\(\int \frac {(a+b x^2+c x^4)^{3/2}}{x^6} \, dx\) [980]

Optimal result

Integrand size = 20, antiderivative size = 365 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^6} \, dx=-\frac {\left (b^2+12 a c\right ) \sqrt {a+b x^2+c x^4}}{5 \sqrt {a} x \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^3}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}-\frac {\sqrt [4]{c} \left (b^2+12 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{5 a^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\sqrt [4]{c} \left (b^2+8 \sqrt {a} b \sqrt {c}+12 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{10 a^{3/4} \sqrt {a+b x^2+c x^4}} \] Output:

-1/5*(12*a*c+b^2)*(c*x^4+b*x^2+a)^(1/2)/a^(1/2)/x/(a^(1/2)+c^(1/2)*x^2)-1/ 
5*(-6*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/x^3-1/5*(c*x^4+b*x^2+a)^(3/2)/x^5-1/5 
*c^(1/4)*(12*a*c+b^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1 
/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1 
/2)/c^(1/2))^(1/2))/a^(3/4)/(c*x^4+b*x^2+a)^(1/2)+1/10*c^(1/4)*(b^2+8*a^(1 
/2)*b*c^(1/2)+12*a*c)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1 
/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^( 
1/2)/c^(1/2))^(1/2))/a^(3/4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.88 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^6} \, dx=\frac {-4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (a^3+b^2 x^6 \left (b+c x^2\right )+a^2 \left (3 b x^2+8 c x^4\right )+a \left (3 b^2 x^4+9 b c x^6+7 c^2 x^8\right )\right )+i \left (b^2+12 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x^5 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}+12 a c \sqrt {b^2-4 a c}\right ) x^5 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{20 a \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x^5 \sqrt {a+b x^2+c x^4}} \] Input:

Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^6,x]
 

Output:

(-4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a^3 + b^2*x^6*(b + c*x^2) + a^2*(3*b* 
x^2 + 8*c*x^4) + a*(3*b^2*x^4 + 9*b*c*x^6 + 7*c^2*x^8)) + I*(b^2 + 12*a*c) 
*(-b + Sqrt[b^2 - 4*a*c])*x^5*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + 
Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b 
^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]* 
x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(-b^3 + 4*a*b*c + 
 b^2*Sqrt[b^2 - 4*a*c] + 12*a*c*Sqrt[b^2 - 4*a*c])*x^5*Sqrt[(b + Sqrt[b^2 
- 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a* 
c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/ 
(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c 
])])/(20*a*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^5*Sqrt[a + b*x^2 + c*x^4])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1437, 1594, 25, 1604, 25, 27, 1511, 27, 1416, 1509}

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.

\(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^6} \, dx\)

\(\Big \downarrow \) 1437

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

\(\Big \downarrow \) 1594

\(\displaystyle \frac {3}{5} \left (-\frac {1}{3} \int -\frac {b^2+8 c x^2 b+12 a c}{x^2 \sqrt {c x^4+b x^2+a}}dx-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{5} \left (\frac {1}{3} \int \frac {b^2+8 c x^2 b+12 a c}{x^2 \sqrt {c x^4+b x^2+a}}dx-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {3}{5} \left (\frac {1}{3} \left (-\frac {\int -\frac {c \left (\left (b^2+12 a c\right ) x^2+8 a b\right )}{\sqrt {c x^4+b x^2+a}}dx}{a}-\frac {\left (12 a c+b^2\right ) \sqrt {a+b x^2+c x^4}}{a x}\right )-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {3}{5} \left (\frac {1}{3} \left (\frac {\int \frac {c \left (\left (b^2+12 a c\right ) x^2+8 a b\right )}{\sqrt {c x^4+b x^2+a}}dx}{a}-\frac {\left (12 a c+b^2\right ) \sqrt {a+b x^2+c x^4}}{a x}\right )-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{5} \left (\frac {1}{3} \left (\frac {c \int \frac {\left (b^2+12 a c\right ) x^2+8 a b}{\sqrt {c x^4+b x^2+a}}dx}{a}-\frac {\left (12 a c+b^2\right ) \sqrt {a+b x^2+c x^4}}{a x}\right )-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {3}{5} \left (\frac {1}{3} \left (\frac {c \left (\frac {\sqrt {a} \left (8 \sqrt {a} b \sqrt {c}+12 a c+b^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (12 a c+b^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a}-\frac {\left (12 a c+b^2\right ) \sqrt {a+b x^2+c x^4}}{a x}\right )-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{5} \left (\frac {1}{3} \left (\frac {c \left (\frac {\sqrt {a} \left (8 \sqrt {a} b \sqrt {c}+12 a c+b^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (12 a c+b^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a}-\frac {\left (12 a c+b^2\right ) \sqrt {a+b x^2+c x^4}}{a x}\right )-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {3}{5} \left (\frac {1}{3} \left (\frac {c \left (\frac {\sqrt [4]{a} \left (8 \sqrt {a} b \sqrt {c}+12 a c+b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (12 a c+b^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a}-\frac {\left (12 a c+b^2\right ) \sqrt {a+b x^2+c x^4}}{a x}\right )-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {3}{5} \left (\frac {1}{3} \left (\frac {c \left (\frac {\sqrt [4]{a} \left (8 \sqrt {a} b \sqrt {c}+12 a c+b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (12 a c+b^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{a}-\frac {\left (12 a c+b^2\right ) \sqrt {a+b x^2+c x^4}}{a x}\right )-\frac {\left (b-6 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x^3}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}\)

Input:

Int[(a + b*x^2 + c*x^4)^(3/2)/x^6,x]
 

Output:

-1/5*(a + b*x^2 + c*x^4)^(3/2)/x^5 + (3*(-1/3*((b - 6*c*x^2)*Sqrt[a + b*x^ 
2 + c*x^4])/x^3 + (-(((b^2 + 12*a*c)*Sqrt[a + b*x^2 + c*x^4])/(a*x)) + (c* 
(-(((b^2 + 12*a*c)*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) 
 + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sq 
rt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqr 
t[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a^(1/4)*(b^2 + 
8*Sqrt[a]*b*Sqrt[c] + 12*a*c)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c* 
x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], ( 
2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4)*Sqrt[a + b*x^2 + c*x^4])))/a)/3))/ 
5
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1437

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*x^2 + c*x^4)^p/(d*(m + 1))), x] - Simp[2*(p/( 
d^2*(m + 1)))   Int[(d*x)^(m + 2)*(b + 2*c*x^2)*(a + b*x^2 + c*x^4)^(p - 1) 
, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && L 
tQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1509

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

rule 1511

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

rule 1594

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(a + b*x^2 + c*x^4)^p*((d*(m 
+ 4*p + 3) + e*(m + 1)*x^2)/(f*(m + 1)*(m + 4*p + 3))), x] + Simp[2*(p/(f^2 
*(m + 1)*(m + 4*p + 3)))   Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^(p - 1)*Si 
mp[2*a*e*(m + 1) - b*d*(m + 4*p + 3) + (b*e*(m + 1) - 2*c*d*(m + 4*p + 3))* 
x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && G 
tQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && IntegerQ[2*p] && (IntegerQ[p] 
 || IntegerQ[m])
 

rule 1604

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) 
/(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1))   Int[(f*x)^(m + 2)*(a + b*x^2 
 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ 
m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

Maple [A] (verified)

Time = 3.22 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.16

method	result	size
risch	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (7 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}{5 x^{5} a}+\frac {c \left (-\frac {\left (12 a c +b^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}+\frac {2 a b \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{5 a}\)	\(424\)
default	\(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{5}}-\frac {2 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{3}}-\frac {\left (7 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 a x}+\frac {2 b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{5 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (c^{2}+\frac {c \left (7 a c +b^{2}\right )}{5 a}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(450\)
elliptic	\(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{5}}-\frac {2 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 x^{3}}-\frac {\left (7 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 a x}+\frac {2 b c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{5 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (c^{2}+\frac {c \left (7 a c +b^{2}\right )}{5 a}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(450\)

Input:

int((c*x^4+b*x^2+a)^(3/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/5*(c*x^4+b*x^2+a)^(1/2)*(7*a*c*x^4+b^2*x^4+2*a*b*x^2+a^2)/x^5/a+1/5*c/a 
*(-1/2*(12*a*c+b^2)*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+( 
-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/( 
c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/2)*((-b+ 
(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2) 
)-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b 
+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))+2*a*b*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a 
)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/ 
2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a* 
c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))
 

Fricas [F]

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

integrate((c*x^4+b*x^2+a)^(3/2)/x^6,x, algorithm="fricas")
 

Output:

integral((c*x^4 + b*x^2 + a)^(3/2)/x^6, x)
 

Sympy [F]

\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^6} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{6}}\, dx \] Input:

integrate((c*x**4+b*x**2+a)**(3/2)/x**6,x)
 

Output:

Integral((a + b*x**2 + c*x**4)**(3/2)/x**6, x)
 

Maxima [F]

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

integrate((c*x^4+b*x^2+a)^(3/2)/x^6,x, algorithm="maxima")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^6, x)
 

Giac [F]

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

integrate((c*x^4+b*x^2+a)^(3/2)/x^6,x, algorithm="giac")
 

Output:

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^6} \,d x \] Input:

int((a + b*x^2 + c*x^4)^(3/2)/x^6,x)
                                                                                    
                                                                                    
 

Output:

int((a + b*x^2 + c*x^4)^(3/2)/x^6, x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^6} \, dx=\frac {-\sqrt {c \,x^{4}+b \,x^{2}+a}\, a^{2}-10 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a b \,x^{2}-7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a c \,x^{4}+15 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} x^{4}-24 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{8}+b \,x^{6}+a \,x^{4}}d x \right ) a^{2} b \,x^{5}+12 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,c^{2} x^{5}-15 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) b^{2} c \,x^{5}}{5 a \,x^{5}} \] Input:

int((c*x^4+b*x^2+a)^(3/2)/x^6,x)
 

Output:

( - sqrt(a + b*x**2 + c*x**4)*a**2 - 10*sqrt(a + b*x**2 + c*x**4)*a*b*x**2 
 - 7*sqrt(a + b*x**2 + c*x**4)*a*c*x**4 + 15*sqrt(a + b*x**2 + c*x**4)*b** 
2*x**4 - 24*int(sqrt(a + b*x**2 + c*x**4)/(a*x**4 + b*x**6 + c*x**8),x)*a* 
*2*b*x**5 + 12*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4), 
x)*a*c**2*x**5 - 15*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x 
**4),x)*b**2*c*x**5)/(5*a*x**5)