\(\int \sqrt {d+e x^2} (a-c x^4)^2 \, dx\) [333]

Optimal result

Integrand size = 22, antiderivative size = 221 \[ \int \sqrt {d+e x^2} \left (a-c x^4\right )^2 \, dx=\frac {\left (7 c^2 d^4-32 a c d^2 e^2+128 a^2 e^4\right ) x \sqrt {d+e x^2}}{256 e^4}-\frac {c d \left (7 c d^2-32 a e^2\right ) x \left (d+e x^2\right )^{3/2}}{128 e^4}+\frac {c \left (7 c d^2-32 a e^2\right ) x^3 \left (d+e x^2\right )^{3/2}}{96 e^3}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{80 e^2}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}+\frac {d \left (7 c^2 d^4-32 a c d^2 e^2+128 a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{256 e^{9/2}} \] Output:

1/256*(128*a^2*e^4-32*a*c*d^2*e^2+7*c^2*d^4)*x*(e*x^2+d)^(1/2)/e^4-1/128*c 
*d*(-32*a*e^2+7*c*d^2)*x*(e*x^2+d)^(3/2)/e^4+1/96*c*(-32*a*e^2+7*c*d^2)*x^ 
3*(e*x^2+d)^(3/2)/e^3-7/80*c^2*d*x^5*(e*x^2+d)^(3/2)/e^2+1/10*c^2*x^7*(e*x 
^2+d)^(3/2)/e+1/256*d*(128*a^2*e^4-32*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 = 0.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.74 \[ \int \sqrt {d+e x^2} \left (a-c x^4\right )^2 \, dx=\frac {\sqrt {e} x \sqrt {d+e x^2} \left (1920 a^2 e^4-160 a c e^2 \left (-3 d^2+2 d e x^2+8 e^2 x^4\right )+c^2 \left (-105 d^4+70 d^3 e x^2-56 d^2 e^2 x^4+48 d e^3 x^6+384 e^4 x^8\right )\right )-15 \left (7 c^2 d^5-32 a c d^3 e^2+128 a^2 d e^4\right ) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )}{3840 e^{9/2}} \] Input:

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

Output:

(Sqrt[e]*x*Sqrt[d + e*x^2]*(1920*a^2*e^4 - 160*a*c*e^2*(-3*d^2 + 2*d*e*x^2 
 + 8*e^2*x^4) + c^2*(-105*d^4 + 70*d^3*e*x^2 - 56*d^2*e^2*x^4 + 48*d*e^3*x 
^6 + 384*e^4*x^8)) - 15*(7*c^2*d^5 - 32*a*c*d^3*e^2 + 128*a^2*d*e^4)*Log[- 
(Sqrt[e]*x) + Sqrt[d + e*x^2]])/(3840*e^(9/2))
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1474, 2346, 27, 1474, 27, 299, 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.

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

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\int \sqrt {e x^2+d} \left (-7 c^2 d x^6-20 a c e x^4+10 a^2 e\right )dx}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\int 5 \sqrt {e x^2+d} \left (c \left (7 c d^2-32 a e^2\right ) x^4+16 a^2 e^2\right )dx}{8 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{8 e}}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {5 \int \sqrt {e x^2+d} \left (c \left (7 c d^2-32 a e^2\right ) x^4+16 a^2 e^2\right )dx}{8 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{8 e}}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\frac {5 \left (\frac {\int 3 \sqrt {e x^2+d} \left (32 a^2 e^3-c d \left (7 c d^2-32 a e^2\right ) x^2\right )dx}{6 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{6 e}\right )}{8 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{8 e}}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {5 \left (\frac {\int \sqrt {e x^2+d} \left (32 a^2 e^3-c d \left (7 c d^2-32 a e^2\right ) x^2\right )dx}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{6 e}\right )}{8 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{8 e}}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {5 \left (\frac {\frac {\left (128 a^2 e^4-32 a c d^2 e^2+7 c^2 d^4\right ) \int \sqrt {e x^2+d}dx}{4 e}-\frac {c d x \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{4 e}}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{6 e}\right )}{8 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{8 e}}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {5 \left (\frac {\frac {\left (128 a^2 e^4-32 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {1}{2} d \int \frac {1}{\sqrt {e x^2+d}}dx+\frac {1}{2} x \sqrt {d+e x^2}\right )}{4 e}-\frac {c d x \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{4 e}}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{6 e}\right )}{8 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{8 e}}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {5 \left (\frac {\frac {\left (128 a^2 e^4-32 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {1}{2} d \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {1}{2} x \sqrt {d+e x^2}\right )}{4 e}-\frac {c d x \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{4 e}}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{6 e}\right )}{8 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{8 e}}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {5 \left (\frac {\frac {\left (128 a^2 e^4-32 a c d^2 e^2+7 c^2 d^4\right ) \left (\frac {d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d+e x^2}\right )}{4 e}-\frac {c d x \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{4 e}}{2 e}+\frac {c x^3 \left (d+e x^2\right )^{3/2} \left (7 c d^2-32 a e^2\right )}{6 e}\right )}{8 e}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{3/2}}{8 e}}{10 e}+\frac {c^2 x^7 \left (d+e x^2\right )^{3/2}}{10 e}\)

Input:

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

Output:

(c^2*x^7*(d + e*x^2)^(3/2))/(10*e) + ((-7*c^2*d*x^5*(d + e*x^2)^(3/2))/(8* 
e) + (5*((c*(7*c*d^2 - 32*a*e^2)*x^3*(d + e*x^2)^(3/2))/(6*e) + (-1/4*(c*d 
*(7*c*d^2 - 32*a*e^2)*x*(d + e*x^2)^(3/2))/e + ((7*c^2*d^4 - 32*a*c*d^2*e^ 
2 + 128*a^2*e^4)*((x*Sqrt[d + e*x^2])/2 + (d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + 
e*x^2]])/(2*Sqrt[e])))/(4*e))/(2*e)))/(8*e))/(10*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 = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(\frac {d \left (a^{2} e^{4}-\frac {1}{4} a c \,d^{2} e^{2}+\frac {7}{128} c^{2} d^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+x \left (\left (\frac {1}{5} c^{2} x^{8}-\frac {2}{3} a c \,x^{4}+a^{2}\right ) e^{\frac {9}{2}}+\frac {\left (d \left (-\frac {7 c \,x^{4}}{60}+a \right ) e^{\frac {5}{2}}-\frac {2 x^{2} \left (-\frac {3 c \,x^{4}}{20}+a \right ) e^{\frac {7}{2}}}{3}-\frac {7 d^{2} c \left (-\frac {2 e^{\frac {3}{2}} x^{2}}{3}+\sqrt {e}\, d \right )}{32}\right ) d c}{4}\right ) \sqrt {e \,x^{2}+d}}{2 e^{\frac {9}{2}}}\)	\(141\)
risch	\(\frac {x \left (384 c^{2} e^{4} x^{8}+48 d \,c^{2} e^{3} x^{6}-1280 a c \,e^{4} x^{4}-56 c^{2} d^{2} e^{2} x^{4}-320 a c d \,e^{3} x^{2}+70 c^{2} d^{3} e \,x^{2}+1920 a^{2} e^{4}+480 a c \,d^{2} e^{2}-105 c^{2} d^{4}\right ) \sqrt {e \,x^{2}+d}}{3840 e^{4}}+\frac {d \left (128 a^{2} e^{4}-32 a c \,d^{2} e^{2}+7 c^{2} d^{4}\right ) \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{256 e^{\frac {9}{2}}}\)	\(163\)
default	\(a^{2} \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )+c^{2} \left (\frac {x^{7} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{10 e}-\frac {7 d \left (\frac {x^{5} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{8 e}-\frac {5 d \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )}{8 e}\right )}{10 e}\right )-2 a c \left (\frac {x^{3} \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{6 e}-\frac {d \left (\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{4 e}-\frac {d \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (x \sqrt {e}+\sqrt {e \,x^{2}+d}\right )}{2 \sqrt {e}}\right )}{4 e}\right )}{2 e}\right )\)	\(259\)

Input:

int((e*x^2+d)^(1/2)*(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/2/e^(9/2)*(d*(a^2*e^4-1/4*a*c*d^2*e^2+7/128*c^2*d^4)*arctanh((e*x^2+d)^( 
1/2)/x/e^(1/2))+x*((1/5*c^2*x^8-2/3*a*c*x^4+a^2)*e^(9/2)+1/4*(d*(-7/60*c*x 
^4+a)*e^(5/2)-2/3*x^2*(-3/20*c*x^4+a)*e^(7/2)-7/32*d^2*c*(-2/3*e^(3/2)*x^2 
+e^(1/2)*d))*d*c)*(e*x^2+d)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.60 \[ \int \sqrt {d+e x^2} \left (a-c x^4\right )^2 \, dx=\left [\frac {15 \, {\left (7 \, c^{2} d^{5} - 32 \, a c d^{3} e^{2} + 128 \, a^{2} d e^{4}\right )} \sqrt {e} \log \left (-2 \, e x^{2} - 2 \, \sqrt {e x^{2} + d} \sqrt {e} x - d\right ) + 2 \, {\left (384 \, c^{2} e^{5} x^{9} + 48 \, c^{2} d e^{4} x^{7} - 8 \, {\left (7 \, c^{2} d^{2} e^{3} + 160 \, a c e^{5}\right )} x^{5} + 10 \, {\left (7 \, c^{2} d^{3} e^{2} - 32 \, a c d e^{4}\right )} x^{3} - 15 \, {\left (7 \, c^{2} d^{4} e - 32 \, a c d^{2} e^{3} - 128 \, a^{2} e^{5}\right )} x\right )} \sqrt {e x^{2} + d}}{7680 \, e^{5}}, -\frac {15 \, {\left (7 \, c^{2} d^{5} - 32 \, a c d^{3} e^{2} + 128 \, a^{2} d e^{4}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {-e} x}{\sqrt {e x^{2} + d}}\right ) - {\left (384 \, c^{2} e^{5} x^{9} + 48 \, c^{2} d e^{4} x^{7} - 8 \, {\left (7 \, c^{2} d^{2} e^{3} + 160 \, a c e^{5}\right )} x^{5} + 10 \, {\left (7 \, c^{2} d^{3} e^{2} - 32 \, a c d e^{4}\right )} x^{3} - 15 \, {\left (7 \, c^{2} d^{4} e - 32 \, a c d^{2} e^{3} - 128 \, a^{2} e^{5}\right )} x\right )} \sqrt {e x^{2} + d}}{3840 \, e^{5}}\right ] \] Input:

integrate((e*x^2+d)^(1/2)*(-c*x^4+a)^2,x, algorithm="fricas")
 

Output:

[1/7680*(15*(7*c^2*d^5 - 32*a*c*d^3*e^2 + 128*a^2*d*e^4)*sqrt(e)*log(-2*e* 
x^2 - 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + 2*(384*c^2*e^5*x^9 + 48*c^2*d*e^4 
*x^7 - 8*(7*c^2*d^2*e^3 + 160*a*c*e^5)*x^5 + 10*(7*c^2*d^3*e^2 - 32*a*c*d* 
e^4)*x^3 - 15*(7*c^2*d^4*e - 32*a*c*d^2*e^3 - 128*a^2*e^5)*x)*sqrt(e*x^2 + 
 d))/e^5, -1/3840*(15*(7*c^2*d^5 - 32*a*c*d^3*e^2 + 128*a^2*d*e^4)*sqrt(-e 
)*arctan(sqrt(-e)*x/sqrt(e*x^2 + d)) - (384*c^2*e^5*x^9 + 48*c^2*d*e^4*x^7 
 - 8*(7*c^2*d^2*e^3 + 160*a*c*e^5)*x^5 + 10*(7*c^2*d^3*e^2 - 32*a*c*d*e^4) 
*x^3 - 15*(7*c^2*d^4*e - 32*a*c*d^2*e^3 - 128*a^2*e^5)*x)*sqrt(e*x^2 + d)) 
/e^5]
 

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.27 \[ \int \sqrt {d+e x^2} \left (a-c x^4\right )^2 \, dx=\begin {cases} \sqrt {d + e x^{2}} \left (\frac {c^{2} d x^{7}}{80 e} + \frac {c^{2} x^{9}}{10} + \frac {x^{5} \left (- 2 a c e - \frac {7 c^{2} d^{2}}{80 e}\right )}{6 e} + \frac {x^{3} \left (- 2 a c d - \frac {5 d \left (- 2 a c e - \frac {7 c^{2} d^{2}}{80 e}\right )}{6 e}\right )}{4 e} + \frac {x \left (a^{2} e - \frac {3 d \left (- 2 a c d - \frac {5 d \left (- 2 a c e - \frac {7 c^{2} d^{2}}{80 e}\right )}{6 e}\right )}{4 e}\right )}{2 e}\right ) + \left (a^{2} d - \frac {d \left (a^{2} e - \frac {3 d \left (- 2 a c d - \frac {5 d \left (- 2 a c e - \frac {7 c^{2} d^{2}}{80 e}\right )}{6 e}\right )}{4 e}\right )}{2 e}\right ) \left (\begin {cases} \frac {\log {\left (2 \sqrt {e} \sqrt {d + e x^{2}} + 2 e x \right )}}{\sqrt {e}} & \text {for}\: d \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {e x^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: e \neq 0 \\\sqrt {d} \left (a^{2} x - \frac {2 a c x^{5}}{5} + \frac {c^{2} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \] Input:

integrate((e*x**2+d)**(1/2)*(-c*x**4+a)**2,x)
 

Output:

Piecewise((sqrt(d + e*x**2)*(c**2*d*x**7/(80*e) + c**2*x**9/10 + x**5*(-2* 
a*c*e - 7*c**2*d**2/(80*e))/(6*e) + x**3*(-2*a*c*d - 5*d*(-2*a*c*e - 7*c** 
2*d**2/(80*e))/(6*e))/(4*e) + x*(a**2*e - 3*d*(-2*a*c*d - 5*d*(-2*a*c*e - 
7*c**2*d**2/(80*e))/(6*e))/(4*e))/(2*e)) + (a**2*d - d*(a**2*e - 3*d*(-2*a 
*c*d - 5*d*(-2*a*c*e - 7*c**2*d**2/(80*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)/sqr 
t(e*x**2), True)), Ne(e, 0)), (sqrt(d)*(a**2*x - 2*a*c*x**5/5 + c**2*x**9/ 
9), True))
 

Maxima [F(-2)]

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

integrate((e*x^2+d)^(1/2)*(-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 = 183, normalized size of antiderivative = 0.83 \[ \int \sqrt {d+e x^2} \left (a-c x^4\right )^2 \, dx=\frac {1}{3840} \, {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, c^{2} x^{2} + \frac {c^{2} d}{e}\right )} x^{2} - \frac {7 \, c^{2} d^{2} e^{6} + 160 \, a c e^{8}}{e^{8}}\right )} x^{2} + \frac {5 \, {\left (7 \, c^{2} d^{3} e^{5} - 32 \, a c d e^{7}\right )}}{e^{8}}\right )} x^{2} - \frac {15 \, {\left (7 \, c^{2} d^{4} e^{4} - 32 \, a c d^{2} e^{6} - 128 \, a^{2} e^{8}\right )}}{e^{8}}\right )} \sqrt {e x^{2} + d} x - \frac {{\left (7 \, c^{2} d^{5} - 32 \, a c d^{3} e^{2} + 128 \, a^{2} d e^{4}\right )} \log \left ({\left | -\sqrt {e} x + \sqrt {e x^{2} + d} \right |}\right )}{256 \, e^{\frac {9}{2}}} \] Input:

integrate((e*x^2+d)^(1/2)*(-c*x^4+a)^2,x, algorithm="giac")
 

Output:

1/3840*(2*(4*(6*(8*c^2*x^2 + c^2*d/e)*x^2 - (7*c^2*d^2*e^6 + 160*a*c*e^8)/ 
e^8)*x^2 + 5*(7*c^2*d^3*e^5 - 32*a*c*d*e^7)/e^8)*x^2 - 15*(7*c^2*d^4*e^4 - 
 32*a*c*d^2*e^6 - 128*a^2*e^8)/e^8)*sqrt(e*x^2 + d)*x - 1/256*(7*c^2*d^5 - 
 32*a*c*d^3*e^2 + 128*a^2*d*e^4)*log(abs(-sqrt(e)*x + sqrt(e*x^2 + d)))/e^ 
(9/2)
 

Mupad [F(-1)]

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

int((a - c*x^4)^2*(d + e*x^2)^(1/2),x)
 

Output:

int((a - c*x^4)^2*(d + e*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.22 \[ \int \sqrt {d+e x^2} \left (a-c x^4\right )^2 \, dx=\frac {1920 \sqrt {e \,x^{2}+d}\, a^{2} e^{5} x +480 \sqrt {e \,x^{2}+d}\, a c \,d^{2} e^{3} x -320 \sqrt {e \,x^{2}+d}\, a c d \,e^{4} x^{3}-1280 \sqrt {e \,x^{2}+d}\, a c \,e^{5} x^{5}-105 \sqrt {e \,x^{2}+d}\, c^{2} d^{4} e x +70 \sqrt {e \,x^{2}+d}\, c^{2} d^{3} e^{2} x^{3}-56 \sqrt {e \,x^{2}+d}\, c^{2} d^{2} e^{3} x^{5}+48 \sqrt {e \,x^{2}+d}\, c^{2} d \,e^{4} x^{7}+384 \sqrt {e \,x^{2}+d}\, c^{2} e^{5} x^{9}+1920 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a^{2} d \,e^{4}-480 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) a c \,d^{3} e^{2}+105 \sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) c^{2} d^{5}}{3840 e^{5}} \] Input:

int((e*x^2+d)^(1/2)*(-c*x^4+a)^2,x)
 

Output:

(1920*sqrt(d + e*x**2)*a**2*e**5*x + 480*sqrt(d + e*x**2)*a*c*d**2*e**3*x 
- 320*sqrt(d + e*x**2)*a*c*d*e**4*x**3 - 1280*sqrt(d + e*x**2)*a*c*e**5*x* 
*5 - 105*sqrt(d + e*x**2)*c**2*d**4*e*x + 70*sqrt(d + e*x**2)*c**2*d**3*e* 
*2*x**3 - 56*sqrt(d + e*x**2)*c**2*d**2*e**3*x**5 + 48*sqrt(d + e*x**2)*c* 
*2*d*e**4*x**7 + 384*sqrt(d + e*x**2)*c**2*e**5*x**9 + 1920*sqrt(e)*log((s 
qrt(d + e*x**2) + sqrt(e)*x)/sqrt(d))*a**2*d*e**4 - 480*sqrt(e)*log((sqrt( 
d + e*x**2) + sqrt(e)*x)/sqrt(d))*a*c*d**3*e**2 + 105*sqrt(e)*log((sqrt(d 
+ e*x**2) + sqrt(e)*x)/sqrt(d))*c**2*d**5)/(3840*e**5)