\(\int \sqrt {a-b x^4} (A+B x^2+C x^4) \, dx\) [21]

Optimal result

Integrand size = 25, antiderivative size = 195 \[ \int \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \left (5 (7 A b+a C)+21 b B x^2\right ) \sqrt {a-b x^4}}{105 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}+\frac {2 a^{7/4} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{5 b^{3/4} \sqrt {a-b x^4}}+\frac {2 a^{5/4} \left (35 A b-21 \sqrt {a} \sqrt {b} B+5 a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{105 b^{5/4} \sqrt {a-b x^4}} \] Output:

1/105*x*(21*B*b*x^2+35*A*b+5*C*a)*(-b*x^4+a)^(1/2)/b-1/7*C*x*(-b*x^4+a)^(3 
/2)/b+2/5*a^(7/4)*B*(1-b*x^4/a)^(1/2)*EllipticE(b^(1/4)*x/a^(1/4),I)/b^(3/ 
4)/(-b*x^4+a)^(1/2)+2/105*a^(5/4)*(35*A*b-21*a^(1/2)*b^(1/2)*B+5*C*a)*(1-b 
*x^4/a)^(1/2)*EllipticF(b^(1/4)*x/a^(1/4),I)/b^(5/4)/(-b*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.59 \[ \int \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \sqrt {a-b x^4} \left (-3 C \left (a-b x^4\right ) \sqrt {1-\frac {b x^4}{a}}+3 (7 A b+a C) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {b x^4}{a}\right )+7 b B x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )\right )}{21 b \sqrt {1-\frac {b x^4}{a}}} \] Input:

Integrate[Sqrt[a - b*x^4]*(A + B*x^2 + C*x^4),x]
 

Output:

(x*Sqrt[a - b*x^4]*(-3*C*(a - b*x^4)*Sqrt[1 - (b*x^4)/a] + 3*(7*A*b + a*C) 
*Hypergeometric2F1[-1/2, 1/4, 5/4, (b*x^4)/a] + 7*b*B*x^2*Hypergeometric2F 
1[-1/2, 3/4, 7/4, (b*x^4)/a]))/(21*b*Sqrt[1 - (b*x^4)/a])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2427, 25, 1491, 27, 1513, 27, 765, 762, 1390, 1389, 327}

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.

\(\displaystyle \int \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx\)

\(\Big \downarrow \) 2427

\(\displaystyle -\frac {\int -\left (\left (7 b B x^2+7 A b+a C\right ) \sqrt {a-b x^4}\right )dx}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \left (7 b B x^2+7 A b+a C\right ) \sqrt {a-b x^4}dx}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 1491

\(\displaystyle \frac {\frac {1}{15} \int \frac {2 a \left (21 b B x^2+5 (7 A b+a C)\right )}{\sqrt {a-b x^4}}dx+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2}{15} a \int \frac {21 b B x^2+5 (7 A b+a C)}{\sqrt {a-b x^4}}dx+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 1513

\(\displaystyle \frac {\frac {2}{15} a \left (\left (-21 \sqrt {a} \sqrt {b} B+5 a C+35 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+21 \sqrt {a} \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-b x^4}}dx\right )+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2}{15} a \left (\left (-21 \sqrt {a} \sqrt {b} B+5 a C+35 A b\right ) \int \frac {1}{\sqrt {a-b x^4}}dx+21 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\frac {2}{15} a \left (\frac {\sqrt {1-\frac {b x^4}{a}} \left (-21 \sqrt {a} \sqrt {b} B+5 a C+35 A b\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+21 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx\right )+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\frac {2}{15} a \left (21 \sqrt {b} B \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a-b x^4}}dx+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-21 \sqrt {a} \sqrt {b} B+5 a C+35 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {\frac {2}{15} a \left (\frac {21 \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {1-\frac {b x^4}{a}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-21 \sqrt {a} \sqrt {b} B+5 a C+35 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\frac {2}{15} a \left (\frac {21 \sqrt {a} \sqrt {b} B \sqrt {1-\frac {b x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {b} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}}dx}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-21 \sqrt {a} \sqrt {b} B+5 a C+35 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {2}{15} a \left (\frac {21 a^{3/4} \sqrt [4]{b} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} \left (-21 \sqrt {a} \sqrt {b} B+5 a C+35 A b\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\right )+\frac {1}{15} x \sqrt {a-b x^4} \left (5 (a C+7 A b)+21 b B x^2\right )}{7 b}-\frac {C x \left (a-b x^4\right )^{3/2}}{7 b}\)

Input:

Int[Sqrt[a - b*x^4]*(A + B*x^2 + C*x^4),x]
 

Output:

-1/7*(C*x*(a - b*x^4)^(3/2))/b + ((x*(5*(7*A*b + a*C) + 21*b*B*x^2)*Sqrt[a 
 - b*x^4])/15 + (2*a*((21*a^(3/4)*b^(1/4)*B*Sqrt[1 - (b*x^4)/a]*EllipticE[ 
ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - b*x^4] + (a^(1/4)*(35*A*b - 21* 
Sqrt[a]*Sqrt[b]*B + 5*a*C)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x 
)/a^(1/4)], -1])/(b^(1/4)*Sqrt[a - b*x^4])))/15)/(7*b)
 

Defintions of rubi rules used

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

rule 1491

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*( 
d*(4*p + 3) + e*(4*p + 1)*x^2)*((a + c*x^4)^p/((4*p + 1)*(4*p + 3))), x] + 
Simp[2*(p/((4*p + 1)*(4*p + 3)))   Int[Simp[2*a*d*(4*p + 3) + (2*a*e*(4*p + 
 1))*x^2, x]*(a + c*x^4)^(p - 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c 
*d^2 + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

rule 1513

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[-c/a, 2]}, Simp[(d*q - e)/q   Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q 
  Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg 
Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
 

rule 2427

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x 
]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 
1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1))   Int[ExpandToSum[b*(q 
 + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, 
 x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ 
[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
 

Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.17

method	result	size
elliptic	\(\frac {C \,x^{5} \sqrt {-b \,x^{4}+a}}{7}+\frac {B \,x^{3} \sqrt {-b \,x^{4}+a}}{5}-\frac {\left (-A b +\frac {2 C a}{7}\right ) x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (A a +\frac {a \left (-A b +\frac {2 C a}{7}\right )}{3 b}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {2 B \,a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\)	\(229\)
risch	\(\frac {x \left (15 C b \,x^{4}+21 B b \,x^{2}+35 A b -10 C a \right ) \sqrt {-b \,x^{4}+a}}{105 b}+\frac {2 a \left (\frac {35 A b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {5 C a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {21 B \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )}{105 b}\)	\(268\)
default	\(A \left (\frac {x \sqrt {-b \,x^{4}+a}}{3}+\frac {2 a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )+B \left (\frac {x^{3} \sqrt {-b \,x^{4}+a}}{5}-\frac {2 a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{5 \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\right )+C \left (\frac {x^{5} \sqrt {-b \,x^{4}+a}}{7}-\frac {2 a x \sqrt {-b \,x^{4}+a}}{21 b}+\frac {2 a^{2} \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{21 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\)	\(293\)

Input:

int((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/7*C*x^5*(-b*x^4+a)^(1/2)+1/5*B*x^3*(-b*x^4+a)^(1/2)-1/3*(-A*b+2/7*C*a)/b 
*x*(-b*x^4+a)^(1/2)+(A*a+1/3*a/b*(-A*b+2/7*C*a))/(1/a^(1/2)*b^(1/2))^(1/2) 
*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b*x^4+a)^(1 
/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-2/5*B*a^(3/2)/(1/a^(1/2)*b^(1 
/2))^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b 
*x^4+a)^(1/2)/b^(1/2)*(EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE( 
x*(1/a^(1/2)*b^(1/2))^(1/2),I))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.66 \[ \int \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=-\frac {42 \, B a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - 2 \, {\left ({\left (21 \, B + 5 \, C\right )} a + 35 \, A b\right )} \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (15 \, C b x^{6} + 21 \, B b x^{4} - 5 \, {\left (2 \, C a - 7 \, A b\right )} x^{2} - 42 \, B a\right )} \sqrt {-b x^{4} + a}}{105 \, b x} \] Input:

integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="fricas")
 

Output:

-1/105*(42*B*a*sqrt(-b)*x*(a/b)^(3/4)*elliptic_e(arcsin((a/b)^(1/4)/x), -1 
) - 2*((21*B + 5*C)*a + 35*A*b)*sqrt(-b)*x*(a/b)^(3/4)*elliptic_f(arcsin(( 
a/b)^(1/4)/x), -1) - (15*C*b*x^6 + 21*B*b*x^4 - 5*(2*C*a - 7*A*b)*x^2 - 42 
*B*a)*sqrt(-b*x^4 + a))/(b*x)
 

Sympy [A] (verification not implemented)

Time = 1.49 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.66 \[ \int \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {A \sqrt {a} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {B \sqrt {a} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {C \sqrt {a} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \] Input:

integrate((-b*x**4+a)**(1/2)*(C*x**4+B*x**2+A),x)
 

Output:

A*sqrt(a)*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**4*exp_polar(2*I*pi) 
/a)/(4*gamma(5/4)) + B*sqrt(a)*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), 
b*x**4*exp_polar(2*I*pi)/a)/(4*gamma(7/4)) + C*sqrt(a)*x**5*gamma(5/4)*hyp 
er((-1/2, 5/4), (9/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*gamma(9/4))
 

Maxima [F]

\[ \int \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a} \,d x } \] Input:

integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a), x)
 

Giac [F]

\[ \int \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} \sqrt {-b x^{4} + a} \,d x } \] Input:

integrate((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x, algorithm="giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)*sqrt(-b*x^4 + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

int((a - b*x^4)^(1/2)*(A + B*x^2 + C*x^4),x)
 

Output:

int((a - b*x^4)^(1/2)*(A + B*x^2 + C*x^4), x)
 

Reduce [F]

\[ \int \sqrt {a-b x^4} \left (A+B x^2+C x^4\right ) \, dx=\frac {35 \sqrt {-b \,x^{4}+a}\, a b x -10 \sqrt {-b \,x^{4}+a}\, a c x +21 \sqrt {-b \,x^{4}+a}\, b^{2} x^{3}+15 \sqrt {-b \,x^{4}+a}\, b c \,x^{5}+70 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a^{2} b +10 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a^{2} c +42 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{-b \,x^{4}+a}d x \right ) a \,b^{2}}{105 b} \] Input:

int((-b*x^4+a)^(1/2)*(C*x^4+B*x^2+A),x)
 

Output:

(35*sqrt(a - b*x**4)*a*b*x - 10*sqrt(a - b*x**4)*a*c*x + 21*sqrt(a - b*x** 
4)*b**2*x**3 + 15*sqrt(a - b*x**4)*b*c*x**5 + 70*int(sqrt(a - b*x**4)/(a - 
 b*x**4),x)*a**2*b + 10*int(sqrt(a - b*x**4)/(a - b*x**4),x)*a**2*c + 42*i 
nt((sqrt(a - b*x**4)*x**2)/(a - b*x**4),x)*a*b**2)/(105*b)