\(\int \frac {(a+c x^4)^2}{(d+e x^2)^{11/2}} \, dx\) [380]

Optimal result

Integrand size = 21, antiderivative size = 228 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {\left (c d^2+a e^2\right )^2 x}{9 d e^4 \left (d+e x^2\right )^{9/2}}-\frac {4 \left (7 c d^2-2 a e^2\right ) \left (c d^2+a e^2\right ) x}{63 d^2 e^4 \left (d+e x^2\right )^{7/2}}+\frac {2 \left (35 c^2 d^4+a c d^2 e^2+8 a^2 e^4\right ) x}{105 d^3 e^4 \left (d+e x^2\right )^{5/2}}-\frac {4 \left (35 c^2 d^4-2 a c d^2 e^2-16 a^2 e^4\right ) x}{315 d^4 e^4 \left (d+e x^2\right )^{3/2}}+\frac {\left (35 c^2 d^4+16 a c d^2 e^2+128 a^2 e^4\right ) x}{315 d^5 e^4 \sqrt {d+e x^2}} \] Output:

1/9*(a*e^2+c*d^2)^2*x/d/e^4/(e*x^2+d)^(9/2)-4/63*(-2*a*e^2+7*c*d^2)*(a*e^2 
+c*d^2)*x/d^2/e^4/(e*x^2+d)^(7/2)+2/105*(8*a^2*e^4+a*c*d^2*e^2+35*c^2*d^4) 
*x/d^3/e^4/(e*x^2+d)^(5/2)-4/315*(-16*a^2*e^4-2*a*c*d^2*e^2+35*c^2*d^4)*x/ 
d^4/e^4/(e*x^2+d)^(3/2)+1/315*(128*a^2*e^4+16*a*c*d^2*e^2+35*c^2*d^4)*x/d^ 
5/e^4/(e*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.48 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {35 c^2 d^4 x^9+2 a c d^2 x^5 \left (63 d^2+36 d e x^2+8 e^2 x^4\right )+a^2 \left (315 d^4 x+840 d^3 e x^3+1008 d^2 e^2 x^5+576 d e^3 x^7+128 e^4 x^9\right )}{315 d^5 \left (d+e x^2\right )^{9/2}} \] Input:

Integrate[(a + c*x^4)^2/(d + e*x^2)^(11/2),x]
 

Output:

(35*c^2*d^4*x^9 + 2*a*c*d^2*x^5*(63*d^2 + 36*d*e*x^2 + 8*e^2*x^4) + a^2*(3 
15*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))
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.94, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {1470, 2334, 27, 2090, 1587, 9, 25, 25, 27, 362, 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.

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

\(\Big \downarrow \) 1470

\(\displaystyle \frac {\int \frac {x^2 \left (8 e a^2+d \left (c^2 x^6+2 a c x^2\right )\right )}{\left (e x^2+d\right )^{11/2}}dx}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 2334

\(\displaystyle \frac {\frac {\int \frac {3 x^4 \left (c \left (c x^4+2 a\right ) d^2+16 a^2 e^2\right )}{\left (e x^2+d\right )^{11/2}}dx}{3 d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {x^4 \left (c \left (c x^4+2 a\right ) d^2+16 a^2 e^2\right )}{\left (e x^2+d\right )^{11/2}}dx}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 2090

\(\displaystyle \frac {\frac {\int \frac {x^4 \left (c^2 d^2 x^4+2 a \left (c d^2+8 a e^2\right )\right )}{\left (e x^2+d\right )^{11/2}}dx}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 1587

\(\displaystyle \frac {\frac {\frac {x^5 \left (16 a^2 e^2+2 a c d^2+\frac {c^2 d^4}{e^2}\right )}{9 d \left (d+e x^2\right )^{9/2}}-\frac {\int -\frac {x^3 \left (\frac {9 c^2 d^3 x^3}{e}+\left (-\frac {5 c^2 d^4}{e^2}+8 a c d^2+64 a^2 e^2\right ) x\right )}{\left (e x^2+d\right )^{9/2}}dx}{9 d}}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 9

\(\displaystyle \frac {\frac {\frac {x^5 \left (16 a^2 e^2+2 a c d^2+\frac {c^2 d^4}{e^2}\right )}{9 d \left (d+e x^2\right )^{9/2}}-\frac {\int -\frac {x^4 \left (9 c^2 x^2 d^3+e \left (-\frac {5 c^2 d^4}{e^2}+8 a c d^2+64 a^2 e^2\right )\right )}{e \left (e x^2+d\right )^{9/2}}dx}{9 d}}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int -\frac {x^4 \left (\frac {5 c^2 d^4}{e}-9 c^2 x^2 d^3-8 a c e d^2-64 a^2 e^3\right )}{e \left (e x^2+d\right )^{9/2}}dx}{9 d}+\frac {x^5 \left (16 a^2 e^2+2 a c d^2+\frac {c^2 d^4}{e^2}\right )}{9 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {x^5 \left (16 a^2 e^2+2 a c d^2+\frac {c^2 d^4}{e^2}\right )}{9 d \left (d+e x^2\right )^{9/2}}-\frac {\int \frac {x^4 \left (\frac {5 c^2 d^4}{e}-9 c^2 x^2 d^3-8 a c e d^2-64 a^2 e^3\right )}{e \left (e x^2+d\right )^{9/2}}dx}{9 d}}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {x^5 \left (16 a^2 e^2+2 a c d^2+\frac {c^2 d^4}{e^2}\right )}{9 d \left (d+e x^2\right )^{9/2}}-\frac {\int \frac {x^4 \left (\frac {5 c^2 d^4}{e}-9 c^2 x^2 d^3-8 a c e d^2-64 a^2 e^3\right )}{\left (e x^2+d\right )^{9/2}}dx}{9 d e}}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 362

\(\displaystyle \frac {\frac {\frac {x^5 \left (16 a^2 e^2+2 a c d^2+\frac {c^2 d^4}{e^2}\right )}{9 d \left (d+e x^2\right )^{9/2}}-\frac {\frac {2 x^5 \left (-32 a^2 e^4-4 a c d^2 e^2+7 c^2 d^4\right )}{7 d e \left (d+e x^2\right )^{7/2}}-\frac {\left (128 a^2 e^4+16 a c d^2 e^2+35 c^2 d^4\right ) \int \frac {x^4}{\left (e x^2+d\right )^{7/2}}dx}{7 d e}}{9 d e}}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\frac {\frac {x^5 \left (16 a^2 e^2+2 a c d^2+\frac {c^2 d^4}{e^2}\right )}{9 d \left (d+e x^2\right )^{9/2}}-\frac {\frac {2 x^5 \left (-32 a^2 e^4-4 a c d^2 e^2+7 c^2 d^4\right )}{7 d e \left (d+e x^2\right )^{7/2}}-\frac {x^5 \left (128 a^2 e^4+16 a c d^2 e^2+35 c^2 d^4\right )}{35 d^2 e \left (d+e x^2\right )^{5/2}}}{9 d e}}{d}+\frac {8 a^2 e x^3}{3 d \left (d+e x^2\right )^{9/2}}}{d}+\frac {a^2 x}{d \left (d+e x^2\right )^{9/2}}\)

Input:

Int[(a + c*x^4)^2/(d + e*x^2)^(11/2),x]
 

Output:

(a^2*x)/(d*(d + e*x^2)^(9/2)) + ((8*a^2*e*x^3)/(3*d*(d + e*x^2)^(9/2)) + ( 
((2*a*c*d^2 + (c^2*d^4)/e^2 + 16*a^2*e^2)*x^5)/(9*d*(d + e*x^2)^(9/2)) - ( 
(2*(7*c^2*d^4 - 4*a*c*d^2*e^2 - 32*a^2*e^4)*x^5)/(7*d*e*(d + e*x^2)^(7/2)) 
 - ((35*c^2*d^4 + 16*a*c*d^2*e^2 + 128*a^2*e^4)*x^5)/(35*d^2*e*(d + e*x^2) 
^(5/2)))/(9*d*e))/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 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 = 0.18 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.44

method	result	size
pseudoelliptic	\(\frac {x \left (\left (\frac {1}{9} c^{2} x^{8}+\frac {2}{5} a c \,x^{4}+a^{2}\right ) d^{4}+\frac {8 x^{2} a \left (\frac {3 c \,x^{4}}{35}+a \right ) e \,d^{3}}{3}+\frac {16 x^{4} a \left (\frac {c \,x^{4}}{63}+a \right ) e^{2} d^{2}}{5}+\frac {64 a^{2} d \,e^{3} x^{6}}{35}+\frac {128 a^{2} e^{4} x^{8}}{315}\right )}{\left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\)	\(100\)
gosper	\(\frac {x \left (128 a^{2} e^{4} x^{8}+16 a c \,d^{2} e^{2} x^{8}+35 c^{2} d^{4} x^{8}+576 a^{2} d \,e^{3} x^{6}+72 a c \,d^{3} e \,x^{6}+1008 a^{2} d^{2} e^{2} x^{4}+126 a c \,d^{4} x^{4}+840 a^{2} e \,x^{2} d^{3}+315 a^{2} d^{4}\right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\)	\(119\)
trager	\(\frac {x \left (128 a^{2} e^{4} x^{8}+16 a c \,d^{2} e^{2} x^{8}+35 c^{2} d^{4} x^{8}+576 a^{2} d \,e^{3} x^{6}+72 a c \,d^{3} e \,x^{6}+1008 a^{2} d^{2} e^{2} x^{4}+126 a c \,d^{4} x^{4}+840 a^{2} e \,x^{2} d^{3}+315 a^{2} d^{4}\right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\)	\(119\)
orering	\(\frac {x \left (128 a^{2} e^{4} x^{8}+16 a c \,d^{2} e^{2} x^{8}+35 c^{2} d^{4} x^{8}+576 a^{2} d \,e^{3} x^{6}+72 a c \,d^{3} e \,x^{6}+1008 a^{2} d^{2} e^{2} x^{4}+126 a c \,d^{4} x^{4}+840 a^{2} e \,x^{2} d^{3}+315 a^{2} d^{4}\right )}{315 \left (e \,x^{2}+d \right )^{\frac {9}{2}} d^{5}}\)	\(119\)
default	\(a^{2} \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )+c^{2} \left (-\frac {x^{7}}{2 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {7 d \left (-\frac {x^{5}}{4 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {5 d \left (-\frac {x^{3}}{6 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (-\frac {x}{8 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )}{8 e}\right )}{2 e}\right )}{4 e}\right )}{2 e}\right )+2 a c \left (-\frac {x^{3}}{6 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (-\frac {x}{8 e \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {d \left (\frac {x}{9 d \left (e \,x^{2}+d \right )^{\frac {9}{2}}}+\frac {\frac {8 x}{63 d \left (e \,x^{2}+d \right )^{\frac {7}{2}}}+\frac {8 \left (\frac {6 x}{35 d \left (e \,x^{2}+d \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 d \left (e \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{2} \sqrt {e \,x^{2}+d}}\right )}{7 d}\right )}{9 d}}{d}\right )}{8 e}\right )}{2 e}\right )\)	\(436\)

Input:

int((c*x^4+a)^2/(e*x^2+d)^(11/2),x,method=_RETURNVERBOSE)
 

Output:

x/(e*x^2+d)^(9/2)*((1/9*c^2*x^8+2/5*a*c*x^4+a^2)*d^4+8/3*x^2*a*(3/35*c*x^4 
+a)*e*d^3+16/5*x^4*a*(1/63*c*x^4+a)*e^2*d^2+64/35*a^2*d*e^3*x^6+128/315*a^ 
2*e^4*x^8)/d^5
 

Fricas [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {{\left ({\left (35 \, c^{2} d^{4} + 16 \, a c d^{2} e^{2} + 128 \, a^{2} e^{4}\right )} x^{9} + 840 \, a^{2} d^{3} e x^{3} + 72 \, {\left (a c d^{3} e + 8 \, a^{2} d e^{3}\right )} x^{7} + 315 \, a^{2} d^{4} x + 126 \, {\left (a c d^{4} + 8 \, a^{2} d^{2} e^{2}\right )} x^{5}\right )} \sqrt {e x^{2} + d}}{315 \, {\left (d^{5} e^{5} x^{10} + 5 \, d^{6} e^{4} x^{8} + 10 \, d^{7} e^{3} x^{6} + 10 \, d^{8} e^{2} x^{4} + 5 \, d^{9} e x^{2} + d^{10}\right )}} \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(11/2),x, algorithm="fricas")
 

Output:

1/315*((35*c^2*d^4 + 16*a*c*d^2*e^2 + 128*a^2*e^4)*x^9 + 840*a^2*d^3*e*x^3 
 + 72*(a*c*d^3*e + 8*a^2*d*e^3)*x^7 + 315*a^2*d^4*x + 126*(a*c*d^4 + 8*a^2 
*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)
 

Sympy [F]

\[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{11/2}} \, dx=\int \frac {\left (a + c x^{4}\right )^{2}}{\left (d + e x^{2}\right )^{\frac {11}{2}}}\, dx \] Input:

integrate((c*x**4+a)**2/(e*x**2+d)**(11/2),x)
 

Output:

Integral((a + c*x**4)**2/(d + e*x**2)**(11/2), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{11/2}} \, dx=-\frac {c^{2} x^{7}}{2 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} - \frac {7 \, c^{2} d x^{5}}{8 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} - \frac {35 \, c^{2} d^{2} x^{3}}{48 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{3}} - \frac {a c x^{3}}{3 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e} + \frac {128 \, a^{2} x}{315 \, \sqrt {e x^{2} + d} d^{5}} + \frac {64 \, a^{2} x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{4}} + \frac {16 \, a^{2} x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d^{3}} + \frac {8 \, a^{2} x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} d^{2}} + \frac {a^{2} x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} d} + \frac {c^{2} x}{18 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} e^{4}} + \frac {c^{2} x}{9 \, \sqrt {e x^{2} + d} d e^{4}} + \frac {c^{2} d x}{24 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} e^{4}} + \frac {5 \, c^{2} d^{2} x}{144 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{4}} - \frac {35 \, c^{2} d^{3} x}{144 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{4}} + \frac {a c x}{63 \, {\left (e x^{2} + d\right )}^{\frac {7}{2}} e^{2}} + \frac {16 \, a c x}{315 \, \sqrt {e x^{2} + d} d^{3} e^{2}} + \frac {8 \, a c x}{315 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}} d^{2} e^{2}} + \frac {2 \, a c x}{105 \, {\left (e x^{2} + d\right )}^{\frac {5}{2}} d e^{2}} - \frac {a c d x}{9 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}} e^{2}} \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(11/2),x, algorithm="maxima")
 

Output:

-1/2*c^2*x^7/((e*x^2 + d)^(9/2)*e) - 7/8*c^2*d*x^5/((e*x^2 + d)^(9/2)*e^2) 
 - 35/48*c^2*d^2*x^3/((e*x^2 + d)^(9/2)*e^3) - 1/3*a*c*x^3/((e*x^2 + d)^(9 
/2)*e) + 128/315*a^2*x/(sqrt(e*x^2 + d)*d^5) + 64/315*a^2*x/((e*x^2 + d)^( 
3/2)*d^4) + 16/105*a^2*x/((e*x^2 + d)^(5/2)*d^3) + 8/63*a^2*x/((e*x^2 + d) 
^(7/2)*d^2) + 1/9*a^2*x/((e*x^2 + d)^(9/2)*d) + 1/18*c^2*x/((e*x^2 + d)^(3 
/2)*e^4) + 1/9*c^2*x/(sqrt(e*x^2 + d)*d*e^4) + 1/24*c^2*d*x/((e*x^2 + d)^( 
5/2)*e^4) + 5/144*c^2*d^2*x/((e*x^2 + d)^(7/2)*e^4) - 35/144*c^2*d^3*x/((e 
*x^2 + d)^(9/2)*e^4) + 1/63*a*c*x/((e*x^2 + d)^(7/2)*e^2) + 16/315*a*c*x/( 
sqrt(e*x^2 + d)*d^3*e^2) + 8/315*a*c*x/((e*x^2 + d)^(3/2)*d^2*e^2) + 2/105 
*a*c*x/((e*x^2 + d)^(5/2)*d*e^2) - 1/9*a*c*d*x/((e*x^2 + d)^(9/2)*e^2)
 

Giac [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.62 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {{\left ({\left ({\left (x^{2} {\left (\frac {{\left (35 \, c^{2} d^{4} e^{4} + 16 \, a c d^{2} e^{6} + 128 \, a^{2} e^{8}\right )} x^{2}}{d^{5} e^{4}} + \frac {72 \, {\left (a c d^{3} e^{5} + 8 \, a^{2} d e^{7}\right )}}{d^{5} e^{4}}\right )} + \frac {126 \, {\left (a c d^{4} e^{4} + 8 \, a^{2} d^{2} e^{6}\right )}}{d^{5} e^{4}}\right )} x^{2} + \frac {840 \, a^{2} e}{d^{2}}\right )} x^{2} + \frac {315 \, a^{2}}{d}\right )} x}{315 \, {\left (e x^{2} + d\right )}^{\frac {9}{2}}} \] Input:

integrate((c*x^4+a)^2/(e*x^2+d)^(11/2),x, algorithm="giac")
 

Output:

1/315*(((x^2*((35*c^2*d^4*e^4 + 16*a*c*d^2*e^6 + 128*a^2*e^8)*x^2/(d^5*e^4 
) + 72*(a*c*d^3*e^5 + 8*a^2*d*e^7)/(d^5*e^4)) + 126*(a*c*d^4*e^4 + 8*a^2*d 
^2*e^6)/(d^5*e^4))*x^2 + 840*a^2*e/d^2)*x^2 + 315*a^2/d)*x/(e*x^2 + d)^(9/ 
2)
 

Mupad [B] (verification not implemented)

Time = 17.43 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {x\,\left (\frac {a^2}{9\,d}+\frac {d^2\,\left (\frac {c^2\,d}{9\,e^2}+\frac {2\,a\,c}{9\,d}\right )}{e^2}\right )}{{\left (e\,x^2+d\right )}^{9/2}}-\frac {x\,\left (\frac {-8\,a^2\,e^4+2\,a\,c\,d^2\,e^2+c^2\,d^4}{63\,d^2\,e^4}+\frac {d\,\left (\frac {2\,c^2\,d}{7\,e^3}+\frac {c\,\left (c\,d^2+2\,a\,e^2\right )}{7\,d\,e^3}\right )}{e}\right )}{{\left (e\,x^2+d\right )}^{7/2}}-\frac {x\,\left (\frac {c^2}{3\,e^4}-\frac {64\,a^2\,e^4+8\,a\,c\,d^2\,e^2-35\,c^2\,d^4}{315\,d^4\,e^4}\right )}{{\left (e\,x^2+d\right )}^{3/2}}+\frac {x\,\left (\frac {3\,c^2\,d}{5\,e^4}+\frac {16\,a^2\,e^4+2\,a\,c\,d^2\,e^2+7\,c^2\,d^4}{105\,d^3\,e^4}\right )}{{\left (e\,x^2+d\right )}^{5/2}}+\frac {x\,\left (128\,a^2\,e^4+16\,a\,c\,d^2\,e^2+35\,c^2\,d^4\right )}{315\,d^5\,e^4\,\sqrt {e\,x^2+d}} \] Input:

int((a + c*x^4)^2/(d + e*x^2)^(11/2),x)
 

Output:

(x*(a^2/(9*d) + (d^2*((c^2*d)/(9*e^2) + (2*a*c)/(9*d)))/e^2))/(d + e*x^2)^ 
(9/2) - (x*((c^2*d^4 - 8*a^2*e^4 + 2*a*c*d^2*e^2)/(63*d^2*e^4) + (d*((2*c^ 
2*d)/(7*e^3) + (c*(2*a*e^2 + c*d^2))/(7*d*e^3)))/e))/(d + e*x^2)^(7/2) - ( 
x*(c^2/(3*e^4) - (64*a^2*e^4 - 35*c^2*d^4 + 8*a*c*d^2*e^2)/(315*d^4*e^4))) 
/(d + e*x^2)^(3/2) + (x*((3*c^2*d)/(5*e^4) + (16*a^2*e^4 + 7*c^2*d^4 + 2*a 
*c*d^2*e^2)/(105*d^3*e^4)))/(d + e*x^2)^(5/2) + (x*(128*a^2*e^4 + 35*c^2*d 
^4 + 16*a*c*d^2*e^2))/(315*d^5*e^4*(d + e*x^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 513, normalized size of antiderivative = 2.25 \[ \int \frac {\left (a+c x^4\right )^2}{\left (d+e x^2\right )^{11/2}} \, dx=\frac {126 \sqrt {e \,x^{2}+d}\, a c \,d^{4} e^{5} x^{5}+72 \sqrt {e \,x^{2}+d}\, a c \,d^{3} e^{6} x^{7}+16 \sqrt {e \,x^{2}+d}\, a c \,d^{2} e^{7} x^{9}-80 \sqrt {e}\, a c \,d^{6} e^{3} x^{2}-160 \sqrt {e}\, a c \,d^{5} e^{4} x^{4}-160 \sqrt {e}\, a c \,d^{4} e^{5} x^{6}-80 \sqrt {e}\, a c \,d^{3} e^{6} x^{8}-16 \sqrt {e}\, a c \,d^{2} e^{7} x^{10}+315 \sqrt {e \,x^{2}+d}\, a^{2} d^{4} e^{5} x +840 \sqrt {e \,x^{2}+d}\, a^{2} d^{3} e^{6} x^{3}+1008 \sqrt {e \,x^{2}+d}\, a^{2} d^{2} e^{7} x^{5}+576 \sqrt {e \,x^{2}+d}\, a^{2} d \,e^{8} x^{7}+35 \sqrt {e \,x^{2}+d}\, c^{2} d^{4} e^{5} x^{9}-640 \sqrt {e}\, a^{2} d^{4} e^{5} x^{2}-1280 \sqrt {e}\, a^{2} d^{3} e^{6} x^{4}-1280 \sqrt {e}\, a^{2} d^{2} e^{7} x^{6}-640 \sqrt {e}\, a^{2} d \,e^{8} x^{8}-16 \sqrt {e}\, a c \,d^{7} e^{2}+175 \sqrt {e}\, c^{2} d^{8} e \,x^{2}+350 \sqrt {e}\, c^{2} d^{7} e^{2} x^{4}+350 \sqrt {e}\, c^{2} d^{6} e^{3} x^{6}+175 \sqrt {e}\, c^{2} d^{5} e^{4} x^{8}+35 \sqrt {e}\, c^{2} d^{4} e^{5} x^{10}+128 \sqrt {e \,x^{2}+d}\, a^{2} e^{9} x^{9}-128 \sqrt {e}\, a^{2} d^{5} e^{4}-128 \sqrt {e}\, a^{2} e^{9} x^{10}+35 \sqrt {e}\, c^{2} d^{9}}{315 d^{5} e^{5} \left (e^{5} x^{10}+5 d \,e^{4} x^{8}+10 d^{2} e^{3} x^{6}+10 d^{3} e^{2} x^{4}+5 d^{4} e \,x^{2}+d^{5}\right )} \] Input:

int((c*x^4+a)^2/(e*x^2+d)^(11/2),x)
 

Output:

(315*sqrt(d + e*x**2)*a**2*d**4*e**5*x + 840*sqrt(d + e*x**2)*a**2*d**3*e* 
*6*x**3 + 1008*sqrt(d + e*x**2)*a**2*d**2*e**7*x**5 + 576*sqrt(d + e*x**2) 
*a**2*d*e**8*x**7 + 128*sqrt(d + e*x**2)*a**2*e**9*x**9 + 126*sqrt(d + e*x 
**2)*a*c*d**4*e**5*x**5 + 72*sqrt(d + e*x**2)*a*c*d**3*e**6*x**7 + 16*sqrt 
(d + e*x**2)*a*c*d**2*e**7*x**9 + 35*sqrt(d + e*x**2)*c**2*d**4*e**5*x**9 
- 128*sqrt(e)*a**2*d**5*e**4 - 640*sqrt(e)*a**2*d**4*e**5*x**2 - 1280*sqrt 
(e)*a**2*d**3*e**6*x**4 - 1280*sqrt(e)*a**2*d**2*e**7*x**6 - 640*sqrt(e)*a 
**2*d*e**8*x**8 - 128*sqrt(e)*a**2*e**9*x**10 - 16*sqrt(e)*a*c*d**7*e**2 - 
 80*sqrt(e)*a*c*d**6*e**3*x**2 - 160*sqrt(e)*a*c*d**5*e**4*x**4 - 160*sqrt 
(e)*a*c*d**4*e**5*x**6 - 80*sqrt(e)*a*c*d**3*e**6*x**8 - 16*sqrt(e)*a*c*d* 
*2*e**7*x**10 + 35*sqrt(e)*c**2*d**9 + 175*sqrt(e)*c**2*d**8*e*x**2 + 350* 
sqrt(e)*c**2*d**7*e**2*x**4 + 350*sqrt(e)*c**2*d**6*e**3*x**6 + 175*sqrt(e 
)*c**2*d**5*e**4*x**8 + 35*sqrt(e)*c**2*d**4*e**5*x**10)/(315*d**5*e**5*(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))