\(\int \frac {A+B x^2+C x^4}{x^2 (a+b x^4)^{3/2}} \, dx\) [40]

Optimal result

Integrand size = 27, antiderivative size = 313 \[ \int \frac {A+B x^2+C x^4}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\frac {x \left (a B-(A b-a C) x^2\right )}{2 a^2 \sqrt {a+b x^4}}-\frac {A \sqrt {a+b x^4}}{a^2 x}+\frac {(3 A b-a C) x \sqrt {a+b x^4}}{2 a^2 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x^2\right )}-\frac {(3 A b-a C) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{7/4} b^{3/4} \sqrt {a+b x^4}}+\frac {\left (3 A b+\sqrt {a} \sqrt {b} B-a C\right ) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{7/4} b^{3/4} \sqrt {a+b x^4}} \] Output:

1/2*x*(a*B-(A*b-C*a)*x^2)/a^2/(b*x^4+a)^(1/2)-A*(b*x^4+a)^(1/2)/a^2/x+1/2* 
(3*A*b-C*a)*x*(b*x^4+a)^(1/2)/a^2/b^(1/2)/(a^(1/2)+b^(1/2)*x^2)-1/2*(3*A*b 
-C*a)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*Elli 
pticE(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(7/4)/b^(3/4)/(b*x^4 
+a)^(1/2)+1/4*(3*A*b+a^(1/2)*b^(1/2)*B-C*a)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+ 
a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/ 
4)),1/2*2^(1/2))/a^(7/4)/b^(3/4)/(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 = 10.11 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.45 \[ \int \frac {A+B x^2+C x^4}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\frac {-6 A \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{2},\frac {3}{4},-\frac {b x^4}{a}\right )+x^2 \left (3 B+3 B \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^4}{a}\right )+2 C x^2 \sqrt {1+\frac {b x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {b x^4}{a}\right )\right )}{6 a x \sqrt {a+b x^4}} \] Input:

Integrate[(A + B*x^2 + C*x^4)/(x^2*(a + b*x^4)^(3/2)),x]
 

Output:

(-6*A*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-1/4, 3/2, 3/4, -((b*x^4)/a)] 
+ x^2*(3*B + 3*B*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b 
*x^4)/a)] + 2*C*x^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, - 
((b*x^4)/a)]))/(6*a*x*Sqrt[a + b*x^4])
 

Rubi [A] (verified)

Time = 1.03 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2368, 25, 2374, 9, 27, 1512, 27, 761, 1510}

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.

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

\(\Big \downarrow \) 2368

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2374

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

\(\Big \downarrow \) 9

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1512

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

Input:

Int[(A + B*x^2 + C*x^4)/(x^2*(a + b*x^4)^(3/2)),x]
 

Output:

(x*(a*B - (A*b - a*C)*x^2))/(2*a^2*Sqrt[a + b*x^4]) + ((-2*A*b*Sqrt[a + b* 
x^4])/(a*x) + (b*(-(((3*A*b - a*C)*(-((x*Sqrt[a + b*x^4])/(Sqrt[a] + Sqrt[ 
b]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sq 
rt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(b^(1/4)*Sqrt 
[a + b*x^4])))/Sqrt[b]) + (a^(1/4)*(3*A*b + Sqrt[a]*Sqrt[b]*B - a*C)*(Sqrt 
[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2 
*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*b^(3/4)*Sqrt[a + b*x^4])))/a)/(2*a* 
b)
 

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, x]
 

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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1510

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

rule 1512

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

rule 2368

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = 
Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ 
m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m 
*Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( 
Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) 
   Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 
 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F 
reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 2374

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Wit 
h[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c 
*(m + 1))), x] + Simp[1/(2*a*c*(m + 1))   Int[(c*x)^(m + 1)*ExpandToSum[2*a 
*(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b* 
x^n)^p, x], x] /; NeQ[Pq0, 0]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && 
 IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.55 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.82

method	result	size
elliptic	\(-\frac {2 b \left (\frac {\left (A b -C a \right ) x^{3}}{4 a^{2} b}-\frac {B x}{4 b a}\right )}{\sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {A \sqrt {b \,x^{4}+a}}{a^{2} x}+\frac {B \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {i \left (\frac {A b -C a}{2 a^{2}}+\frac {A b}{a^{2}}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\)	\(256\)
default	\(B \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+A \left (-\frac {b \,x^{3}}{2 a^{2} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {\sqrt {b \,x^{4}+a}}{a^{2} x}+\frac {3 i \sqrt {b}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+C \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )\)	\(355\)
risch	\(-\frac {A \sqrt {b \,x^{4}+a}}{a^{2} x}+\frac {a^{2} B \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+b^{2} A \left (-\frac {x^{3}}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {3 i \sqrt {a}\, \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 b^{\frac {3}{2}} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+a^{2} C \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}-\frac {i \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}\, \sqrt {b}}\right )}{a^{2}}\)	\(369\)

Input:

int((C*x^4+B*x^2+A)/x^2/(b*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.55 \[ \int \frac {A+B x^2+C x^4}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\frac {{\left ({\left (C a b - 3 \, A b^{2}\right )} x^{5} + {\left (C a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left ({\left (B + C\right )} a b - 3 \, A b^{2}\right )} x^{5} + {\left ({\left (B + C\right )} a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (B a b x^{2} + {\left (C a b - 3 \, A b^{2}\right )} x^{4} - 2 \, A a b\right )} \sqrt {b x^{4} + a}}{2 \, {\left (a^{2} b^{2} x^{5} + a^{3} b x\right )}} \] Input:

integrate((C*x^4+B*x^2+A)/x^2/(b*x^4+a)^(3/2),x, algorithm="fricas")
 

Output:

1/2*(((C*a*b - 3*A*b^2)*x^5 + (C*a^2 - 3*A*a*b)*x)*sqrt(a)*(-b/a)^(3/4)*el 
liptic_e(arcsin(x*(-b/a)^(1/4)), -1) - (((B + C)*a*b - 3*A*b^2)*x^5 + ((B 
+ C)*a^2 - 3*A*a*b)*x)*sqrt(a)*(-b/a)^(3/4)*elliptic_f(arcsin(x*(-b/a)^(1/ 
4)), -1) + (B*a*b*x^2 + (C*a*b - 3*A*b^2)*x^4 - 2*A*a*b)*sqrt(b*x^4 + a))/ 
(a^2*b^2*x^5 + a^3*b*x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.39 \[ \int \frac {A+B x^2+C x^4}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\frac {A \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} x \Gamma \left (\frac {3}{4}\right )} + \frac {B x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {C x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:

integrate((C*x**4+B*x**2+A)/x**2/(b*x**4+a)**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate((C*x^4+B*x^2+A)/x^2/(b*x^4+a)^(3/2),x, algorithm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)/((b*x^4 + a)^(3/2)*x^2), x)
 

Giac [F]

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

integrate((C*x^4+B*x^2+A)/x^2/(b*x^4+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)/((b*x^4 + a)^(3/2)*x^2), x)
 

Mupad [F(-1)]

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

int((A + B*x^2 + C*x^4)/(x^2*(a + b*x^4)^(3/2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/(x^2*(a + b*x^4)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{x^2 \left (a+b x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) b +\left (\int \frac {\sqrt {b \,x^{4}+a}}{b^{2} x^{10}+2 a b \,x^{6}+a^{2} x^{2}}d x \right ) a +\left (\int \frac {\sqrt {b \,x^{4}+a}\, x^{2}}{b^{2} x^{8}+2 a b \,x^{4}+a^{2}}d x \right ) c \] Input:

int((C*x^4+B*x^2+A)/x^2/(b*x^4+a)^(3/2),x)
 

Output:

int(sqrt(a + b*x**4)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*b + int(sqrt(a + b 
*x**4)/(a**2*x**2 + 2*a*b*x**6 + b**2*x**10),x)*a + int((sqrt(a + b*x**4)* 
x**2)/(a**2 + 2*a*b*x**4 + b**2*x**8),x)*c