\(\int \frac {A+B x^2+C x^4}{\sqrt {a-b x^4}} \, dx\) [29]

Optimal result

Integrand size = 25, antiderivative size = 153 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^4}} \, dx=-\frac {C x \sqrt {a-b x^4}}{3 b}+\frac {a^{3/4} B \sqrt {1-\frac {b x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{b^{3/4} \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \left (3 A b-3 \sqrt {a} \sqrt {b} B+a C\right ) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 b^{5/4} \sqrt {a-b x^4}} \] Output:

-1/3*C*x*(-b*x^4+a)^(1/2)/b+a^(3/4)*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)+1/3*a^(1/4)*(3*A*b-3*a^(1/2)*b^(1/2) 
*B+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 = 10.06 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^4}} \, dx=\frac {-a C x+b C x^5+(3 A b+a C) x \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {b x^4}{a}\right )+b B x^3 \sqrt {1-\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {b x^4}{a}\right )}{3 b \sqrt {a-b x^4}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2427, 25, 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.

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

\(\Big \downarrow \) 2427

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

-1/3*(C*x*Sqrt[a - b*x^4])/b + ((3*a^(3/4)*b^(1/4)*B*Sqrt[1 - (b*x^4)/a]*E 
llipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - b*x^4] + (a^(1/4)*(3*A 
*b - 3*Sqrt[a]*Sqrt[b]*B + 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]))/(3*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 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 = 1.52 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.17

method	result	size
elliptic	\(-\frac {C x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\left (A +\frac {a C}{3 b}\right ) \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {B \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}\)	\(179\)
default	\(\frac {A \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {B \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}\, \sqrt {b}}+C \left (-\frac {x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{3 b \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\right )\)	\(241\)
risch	\(-\frac {C x \sqrt {-b \,x^{4}+a}}{3 b}+\frac {\frac {C a \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {3 A b \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {3 B \sqrt {b}\, \sqrt {a}\, \sqrt {1-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}}{3 b}\)	\(244\)

Input:

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

Output:

-1/3*C*x*(-b*x^4+a)^(1/2)/b+(A+1/3*a*C/b)/(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)*Ell 
ipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-B*a^(1/2)/(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 = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^4}} \, dx=-\frac {3 \, B a \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left ({\left (3 \, B + C\right )} a + 3 \, A b\right )} \sqrt {-b} x \left (\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + \sqrt {-b x^{4} + a} {\left (C a x^{2} + 3 \, B a\right )}}{3 \, a b x} \] Input:

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

Output:

-1/3*(3*B*a*sqrt(-b)*x*(a/b)^(3/4)*elliptic_e(arcsin((a/b)^(1/4)/x), -1) - 
 ((3*B + C)*a + 3*A*b)*sqrt(-b)*x*(a/b)^(3/4)*elliptic_f(arcsin((a/b)^(1/4 
)/x), -1) + sqrt(-b*x^4 + a)*(C*a*x^2 + 3*B*a))/(a*b*x)
 

Sympy [A] (verification not implemented)

Time = 1.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.81 \[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^4}} \, dx=\frac {A x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {B x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {C x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} \] Input:

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

Output:

A*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sq 
rt(a)*gamma(5/4)) + B*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), b*x**4*exp 
_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(7/4)) + C*x**5*gamma(5/4)*hyper((1/2, 5 
/4), (9/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(9/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{\sqrt {a-b x^4}} \, dx=\frac {-\sqrt {-b \,x^{4}+a}\, c x +3 \left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a b +\left (\int \frac {\sqrt {-b \,x^{4}+a}}{-b \,x^{4}+a}d x \right ) a c +3 \left (\int \frac {\sqrt {-b \,x^{4}+a}\, x^{2}}{-b \,x^{4}+a}d x \right ) b^{2}}{3 b} \] Input:

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

Output:

( - sqrt(a - b*x**4)*c*x + 3*int(sqrt(a - b*x**4)/(a - b*x**4),x)*a*b + in 
t(sqrt(a - b*x**4)/(a - b*x**4),x)*a*c + 3*int((sqrt(a - b*x**4)*x**2)/(a 
- b*x**4),x)*b**2)/(3*b)