\(\int \frac {A+B x^2+C x^4}{\sqrt {c+\frac {d}{x^2}} \sqrt {a+b x^2}} \, dx\) [242]

Optimal result

Integrand size = 35, antiderivative size = 221 \[ \int \frac {A+B x^2+C x^4}{\sqrt {c+\frac {d}{x^2}} \sqrt {a+b x^2}} \, dx=\frac {(4 b B c-5 a c C-3 b C d) \sqrt {c+\frac {d}{x^2}} x \sqrt {a+b x^2}}{8 b^2 c^2}+\frac {C \sqrt {c+\frac {d}{x^2}} x \left (a+b x^2\right )^{3/2}}{4 b^2 c}+\frac {\sqrt {d} \left (8 A b^2 c^2+3 a^2 c^2 C-b^2 d (4 B c-3 C d)-2 a b c (2 B c-C d)\right ) \sqrt {1+\frac {c x^2}{d}} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {d} \sqrt {1+\frac {c x^2}{d}}}\right )}{8 b^{5/2} c^{5/2} \sqrt {c+\frac {d}{x^2}} x} \] Output:

1/8*(4*B*b*c-5*C*a*c-3*C*b*d)*(c+d/x^2)^(1/2)*x*(b*x^2+a)^(1/2)/b^2/c^2+1/ 
4*C*(c+d/x^2)^(1/2)*x*(b*x^2+a)^(3/2)/b^2/c+1/8*d^(1/2)*(8*A*b^2*c^2+3*a^2 
*c^2*C-b^2*d*(4*B*c-3*C*d)-2*a*b*c*(2*B*c-C*d))*(1+c*x^2/d)^(1/2)*arctanh( 
c^(1/2)*(b*x^2+a)^(1/2)/b^(1/2)/d^(1/2)/(1+c*x^2/d)^(1/2))/b^(5/2)/c^(5/2) 
/(c+d/x^2)^(1/2)/x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 5.88 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x^2+C x^4}{\sqrt {c+\frac {d}{x^2}} \sqrt {a+b x^2}} \, dx=\frac {\left (d+c x^2\right ) \left (-\sqrt {c} (a c-b d) \sqrt {a+b x^2} \sqrt {\frac {b \left (d+c x^2\right )}{-a c+b d}} \left (3 a c C+b \left (-4 B c+3 C d-2 c C x^2\right )\right )-\sqrt {-a c+b d} \left (8 A b^2 c^2+3 a^2 c^2 C+2 a b c (-2 B c+C d)+b^2 d (-4 B c+3 C d)\right ) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {-a c+b d}}\right )\right )}{8 b^2 c^{5/2} (a c-b d) \sqrt {c+\frac {d}{x^2}} x \sqrt {\frac {b \left (d+c x^2\right )}{-a c+b d}}} \] Input:

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

Output:

((d + c*x^2)*(-(Sqrt[c]*(a*c - b*d)*Sqrt[a + b*x^2]*Sqrt[(b*(d + c*x^2))/( 
-(a*c) + b*d)]*(3*a*c*C + b*(-4*B*c + 3*C*d - 2*c*C*x^2))) - Sqrt[-(a*c) + 
 b*d]*(8*A*b^2*c^2 + 3*a^2*c^2*C + 2*a*b*c*(-2*B*c + C*d) + b^2*d*(-4*B*c 
+ 3*C*d))*ArcSinh[(Sqrt[c]*Sqrt[a + b*x^2])/Sqrt[-(a*c) + b*d]]))/(8*b^2*c 
^(5/2)*(a*c - b*d)*Sqrt[c + d/x^2]*x*Sqrt[(b*(d + c*x^2))/(-(a*c) + b*d)])
 

Rubi [A] (verified)

Time = 2.21 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7272, 7266, 1194, 27, 90, 66, 221}

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.

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

\(\Big \downarrow \) 7272

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

\(\Big \downarrow \) 7266

\(\displaystyle \frac {\sqrt {\frac {c x^2}{d}+1} \int \frac {C x^4+B x^2+A}{\sqrt {b x^2+a} \sqrt {\frac {c x^2}{d}+1}}dx^2}{2 x \sqrt {c+\frac {d}{x^2}}}\)

\(\Big \downarrow \) 1194

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {\sqrt {\frac {c x^2}{d}+1} \left (\frac {\frac {\left (3 a^2 c^2 C-2 a b c (2 B c-C d)+8 A b^2 c^2-b^2 d (4 B c-3 C d)\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {\frac {c x^2}{d}+1}}dx^2}{2 c}+\frac {d \sqrt {a+b x^2} \sqrt {\frac {c x^2}{d}+1} (-5 a c C+4 b B c-3 b C d)}{c}}{4 b^2 c}+\frac {C d \left (a+b x^2\right )^{3/2} \sqrt {\frac {c x^2}{d}+1}}{2 b^2 c}\right )}{2 x \sqrt {c+\frac {d}{x^2}}}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {\sqrt {\frac {c x^2}{d}+1} \left (\frac {\frac {\left (3 a^2 c^2 C-2 a b c (2 B c-C d)+8 A b^2 c^2-b^2 d (4 B c-3 C d)\right ) \int \frac {1}{b-\frac {c x^4}{d}}d\frac {\sqrt {b x^2+a}}{\sqrt {\frac {c x^2}{d}+1}}}{c}+\frac {d \sqrt {a+b x^2} \sqrt {\frac {c x^2}{d}+1} (-5 a c C+4 b B c-3 b C d)}{c}}{4 b^2 c}+\frac {C d \left (a+b x^2\right )^{3/2} \sqrt {\frac {c x^2}{d}+1}}{2 b^2 c}\right )}{2 x \sqrt {c+\frac {d}{x^2}}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {\frac {c x^2}{d}+1} \left (\frac {\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {d} \sqrt {\frac {c x^2}{d}+1}}\right ) \left (3 a^2 c^2 C-2 a b c (2 B c-C d)+8 A b^2 c^2-b^2 d (4 B c-3 C d)\right )}{\sqrt {b} c^{3/2}}+\frac {d \sqrt {a+b x^2} \sqrt {\frac {c x^2}{d}+1} (-5 a c C+4 b B c-3 b C d)}{c}}{4 b^2 c}+\frac {C d \left (a+b x^2\right )^{3/2} \sqrt {\frac {c x^2}{d}+1}}{2 b^2 c}\right )}{2 x \sqrt {c+\frac {d}{x^2}}}\)

Input:

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

Output:

(Sqrt[1 + (c*x^2)/d]*((C*d*(a + b*x^2)^(3/2)*Sqrt[1 + (c*x^2)/d])/(2*b^2*c 
) + ((d*(4*b*B*c - 5*a*c*C - 3*b*C*d)*Sqrt[a + b*x^2]*Sqrt[1 + (c*x^2)/d]) 
/c + (Sqrt[d]*(8*A*b^2*c^2 + 3*a^2*c^2*C - b^2*d*(4*B*c - 3*C*d) - 2*a*b*c 
*(2*B*c - C*d))*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[d]*Sqrt[1 
+ (c*x^2)/d])])/(Sqrt[b]*c^(3/2)))/(4*b^2*c)))/(2*Sqrt[c + d/x^2]*x)
 

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 66

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

rule 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 221

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

rule 1194

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x 
)^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 
*p + 1))   Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 
2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) 
*(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ 
[p, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
 

rule 7266

Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1)   Subst[Int[SubstFor[x^(m 
+ 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function 
OfQ[x^(m + 1), u, x]
 

rule 7272

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( 
a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) 
   Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] &&  ! 
IntegerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
 

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.99

method	result	size
risch	\(\frac {\left (2 C \,x^{2} b c +4 B b c -3 C a c -3 C b d \right ) \left (c \,x^{2}+d \right ) \sqrt {b \,x^{2}+a}}{8 b^{2} c^{2} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x}+\frac {\left (8 A \,b^{2} c^{2}-4 B a b \,c^{2}-4 B \,b^{2} c d +3 a^{2} c^{2} C +2 C a b c d +3 C \,b^{2} d^{2}\right ) \ln \left (\frac {\frac {1}{2} a c +\frac {1}{2} b d +b c \,x^{2}}{\sqrt {b c}}+\sqrt {b c \,x^{4}+\left (a c +b d \right ) x^{2}+a d}\right ) \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}{16 b^{2} c^{2} \sqrt {b c}\, \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \sqrt {b \,x^{2}+a}}\)	\(218\)
default	\(\frac {\left (c \,x^{2}+d \right ) \left (4 C \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}\, b c \,x^{2}+8 A \ln \left (\frac {2 b c \,x^{2}+2 \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}+a c +b d}{2 \sqrt {b c}}\right ) b^{2} c^{2}-4 B \ln \left (\frac {2 b c \,x^{2}+2 \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}+a c +b d}{2 \sqrt {b c}}\right ) a b \,c^{2}-4 B \ln \left (\frac {2 b c \,x^{2}+2 \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}+a c +b d}{2 \sqrt {b c}}\right ) b^{2} c d +3 C \ln \left (\frac {2 b c \,x^{2}+2 \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}+a c +b d}{2 \sqrt {b c}}\right ) a^{2} c^{2}+2 C \ln \left (\frac {2 b c \,x^{2}+2 \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}+a c +b d}{2 \sqrt {b c}}\right ) a b c d +3 C \ln \left (\frac {2 b c \,x^{2}+2 \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}+a c +b d}{2 \sqrt {b c}}\right ) b^{2} d^{2}+8 B \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}\, b c -6 C \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}\, a c -6 C \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}\, \sqrt {b c}\, b d \right ) \sqrt {b \,x^{2}+a}}{16 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, x \sqrt {b c}\, c^{2} b^{2} \sqrt {\left (c \,x^{2}+d \right ) \left (b \,x^{2}+a \right )}}\)	\(501\)

Input:

int((C*x^4+B*x^2+A)/(c+d/x^2)^(1/2)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOS 
E)
 

Output:

1/8/b^2/c^2*(2*C*b*c*x^2+4*B*b*c-3*C*a*c-3*C*b*d)*(c*x^2+d)*(b*x^2+a)^(1/2 
)/((c*x^2+d)/x^2)^(1/2)/x+1/16/b^2/c^2*(8*A*b^2*c^2-4*B*a*b*c^2-4*B*b^2*c* 
d+3*C*a^2*c^2+2*C*a*b*c*d+3*C*b^2*d^2)*ln((1/2*a*c+1/2*b*d+b*c*x^2)/(b*c)^ 
(1/2)+(b*c*x^4+(a*c+b*d)*x^2+a*d)^(1/2))/(b*c)^(1/2)/((c*x^2+d)/x^2)^(1/2) 
/x*((c*x^2+d)*(b*x^2+a))^(1/2)/(b*x^2+a)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.97 \[ \int \frac {A+B x^2+C x^4}{\sqrt {c+\frac {d}{x^2}} \sqrt {a+b x^2}} \, dx=\left [\frac {{\left (3 \, C b^{2} d^{2} + {\left (3 \, C a^{2} - 4 \, B a b + 8 \, A b^{2}\right )} c^{2} + 2 \, {\left (C a b - 2 \, B b^{2}\right )} c d\right )} \sqrt {b c} \log \left (8 \, b^{2} c^{2} x^{4} + a^{2} c^{2} + 6 \, a b c d + b^{2} d^{2} + 8 \, {\left (a b c^{2} + b^{2} c d\right )} x^{2} + 4 \, {\left (2 \, b c x^{3} + {\left (a c + b d\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {b c} \sqrt {\frac {c x^{2} + d}{x^{2}}}\right ) + 4 \, {\left (2 \, C b^{2} c^{2} x^{3} - {\left (3 \, C b^{2} c d + {\left (3 \, C a b - 4 \, B b^{2}\right )} c^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{32 \, b^{3} c^{3}}, -\frac {{\left (3 \, C b^{2} d^{2} + {\left (3 \, C a^{2} - 4 \, B a b + 8 \, A b^{2}\right )} c^{2} + 2 \, {\left (C a b - 2 \, B b^{2}\right )} c d\right )} \sqrt {-b c} \arctan \left (\frac {{\left (2 \, b c x^{3} + {\left (a c + b d\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {-b c} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, {\left (b^{2} c^{2} x^{4} + a b c d + {\left (a b c^{2} + b^{2} c d\right )} x^{2}\right )}}\right ) - 2 \, {\left (2 \, C b^{2} c^{2} x^{3} - {\left (3 \, C b^{2} c d + {\left (3 \, C a b - 4 \, B b^{2}\right )} c^{2}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{16 \, b^{3} c^{3}}\right ] \] Input:

integrate((C*x^4+B*x^2+A)/(c+d/x^2)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="fr 
icas")
 

Output:

[1/32*((3*C*b^2*d^2 + (3*C*a^2 - 4*B*a*b + 8*A*b^2)*c^2 + 2*(C*a*b - 2*B*b 
^2)*c*d)*sqrt(b*c)*log(8*b^2*c^2*x^4 + a^2*c^2 + 6*a*b*c*d + b^2*d^2 + 8*( 
a*b*c^2 + b^2*c*d)*x^2 + 4*(2*b*c*x^3 + (a*c + b*d)*x)*sqrt(b*x^2 + a)*sqr 
t(b*c)*sqrt((c*x^2 + d)/x^2)) + 4*(2*C*b^2*c^2*x^3 - (3*C*b^2*c*d + (3*C*a 
*b - 4*B*b^2)*c^2)*x)*sqrt(b*x^2 + a)*sqrt((c*x^2 + d)/x^2))/(b^3*c^3), -1 
/16*((3*C*b^2*d^2 + (3*C*a^2 - 4*B*a*b + 8*A*b^2)*c^2 + 2*(C*a*b - 2*B*b^2 
)*c*d)*sqrt(-b*c)*arctan(1/2*(2*b*c*x^3 + (a*c + b*d)*x)*sqrt(b*x^2 + a)*s 
qrt(-b*c)*sqrt((c*x^2 + d)/x^2)/(b^2*c^2*x^4 + a*b*c*d + (a*b*c^2 + b^2*c* 
d)*x^2)) - 2*(2*C*b^2*c^2*x^3 - (3*C*b^2*c*d + (3*C*a*b - 4*B*b^2)*c^2)*x) 
*sqrt(b*x^2 + a)*sqrt((c*x^2 + d)/x^2))/(b^3*c^3)]
 

Sympy [F]

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

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

Output:

Integral((A + B*x**2 + C*x**4)/(sqrt(a + b*x**2)*sqrt(c + d/x**2)), x)
 

Maxima [F]

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

integrate((C*x^4+B*x^2+A)/(c+d/x^2)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="ma 
xima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (187) = 374\).

Time = 0.25 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.08 \[ \int \frac {A+B x^2+C x^4}{\sqrt {c+\frac {d}{x^2}} \sqrt {a+b x^2}} \, dx=\frac {{\left (\sqrt {{\left (c x^{2} + d\right )} b c + a c^{2} - b c d} \sqrt {c x^{2} + d} {\left (\frac {2 \, {\left (c x^{2} + d\right )} C}{b c^{3}} - \frac {3 \, C a b c^{6} - 4 \, B b^{2} c^{6} + 5 \, C b^{2} c^{5} d}{b^{3} c^{8}}\right )} - \frac {{\left (3 \, C a^{2} c^{2} - 4 \, B a b c^{2} + 8 \, A b^{2} c^{2} + 2 \, C a b c d - 4 \, B b^{2} c d + 3 \, C b^{2} d^{2}\right )} \log \left ({\left | -\sqrt {c x^{2} + d} \sqrt {b c} + \sqrt {{\left (c x^{2} + d\right )} b c + a c^{2} - b c d} \right |}\right )}{\sqrt {b c} b^{2} c^{2}}\right )} c}{8 \, {\left | c \right |} \mathrm {sgn}\left (x\right )} + \frac {{\left (3 \, C a^{2} c^{3} \log \left ({\left | -\sqrt {b c} \sqrt {d} + \sqrt {a c^{2}} \right |}\right ) - 4 \, B a b c^{3} \log \left ({\left | -\sqrt {b c} \sqrt {d} + \sqrt {a c^{2}} \right |}\right ) + 8 \, A b^{2} c^{3} \log \left ({\left | -\sqrt {b c} \sqrt {d} + \sqrt {a c^{2}} \right |}\right ) + 2 \, C a b c^{2} d \log \left ({\left | -\sqrt {b c} \sqrt {d} + \sqrt {a c^{2}} \right |}\right ) - 4 \, B b^{2} c^{2} d \log \left ({\left | -\sqrt {b c} \sqrt {d} + \sqrt {a c^{2}} \right |}\right ) + 3 \, C b^{2} c d^{2} \log \left ({\left | -\sqrt {b c} \sqrt {d} + \sqrt {a c^{2}} \right |}\right ) + 3 \, \sqrt {a c^{2}} \sqrt {b c} C a c \sqrt {d} - 4 \, \sqrt {a c^{2}} \sqrt {b c} B b c \sqrt {d} + 3 \, \sqrt {a c^{2}} \sqrt {b c} C b d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{8 \, \sqrt {b c} b^{2} c^{2} {\left | c \right |}} \] Input:

integrate((C*x^4+B*x^2+A)/(c+d/x^2)^(1/2)/(b*x^2+a)^(1/2),x, algorithm="gi 
ac")
 

Output:

1/8*(sqrt((c*x^2 + d)*b*c + a*c^2 - b*c*d)*sqrt(c*x^2 + d)*(2*(c*x^2 + d)* 
C/(b*c^3) - (3*C*a*b*c^6 - 4*B*b^2*c^6 + 5*C*b^2*c^5*d)/(b^3*c^8)) - (3*C* 
a^2*c^2 - 4*B*a*b*c^2 + 8*A*b^2*c^2 + 2*C*a*b*c*d - 4*B*b^2*c*d + 3*C*b^2* 
d^2)*log(abs(-sqrt(c*x^2 + d)*sqrt(b*c) + sqrt((c*x^2 + d)*b*c + a*c^2 - b 
*c*d)))/(sqrt(b*c)*b^2*c^2))*c/(abs(c)*sgn(x)) + 1/8*(3*C*a^2*c^3*log(abs( 
-sqrt(b*c)*sqrt(d) + sqrt(a*c^2))) - 4*B*a*b*c^3*log(abs(-sqrt(b*c)*sqrt(d 
) + sqrt(a*c^2))) + 8*A*b^2*c^3*log(abs(-sqrt(b*c)*sqrt(d) + sqrt(a*c^2))) 
 + 2*C*a*b*c^2*d*log(abs(-sqrt(b*c)*sqrt(d) + sqrt(a*c^2))) - 4*B*b^2*c^2* 
d*log(abs(-sqrt(b*c)*sqrt(d) + sqrt(a*c^2))) + 3*C*b^2*c*d^2*log(abs(-sqrt 
(b*c)*sqrt(d) + sqrt(a*c^2))) + 3*sqrt(a*c^2)*sqrt(b*c)*C*a*c*sqrt(d) - 4* 
sqrt(a*c^2)*sqrt(b*c)*B*b*c*sqrt(d) + 3*sqrt(a*c^2)*sqrt(b*c)*C*b*d^(3/2)) 
*sgn(x)/(sqrt(b*c)*b^2*c^2*abs(c))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.35 \[ \int \frac {A+B x^2+C x^4}{\sqrt {c+\frac {d}{x^2}} \sqrt {a+b x^2}} \, dx=\frac {-3 \sqrt {b \,x^{2}+a}\, \sqrt {c \,x^{2}+d}\, a b \,c^{2}+4 \sqrt {b \,x^{2}+a}\, \sqrt {c \,x^{2}+d}\, b^{3} c +2 \sqrt {b \,x^{2}+a}\, \sqrt {c \,x^{2}+d}\, b^{2} c^{2} x^{2}-3 \sqrt {b \,x^{2}+a}\, \sqrt {c \,x^{2}+d}\, b^{2} c d +3 \sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {c}\, \sqrt {c \,x^{2}+d}\, b -\sqrt {b}\, \sqrt {b \,x^{2}+a}\, c \right ) a^{2} c^{2}+4 \sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {c}\, \sqrt {c \,x^{2}+d}\, b -\sqrt {b}\, \sqrt {b \,x^{2}+a}\, c \right ) a \,b^{2} c +2 \sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {c}\, \sqrt {c \,x^{2}+d}\, b -\sqrt {b}\, \sqrt {b \,x^{2}+a}\, c \right ) a b c d -4 \sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {c}\, \sqrt {c \,x^{2}+d}\, b -\sqrt {b}\, \sqrt {b \,x^{2}+a}\, c \right ) b^{3} d +3 \sqrt {c}\, \sqrt {b}\, \mathrm {log}\left (-\sqrt {c}\, \sqrt {c \,x^{2}+d}\, b -\sqrt {b}\, \sqrt {b \,x^{2}+a}\, c \right ) b^{2} d^{2}}{8 b^{3} c^{2}} \] Input:

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

Output:

( - 3*sqrt(a + b*x**2)*sqrt(c*x**2 + d)*a*b*c**2 + 4*sqrt(a + b*x**2)*sqrt 
(c*x**2 + d)*b**3*c + 2*sqrt(a + b*x**2)*sqrt(c*x**2 + d)*b**2*c**2*x**2 - 
 3*sqrt(a + b*x**2)*sqrt(c*x**2 + d)*b**2*c*d + 3*sqrt(c)*sqrt(b)*log( - s 
qrt(c)*sqrt(c*x**2 + d)*b - sqrt(b)*sqrt(a + b*x**2)*c)*a**2*c**2 + 4*sqrt 
(c)*sqrt(b)*log( - sqrt(c)*sqrt(c*x**2 + d)*b - sqrt(b)*sqrt(a + b*x**2)*c 
)*a*b**2*c + 2*sqrt(c)*sqrt(b)*log( - sqrt(c)*sqrt(c*x**2 + d)*b - sqrt(b) 
*sqrt(a + b*x**2)*c)*a*b*c*d - 4*sqrt(c)*sqrt(b)*log( - sqrt(c)*sqrt(c*x** 
2 + d)*b - sqrt(b)*sqrt(a + b*x**2)*c)*b**3*d + 3*sqrt(c)*sqrt(b)*log( - s 
qrt(c)*sqrt(c*x**2 + d)*b - sqrt(b)*sqrt(a + b*x**2)*c)*b**2*d**2)/(8*b**3 
*c**2)