\(\int (d+e x^2)^{3/2} (a-c x^4)^2 \, dx\) [332]

Optimal result

Integrand size = 22, antiderivative size = 270 \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2 \, dx=\frac {d \left (7 c^2 d^4-48 a c d^2 e^2+384 a^2 e^4\right ) x \sqrt {d+e x^2}}{1024 e^4}+\frac {\left (7 c^2 d^4-48 a c d^2 e^2+384 a^2 e^4\right ) x \left (d+e x^2\right )^{3/2}}{1536 e^4}-\frac {c d \left (7 c d^2-48 a e^2\right ) x \left (d+e x^2\right )^{5/2}}{384 e^4}+\frac {c \left (7 c d^2-48 a e^2\right ) x^3 \left (d+e x^2\right )^{5/2}}{192 e^3}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{120 e^2}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}+\frac {d^2 \left (7 c^2 d^4-48 a c d^2 e^2+384 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{1024 e^{9/2}} \] Output:

1/1024*d*(384*a^2*e^4-48*a*c*d^2*e^2+7*c^2*d^4)*x*(e*x^2+d)^(1/2)/e^4+1/15 
36*(384*a^2*e^4-48*a*c*d^2*e^2+7*c^2*d^4)*x*(e*x^2+d)^(3/2)/e^4-1/384*c*d* 
(-48*a*e^2+7*c*d^2)*x*(e*x^2+d)^(5/2)/e^4+1/192*c*(-48*a*e^2+7*c*d^2)*x^3* 
(e*x^2+d)^(5/2)/e^3-7/120*c^2*d*x^5*(e*x^2+d)^(5/2)/e^2+1/12*c^2*x^7*(e*x^ 
2+d)^(5/2)/e+1/1024*d^2*(384*a^2*e^4-48*a*c*d^2*e^2+7*c^2*d^4)*arctanh(e^( 
1/2)*x/(e*x^2+d)^(1/2))/e^(9/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.73 \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2 \, dx=\frac {\sqrt {e} x \sqrt {d+e x^2} \left (1920 a^2 e^4 \left (5 d+2 e x^2\right )-240 a c e^2 \left (-3 d^3+2 d^2 e x^2+24 d e^2 x^4+16 e^3 x^6\right )+c^2 \left (-105 d^5+70 d^4 e x^2-56 d^3 e^2 x^4+48 d^2 e^3 x^6+1664 d e^4 x^8+1280 e^5 x^{10}\right )\right )-15 \left (7 c^2 d^6-48 a c d^4 e^2+384 a^2 d^2 e^4\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{15360 e^{9/2}} \] Input:

Integrate[(d + e*x^2)^(3/2)*(a - c*x^4)^2,x]
 

Output:

(Sqrt[e]*x*Sqrt[d + e*x^2]*(1920*a^2*e^4*(5*d + 2*e*x^2) - 240*a*c*e^2*(-3 
*d^3 + 2*d^2*e*x^2 + 24*d*e^2*x^4 + 16*e^3*x^6) + c^2*(-105*d^5 + 70*d^4*e 
*x^2 - 56*d^3*e^2*x^4 + 48*d^2*e^3*x^6 + 1664*d*e^4*x^8 + 1280*e^5*x^10)) 
- 15*(7*c^2*d^6 - 48*a*c*d^4*e^2 + 384*a^2*d^2*e^4)*Log[-(Sqrt[e]*x) + Sqr 
t[d + e*x^2]])/(15360*e^(9/2))
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1474, 2346, 27, 1474, 27, 299, 211, 211, 224, 219}

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.

\(\displaystyle \int \left (a-c x^4\right )^2 \left (d+e x^2\right )^{3/2} \, dx\)

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\int \left (e x^2+d\right )^{3/2} \left (-7 c^2 d x^6-24 a c e x^4+12 a^2 e\right )dx}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\int 5 \left (e x^2+d\right )^{3/2} \left (c \left (7 c d^2-48 a e^2\right ) x^4+24 a^2 e^2\right )dx}{10 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \left (e x^2+d\right )^{3/2} \left (c \left (7 c d^2-48 a e^2\right ) x^4+24 a^2 e^2\right )dx}{2 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\frac {\frac {\int 3 \left (e x^2+d\right )^{3/2} \left (64 a^2 e^3-c d \left (7 c d^2-48 a e^2\right ) x^2\right )dx}{8 e}+\frac {c x^3 \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{8 e}}{2 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \int \left (e x^2+d\right )^{3/2} \left (64 a^2 e^3-c d \left (7 c d^2-48 a e^2\right ) x^2\right )dx}{8 e}+\frac {c x^3 \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{8 e}}{2 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (384 a^2 e^4-48 a c d^2 e^2+7 c^2 d^4\right ) \int \left (e x^2+d\right )^{3/2}dx}{6 e}-\frac {c d x \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{6 e}\right )}{8 e}+\frac {c x^3 \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{8 e}}{2 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (384 a^2 e^4-48 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {3}{4} d \int \sqrt {e x^2+d}dx+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )}{6 e}-\frac {c d x \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{6 e}\right )}{8 e}+\frac {c x^3 \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{8 e}}{2 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (384 a^2 e^4-48 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )}{6 e}-\frac {c d x \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{6 e}\right )}{8 e}+\frac {c x^3 \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{8 e}}{2 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (384 a^2 e^4-48 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )}{6 e}-\frac {c d x \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{6 e}\right )}{8 e}+\frac {c x^3 \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{8 e}}{2 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {3 \left (\frac {\left (384 a^2 e^4-48 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {3}{4} d \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )+\frac {1}{4} x \left (d+e x^2\right )^{3/2}\right )}{6 e}-\frac {c d x \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{6 e}\right )}{8 e}+\frac {c x^3 \left (d+e x^2\right )^{5/2} \left (7 c d^2-48 a e^2\right )}{8 e}}{2 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{5/2}}{10 e}}{12 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{5/2}}{12 e}\)

Input:

Int[(d + e*x^2)^(3/2)*(a - c*x^4)^2,x]
 

Output:

(c^2*x^7*(d + e*x^2)^(5/2))/(12*e) + ((-7*c^2*d*x^5*(d + e*x^2)^(5/2))/(10 
*e) + ((c*(7*c*d^2 - 48*a*e^2)*x^3*(d + e*x^2)^(5/2))/(8*e) + (3*(-1/6*(c* 
d*(7*c*d^2 - 48*a*e^2)*x*(d + e*x^2)^(5/2))/e + ((7*c^2*d^4 - 48*a*c*d^2*e 
^2 + 384*a^2*e^4)*((x*(d + e*x^2)^(3/2))/4 + (3*d*((x*Sqrt[d + e*x^2])/2 + 
 (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[e])))/4))/(6*e)))/(8*e)) 
/(2*e))/(12*e)
 

Defintions of rubi rules used

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 1474

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si 
mp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))), x] + Simp[1/( 
e*(4*p + 2*q + 1))   Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 2*q + 1)*(a + c 
*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], 
x], x] /; FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] 
&&  !LtQ[q, -1]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(\frac {\frac {3 d^{2} \left (a^{2} e^{4}-\frac {1}{8} a c \,d^{2} e^{2}+\frac {7}{384} c^{2} d^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )}{8}+\frac {5 x \sqrt {e \,x^{2}+d}\, \left (\frac {2 x^{2} \left (\frac {1}{3} c^{2} x^{8}-a c \,x^{4}+a^{2}\right ) e^{\frac {11}{2}}}{5}+\left (\left (\frac {13}{75} c^{2} x^{8}-\frac {3}{5} a c \,x^{4}+a^{2}\right ) e^{\frac {9}{2}}+\frac {3 d \left (d \left (-\frac {7 c \,x^{4}}{90}+a \right ) e^{\frac {5}{2}}-\frac {2 x^{2} \left (-\frac {c \,x^{4}}{10}+a \right ) e^{\frac {7}{2}}}{3}-\frac {7 d^{2} c \left (-\frac {2 e^{\frac {3}{2}} x^{2}}{3}+\sqrt {e}\, d \right )}{48}\right ) c}{40}\right ) d \right )}{8}}{e^{\frac {9}{2}}}\)	\(174\)
risch	\(\frac {x \left (1280 e^{5} c^{2} x^{10}+1664 d \,c^{2} e^{4} x^{8}-3840 a c \,e^{5} x^{6}+48 c^{2} d^{2} e^{3} x^{6}-5760 a c d \,e^{4} x^{4}-56 c^{2} d^{3} e^{2} x^{4}+3840 a^{2} e^{5} x^{2}-480 a c \,d^{2} e^{3} x^{2}+70 c^{2} d^{4} e \,x^{2}+9600 a^{2} d \,e^{4}+720 a c \,d^{3} e^{2}-105 c^{2} d^{5}\right ) \sqrt {e \,x^{2}+d}}{15360 e^{4}}+\frac {d^{2} \left (384 a^{2} e^{4}-48 a c \,d^{2} e^{2}+7 c^{2} d^{4}\right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{1024 e^{\frac {9}{2}}}\)	\(204\)
default	\(a^{2} \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )+c^{2} \left (\frac {x^{7} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{12 e}-\frac {7 d \left (\frac {x^{5} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{10 e}-\frac {d \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{8 e}-\frac {3 d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )}{6 e}\right )}{8 e}\right )}{2 e}\right )}{12 e}\right )-2 a c \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{8 e}-\frac {3 d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {5}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4}+\frac {3 d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4}\right )}{6 e}\right )}{8 e}\right )\)	\(307\)

Input:

int((e*x^2+d)^(3/2)*(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

5/8/e^(9/2)*(3/5*d^2*(a^2*e^4-1/8*a*c*d^2*e^2+7/384*c^2*d^4)*arctanh((e*x^ 
2+d)^(1/2)/x/e^(1/2))+x*(e*x^2+d)^(1/2)*(2/5*x^2*(1/3*c^2*x^8-a*c*x^4+a^2) 
*e^(11/2)+((13/75*c^2*x^8-3/5*a*c*x^4+a^2)*e^(9/2)+3/40*d*(d*(-7/90*c*x^4+ 
a)*e^(5/2)-2/3*x^2*(-1/10*c*x^4+a)*e^(7/2)-7/48*d^2*c*(-2/3*e^(3/2)*x^2+e^ 
(1/2)*d))*c)*d))
 

Fricas [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.59 \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2 \, dx=\left [\frac {15 \, {\left (7 \, c^{2} d^{6} - 48 \, a c d^{4} e^{2} + 384 \, a^{2} d^{2} e^{4}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (1280 \, c^{2} e^{6} x^{11} + 1664 \, c^{2} d e^{5} x^{9} + 48 \, {\left (c^{2} d^{2} e^{4} - 80 \, a c e^{6}\right )} x^{7} - 8 \, {\left (7 \, c^{2} d^{3} e^{3} + 720 \, a c d e^{5}\right )} x^{5} + 10 \, {\left (7 \, c^{2} d^{4} e^{2} - 48 \, a c d^{2} e^{4} + 384 \, a^{2} e^{6}\right )} x^{3} - 15 \, {\left (7 \, c^{2} d^{5} e - 48 \, a c d^{3} e^{3} - 640 \, a^{2} d e^{5}\right )} x\right )} \sqrt {e x^{2} + d}}{30720 \, e^{5}}, -\frac {15 \, {\left (7 \, c^{2} d^{6} - 48 \, a c d^{4} e^{2} + 384 \, a^{2} d^{2} e^{4}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (1280 \, c^{2} e^{6} x^{11} + 1664 \, c^{2} d e^{5} x^{9} + 48 \, {\left (c^{2} d^{2} e^{4} - 80 \, a c e^{6}\right )} x^{7} - 8 \, {\left (7 \, c^{2} d^{3} e^{3} + 720 \, a c d e^{5}\right )} x^{5} + 10 \, {\left (7 \, c^{2} d^{4} e^{2} - 48 \, a c d^{2} e^{4} + 384 \, a^{2} e^{6}\right )} x^{3} - 15 \, {\left (7 \, c^{2} d^{5} e - 48 \, a c d^{3} e^{3} - 640 \, a^{2} d e^{5}\right )} x\right )} \sqrt {e x^{2} + d}}{15360 \, e^{5}}\right ] \] Input:

integrate((e*x^2+d)^(3/2)*(-c*x^4+a)^2,x, algorithm="fricas")
 

Output:

[1/30720*(15*(7*c^2*d^6 - 48*a*c*d^4*e^2 + 384*a^2*d^2*e^4)*sqrt(e)*log(-2 
*e*x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*(1280*c^2*e^6*x^11 + 1664*c^ 
2*d*e^5*x^9 + 48*(c^2*d^2*e^4 - 80*a*c*e^6)*x^7 - 8*(7*c^2*d^3*e^3 + 720*a 
*c*d*e^5)*x^5 + 10*(7*c^2*d^4*e^2 - 48*a*c*d^2*e^4 + 384*a^2*e^6)*x^3 - 15 
*(7*c^2*d^5*e - 48*a*c*d^3*e^3 - 640*a^2*d*e^5)*x)*sqrt(e*x^2 + d))/e^5, - 
1/15360*(15*(7*c^2*d^6 - 48*a*c*d^4*e^2 + 384*a^2*d^2*e^4)*sqrt(-e)*arctan 
(sqrt(-e)*x/sqrt(e*x^2 + d)) - (1280*c^2*e^6*x^11 + 1664*c^2*d*e^5*x^9 + 4 
8*(c^2*d^2*e^4 - 80*a*c*e^6)*x^7 - 8*(7*c^2*d^3*e^3 + 720*a*c*d*e^5)*x^5 + 
 10*(7*c^2*d^4*e^2 - 48*a*c*d^2*e^4 + 384*a^2*e^6)*x^3 - 15*(7*c^2*d^5*e - 
 48*a*c*d^3*e^3 - 640*a^2*d*e^5)*x)*sqrt(e*x^2 + d))/e^5]
 

Sympy [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.44 \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2 \, dx=\begin {cases} \sqrt {d + e x^{2}} \cdot \left (\frac {13 c^{2} d x^{9}}{120} + \frac {c^{2} e x^{11}}{12} + \frac {x^{7} \left (- 2 a c e^{2} + \frac {c^{2} d^{2}}{40}\right )}{8 e} + \frac {x^{5} \left (- 4 a c d e - \frac {7 d \left (- 2 a c e^{2} + \frac {c^{2} d^{2}}{40}\right )}{8 e}\right )}{6 e} + \frac {x^{3} \left (a^{2} e^{2} - 2 a c d^{2} - \frac {5 d \left (- 4 a c d e - \frac {7 d \left (- 2 a c e^{2} + \frac {c^{2} d^{2}}{40}\right )}{8 e}\right )}{6 e}\right )}{4 e} + \frac {x \left (2 a^{2} d e - \frac {3 d \left (a^{2} e^{2} - 2 a c d^{2} - \frac {5 d \left (- 4 a c d e - \frac {7 d \left (- 2 a c e^{2} + \frac {c^{2} d^{2}}{40}\right )}{8 e}\right )}{6 e}\right )}{4 e}\right )}{2 e}\right ) + \left (a^{2} d^{2} - \frac {d \left (2 a^{2} d e - \frac {3 d \left (a^{2} e^{2} - 2 a c d^{2} - \frac {5 d \left (- 4 a c d e - \frac {7 d \left (- 2 a c e^{2} + \frac {c^{2} d^{2}}{40}\right )}{8 e}\right )}{6 e}\right )}{4 e}\right )}{2 e}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {e} \sqrt {d + e x^{2}} + 2 e x \right )}}{\sqrt {e}} & \text {for}\: d \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {e x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: e \neq 0 \\d^{\frac {3}{2}} \left (a^{2} x - \frac {2 a c x^{5}}{5} + \frac {c^{2} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((e*x**2+d)**(3/2)*(-c*x**4+a)**2,x)
 

Output:

Piecewise((sqrt(d + e*x**2)*(13*c**2*d*x**9/120 + c**2*e*x**11/12 + x**7*( 
-2*a*c*e**2 + c**2*d**2/40)/(8*e) + x**5*(-4*a*c*d*e - 7*d*(-2*a*c*e**2 + 
c**2*d**2/40)/(8*e))/(6*e) + x**3*(a**2*e**2 - 2*a*c*d**2 - 5*d*(-4*a*c*d* 
e - 7*d*(-2*a*c*e**2 + c**2*d**2/40)/(8*e))/(6*e))/(4*e) + x*(2*a**2*d*e - 
 3*d*(a**2*e**2 - 2*a*c*d**2 - 5*d*(-4*a*c*d*e - 7*d*(-2*a*c*e**2 + c**2*d 
**2/40)/(8*e))/(6*e))/(4*e))/(2*e)) + (a**2*d**2 - d*(2*a**2*d*e - 3*d*(a* 
*2*e**2 - 2*a*c*d**2 - 5*d*(-4*a*c*d*e - 7*d*(-2*a*c*e**2 + c**2*d**2/40)/ 
(8*e))/(6*e))/(4*e))/(2*e))*Piecewise((log(2*sqrt(e)*sqrt(d + e*x**2) + 2* 
e*x)/sqrt(e), Ne(d, 0)), (x*log(x)/sqrt(e*x**2), True)), Ne(e, 0)), (d**(3 
/2)*(a**2*x - 2*a*c*x**5/5 + c**2*x**9/9), True))
 

Maxima [F(-2)]

Exception generated. \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2 \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x^2+d)^(3/2)*(-c*x^4+a)^2,x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.83 \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2 \, dx=\frac {1}{15360} \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, c^{2} e x^{2} + 13 \, c^{2} d\right )} x^{2} + \frac {3 \, {\left (c^{2} d^{2} e^{9} - 80 \, a c e^{11}\right )}}{e^{10}}\right )} x^{2} - \frac {7 \, c^{2} d^{3} e^{8} + 720 \, a c d e^{10}}{e^{10}}\right )} x^{2} + \frac {5 \, {\left (7 \, c^{2} d^{4} e^{7} - 48 \, a c d^{2} e^{9} + 384 \, a^{2} e^{11}\right )}}{e^{10}}\right )} x^{2} - \frac {15 \, {\left (7 \, c^{2} d^{5} e^{6} - 48 \, a c d^{3} e^{8} - 640 \, a^{2} d e^{10}\right )}}{e^{10}}\right )} \sqrt {e x^{2} + d} x - \frac {{\left (7 \, c^{2} d^{6} - 48 \, a c d^{4} e^{2} + 384 \, a^{2} d^{2} e^{4}\right )} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{1024 \, e^{\frac {9}{2}}} \] Input:

integrate((e*x^2+d)^(3/2)*(-c*x^4+a)^2,x, algorithm="giac")
 

Output:

1/15360*(2*(4*(2*(8*(10*c^2*e*x^2 + 13*c^2*d)*x^2 + 3*(c^2*d^2*e^9 - 80*a* 
c*e^11)/e^10)*x^2 - (7*c^2*d^3*e^8 + 720*a*c*d*e^10)/e^10)*x^2 + 5*(7*c^2* 
d^4*e^7 - 48*a*c*d^2*e^9 + 384*a^2*e^11)/e^10)*x^2 - 15*(7*c^2*d^5*e^6 - 4 
8*a*c*d^3*e^8 - 640*a^2*d*e^10)/e^10)*sqrt(e*x^2 + d)*x - 1/1024*(7*c^2*d^ 
6 - 48*a*c*d^4*e^2 + 384*a^2*d^2*e^4)*log(abs(-sqrt(e)*x + sqrt(e*x^2 + d) 
))/e^(9/2)
 

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2 \, dx=\int {\left (a-c\,x^4\right )}^2\,{\left (e\,x^2+d\right )}^{3/2} \,d x \] Input:

int((a - c*x^4)^2*(d + e*x^2)^(3/2),x)
 

Output:

int((a - c*x^4)^2*(d + e*x^2)^(3/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.24 \[ \int \left (d+e x^2\right )^{3/2} \left (a-c x^4\right )^2 \, dx=\frac {9600 \sqrt {e \,x^{2}+d}\, a^{2} d \,e^{5} x +3840 \sqrt {e \,x^{2}+d}\, a^{2} e^{6} x^{3}+720 \sqrt {e \,x^{2}+d}\, a c \,d^{3} e^{3} x -480 \sqrt {e \,x^{2}+d}\, a c \,d^{2} e^{4} x^{3}-5760 \sqrt {e \,x^{2}+d}\, a c d \,e^{5} x^{5}-3840 \sqrt {e \,x^{2}+d}\, a c \,e^{6} x^{7}-105 \sqrt {e \,x^{2}+d}\, c^{2} d^{5} e x +70 \sqrt {e \,x^{2}+d}\, c^{2} d^{4} e^{2} x^{3}-56 \sqrt {e \,x^{2}+d}\, c^{2} d^{3} e^{3} x^{5}+48 \sqrt {e \,x^{2}+d}\, c^{2} d^{2} e^{4} x^{7}+1664 \sqrt {e \,x^{2}+d}\, c^{2} d \,e^{5} x^{9}+1280 \sqrt {e \,x^{2}+d}\, c^{2} e^{6} x^{11}+5760 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a^{2} d^{2} e^{4}-720 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a c \,d^{4} e^{2}+105 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{6}}{15360 e^{5}} \] Input:

int((e*x^2+d)^(3/2)*(-c*x^4+a)^2,x)
 

Output:

(9600*sqrt(d + e*x**2)*a**2*d*e**5*x + 3840*sqrt(d + e*x**2)*a**2*e**6*x** 
3 + 720*sqrt(d + e*x**2)*a*c*d**3*e**3*x - 480*sqrt(d + e*x**2)*a*c*d**2*e 
**4*x**3 - 5760*sqrt(d + e*x**2)*a*c*d*e**5*x**5 - 3840*sqrt(d + e*x**2)*a 
*c*e**6*x**7 - 105*sqrt(d + e*x**2)*c**2*d**5*e*x + 70*sqrt(d + e*x**2)*c* 
*2*d**4*e**2*x**3 - 56*sqrt(d + e*x**2)*c**2*d**3*e**3*x**5 + 48*sqrt(d + 
e*x**2)*c**2*d**2*e**4*x**7 + 1664*sqrt(d + e*x**2)*c**2*d*e**5*x**9 + 128 
0*sqrt(d + e*x**2)*c**2*e**6*x**11 + 5760*sqrt(e)*log((sqrt(d + e*x**2) + 
sqrt(e)*x)/sqrt(d))*a**2*d**2*e**4 - 720*sqrt(e)*log((sqrt(d + e*x**2) + s 
qrt(e)*x)/sqrt(d))*a*c*d**4*e**2 + 105*sqrt(e)*log((sqrt(d + e*x**2) + sqr 
t(e)*x)/sqrt(d))*c**2*d**6)/(15360*e**5)