*(2*a*e^2 + c*d^2))/(9*d*e^3)))/e))/(d + e*x^2)^(9/2 
) - (x*(c^2/(5*e^4) - (160*a^2*e^4 - 49*c^2*d^4 + 12*a*c*d^2*e^2)/(1155*d^ 
4*e^4)))/(d + e*x^2)^(5/2) + (x*((3*c^2*d)/(7*e^4) + (80*a^2*e^4 + 25*c^2* 
d^4 + 6*a*c*d^2*e^2)/(693*d^3*e^4)))/(d + e*x^2)^(7/2) + (x*(640*a^2*e^4 + 
 35*c^2*d^4 + 48*a*c*d^2*e^2))/(3465*d^5*e^4*(d + e*x^2)^(3/2)) + (x*(1280 
*a^2*e^4 + 70*c^2*d^4 + 96*a*c*d^2*e^2))/(3465*d^6*e^4*(d + e*x^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 636, normalized size of antiderivative = 2.29 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{13/2}} \, dx=\frac {-70 \sqrt {e}\, c^{2} d^{10}+3465 \sqrt {e \,x^{2}+d}\, a^{2} d^{5} e^{5} x +11550 \sqrt {e \,x^{2}+d}\, a^{2} d^{4} e^{6} x^{3}+18480 \sqrt {e \,x^{2}+d}\, a^{2} d^{3} e^{7} x^{5}+15840 \sqrt {e \,x^{2}+d}\, a^{2} d^{2} e^{8} x^{7}+7040 \sqrt {e \,x^{2}+d}\, a^{2} d \,e^{9} x^{9}+385 \sqrt {e \,x^{2}+d}\, c^{2} d^{5} e^{5} x^{9}+70 \sqrt {e \,x^{2}+d}\, c^{2} d^{4} e^{6} x^{11}-7680 \sqrt {e}\, a^{2} d^{5} e^{5} x^{2}-19200 \sqrt {e}\, a^{2} d^{4} e^{6} x^{4}-25600 \sqrt {e}\, a^{2} d^{3} e^{7} x^{6}-19200 \sqrt {e}\, a^{2} d^{2} e^{8} x^{8}-7680 \sqrt {e}\, a^{2} d \,e^{9} x^{10}-96 \sqrt {e}\, a c \,d^{8} e^{2}-420 \sqrt {e}\, c^{2} d^{9} e \,x^{2}-1050 \sqrt {e}\, c^{2} d^{8} e^{2} x^{4}-1400 \sqrt {e}\, c^{2} d^{7} e^{3} x^{6}-1050 \sqrt {e}\, c^{2} d^{6} e^{4} x^{8}-420 \sqrt {e}\, c^{2} d^{5} e^{5} x^{10}-70 \sqrt {e}\, c^{2} d^{4} e^{6} x^{12}+1280 \sqrt {e \,x^{2}+d}\, a^{2} e^{10} x^{11}-1280 \sqrt {e}\, a^{2} d^{6} e^{4}-1280 \sqrt {e}\, a^{2} e^{10} x^{12}-1920 \sqrt {e}\, a c \,d^{5} e^{5} x^{6}-1440 \sqrt {e}\, a c \,d^{4} e^{6} x^{8}-576 \sqrt {e}\, a c \,d^{3} e^{7} x^{10}-96 \sqrt {e}\, a c \,d^{2} e^{8} x^{12}+1386 \sqrt {e \,x^{2}+d}\, a c \,d^{5} e^{5} x^{5}+1188 \sqrt {e \,x^{2}+d}\, a c \,d^{4} e^{6} x^{7}+528 \sqrt {e \,x^{2}+d}\, a c \,d^{3} e^{7} x^{9}+96 \sqrt {e \,x^{2}+d}\, a c \,d^{2} e^{8} x^{11}-576 \sqrt {e}\, a c \,d^{7} e^{3} x^{2}-1440 \sqrt {e}\, a c \,d^{6} e^{4} x^{4}}{3465 d^{6} e^{5} \left (e^{6} x^{12}+6 d \,e^{5} x^{10}+15 d^{2} e^{4} x^{8}+20 d^{3} e^{3} x^{6}+15 d^{4} e^{2} x^{4}+6 d^{5} e \,x^{2}+d^{6}\right )} \] Input:

int((c*x^4+a)^2/(e*x^2+d)^(13/2),x)
 

Output:

(3465*sqrt(d + e*x**2)*a**2*d**5*e**5*x + 11550*sqrt(d + e*x**2)*a**2*d**4 
*e**6*x**3 + 18480*sqrt(d + e*x**2)*a**2*d**3*e**7*x**5 + 15840*sqrt(d + e 
*x**2)*a**2*d**2*e**8*x**7 + 7040*sqrt(d + e*x**2)*a**2*d*e**9*x**9 + 1280 
*sqrt(d + e*x**2)*a**2*e**10*x**11 + 1386*sqrt(d + e*x**2)*a*c*d**5*e**5*x 
**5 + 1188*sqrt(d + e*x**2)*a*c*d**4*e**6*x**7 + 528*sqrt(d + e*x**2)*a*c* 
d**3*e**7*x**9 + 96*sqrt(d + e*x**2)*a*c*d**2*e**8*x**11 + 385*sqrt(d + e* 
x**2)*c**2*d**5*e**5*x**9 + 70*sqrt(d + e*x**2)*c**2*d**4*e**6*x**11 - 128 
0*sqrt(e)*a**2*d**6*e**4 - 7680*sqrt(e)*a**2*d**5*e**5*x**2 - 19200*sqrt(e 
)*a**2*d**4*e**6*x**4 - 25600*sqrt(e)*a**2*d**3*e**7*x**6 - 19200*sqrt(e)* 
a**2*d**2*e**8*x**8 - 7680*sqrt(e)*a**2*d*e**9*x**10 - 1280*sqrt(e)*a**2*e 
**10*x**12 - 96*sqrt(e)*a*c*d**8*e**2 - 576*sqrt(e)*a*c*d**7*e**3*x**2 - 1 
440*sqrt(e)*a*c*d**6*e**4*x**4 - 1920*sqrt(e)*a*c*d**5*e**5*x**6 - 1440*sq 
rt(e)*a*c*d**4*e**6*x**8 - 576*sqrt(e)*a*c*d**3*e**7*x**10 - 96*sqrt(e)*a* 
c*d**2*e**8*x**12 - 70*sqrt(e)*c**2*d**10 - 420*sqrt(e)*c**2*d**9*e*x**2 - 
 1050*sqrt(e)*c**2*d**8*e**2*x**4 - 1400*sqrt(e)*c**2*d**7*e**3*x**6 - 105 
0*sqrt(e)*c**2*d**6*e**4*x**8 - 420*sqrt(e)*c**2*d**5*e**5*x**10 - 70*sqrt 
(e)*c**2*d**4*e**6*x**12)/(3465*d**6*e**5*(d**6 + 6*d**5*e*x**2 + 15*d**4* 
e**2*x**4 + 20*d**3*e**3*x**6 + 15*d**2*e**4*x**8 + 6*d*e**5*x**10 + e**6* 
x**12))