\(\int \frac {(a+b x^2+c x^4)^{3/2}}{x^8} \, dx\) [953]

Optimal result

Integrand size = 20, antiderivative size = 447 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a x^3}+\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a^2 x}-\frac {2 b \sqrt {c} \left (b^2-8 a c\right ) x \sqrt {a+b x^2+c x^4}}{35 a^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 x^5}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac {2 b \sqrt [4]{c} \left (b^2-8 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 a^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (\sqrt {a} \sqrt {c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 a^{7/4} \sqrt {a+b x^2+c x^4}} \]

[Out]

-1/7*(c*x^4+b*x^2+a)^(3/2)/x^7-1/35*(-20*a*c+b^2)*(c*x^4+b*x^2+a)^(1/2)/a/x^3+2/35*b*(-8*a*c+b^2)*(c*x^4+b*x^2
+a)^(1/2)/a^2/x-3/35*(10*c*x^2+b)*(c*x^4+b*x^2+a)^(1/2)/x^5-2/35*b*(-8*a*c+b^2)*x*c^(1/2)*(c*x^4+b*x^2+a)^(1/2
)/a^2/(a^(1/2)+x^2*c^(1/2))+2/35*b*c^(1/4)*(-8*a*c+b^2)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arcta
n(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2
*c^(1/2))*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(7/4)/(c*x^4+b*x^2+a)^(1/2)-1/70*c^(1/4)*(cos(2*ar
ctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2*arctan(c^(1/4)*x/a^(1/4))),
1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*(2*b*(-8*a*c+b^2)+(-20*a*c+b^2)*a^(1/2)*c^(1/2))*((c*x^
4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(7/4)/(c*x^4+b*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1131, 1285, 1295, 1211, 1117, 1209} \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=-\frac {\sqrt [4]{c} \left (\sqrt {a} \sqrt {c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 a^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {2 b \sqrt [4]{c} \left (b^2-8 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 a^{7/4} \sqrt {a+b x^2+c x^4}}+\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a^2 x}-\frac {2 b \sqrt {c} x \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a x^3}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}-\frac {3 \left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 x^5} \]

[In]

Int[(a + b*x^2 + c*x^4)^(3/2)/x^8,x]

[Out]

-1/35*((b^2 - 20*a*c)*Sqrt[a + b*x^2 + c*x^4])/(a*x^3) + (2*b*(b^2 - 8*a*c)*Sqrt[a + b*x^2 + c*x^4])/(35*a^2*x
) - (2*b*Sqrt[c]*(b^2 - 8*a*c)*x*Sqrt[a + b*x^2 + c*x^4])/(35*a^2*(Sqrt[a] + Sqrt[c]*x^2)) - (3*(b + 10*c*x^2)
*Sqrt[a + b*x^2 + c*x^4])/(35*x^5) - (a + b*x^2 + c*x^4)^(3/2)/(7*x^7) + (2*b*c^(1/4)*(b^2 - 8*a*c)*(Sqrt[a] +
Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2
- b/(Sqrt[a]*Sqrt[c]))/4])/(35*a^(7/4)*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*(Sqrt[a]*Sqrt[c]*(b^2 - 20*a*c) + 2
*b*(b^2 - 8*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*ArcT
an[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(70*a^(7/4)*Sqrt[a + b*x^2 + c*x^4])

Rule 1117

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 1131

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] - Dist[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] && LtQ[m, -1] && IntegerQ[2*p] &&
(IntegerQ[p] || IntegerQ[m])

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> 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 1211

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(
e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[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] && PosQ[c/a]

Rule 1285

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] + Dist[2*(p
/(f^2*(m + 1)*(m + 4*p + 3))), Int[(f*x)^(m + 2)*(a + b*x^2 + c*x^4)^(p - 1)*Simp[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] && GtQ[p, 0] && LtQ[m, -1] && m + 4*p + 3 != 0 && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1295

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] + Dist[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])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac {3}{7} \int \frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{x^6} \, dx \\ & = -\frac {3 \left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 x^5}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac {3}{35} \int \frac {b^2-20 a c-8 b c x^2}{x^4 \sqrt {a+b x^2+c x^4}} \, dx \\ & = -\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a x^3}-\frac {3 \left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 x^5}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}-\frac {\int \frac {2 b \left (b^2-8 a c\right )+c \left (b^2-20 a c\right ) x^2}{x^2 \sqrt {a+b x^2+c x^4}} \, dx}{35 a} \\ & = -\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a x^3}+\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a^2 x}-\frac {3 \left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 x^5}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac {\int \frac {-a c \left (b^2-20 a c\right )-2 b c \left (b^2-8 a c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{35 a^2} \\ & = -\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a x^3}+\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a^2 x}-\frac {3 \left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 x^5}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac {\left (2 b \sqrt {c} \left (b^2-8 a c\right )\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{35 a^{3/2}}-\frac {\left (\sqrt {c} \left (\sqrt {a} \sqrt {c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right )\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{35 a^{3/2}} \\ & = -\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a x^3}+\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a^2 x}-\frac {2 b \sqrt {c} \left (b^2-8 a c\right ) x \sqrt {a+b x^2+c x^4}}{35 a^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 x^5}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac {2 b \sqrt [4]{c} \left (b^2-8 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 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 a^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (\sqrt {a} \sqrt {c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\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}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 a^{7/4} \sqrt {a+b x^2+c x^4}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.01 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\frac {-2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (5 a^4-2 b^3 x^8 \left (b+c x^2\right )+a^3 \left (13 b x^2+20 c x^4\right )+a b x^6 \left (-b^2+17 b c x^2+16 c^2 x^4\right )+3 a^2 \left (3 b^2 x^4+13 b c x^6+5 c^2 x^8\right )\right )-i b \left (b^2-8 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x^7 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^4+9 a b^2 c-20 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right ) x^7 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{70 a^2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x^7 \sqrt {a+b x^2+c x^4}} \]

[In]

Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^8,x]

[Out]

(-2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(5*a^4 - 2*b^3*x^8*(b + c*x^2) + a^3*(13*b*x^2 + 20*c*x^4) + a*b*x^6*(-b^2
+ 17*b*c*x^2 + 16*c^2*x^4) + 3*a^2*(3*b^2*x^4 + 13*b*c*x^6 + 5*c^2*x^8)) - I*b*(b^2 - 8*a*c)*(-b + Sqrt[b^2 -
4*a*c])*x^7*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^4 + 9*a*b^2*c - 20*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*Sqr
t[b^2 - 4*a*c])*x^7*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])])/(70*a^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^7*Sqrt[a + b*x^2 + c*x
^4])

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.11

method	result	size
default	\(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{7 x^{7}}-\frac {8 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 x^{5}}-\frac {\left (15 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 a \,x^{3}}-\frac {2 b \left (8 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 a^{2} x}+\frac {\left (c^{2}-\frac {c \left (15 a c +b^{2}\right )}{35 a}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b c \left (8 a c -b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{35 a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(495\)
elliptic	\(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{7 x^{7}}-\frac {8 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 x^{5}}-\frac {\left (15 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 a \,x^{3}}-\frac {2 b \left (8 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 a^{2} x}+\frac {\left (c^{2}-\frac {c \left (15 a c +b^{2}\right )}{35 a}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b c \left (8 a c -b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{35 a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(495\)
risch	\(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (16 a b c \,x^{6}-2 b^{3} x^{6}+15 a^{2} c \,x^{4}+b^{2} x^{4} a +8 a^{2} b \,x^{2}+5 a^{3}\right )}{35 x^{7} a^{2}}+\frac {c \left (-\frac {b^{2} a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {5 c \,a^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (16 a b c -2 b^{3}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right )}{35 a^{2}}\)	\(599\)

[In]

int((c*x^4+b*x^2+a)^(3/2)/x^8,x,method=_RETURNVERBOSE)

[Out]

-1/7*a*(c*x^4+b*x^2+a)^(1/2)/x^7-8/35*b*(c*x^4+b*x^2+a)^(1/2)/x^5-1/35*(15*a*c+b^2)/a*(c*x^4+b*x^2+a)^(1/2)/x^
3-2/35*b*(8*a*c-b^2)/a^2*(c*x^4+b*x^2+a)^(1/2)/x+1/4*(c^2-1/35*c*(15*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)*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
))-1/35*b*c*(8*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)))

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.90 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (a b^{3} - 8 \, a^{2} b c\right )} x^{7} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - {\left (b^{4} - 8 \, a b^{2} c\right )} x^{7}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left (a^{2} b^{2} - 2 \, a b^{3} - 4 \, {\left (5 \, a^{3} - 4 \, a^{2} b\right )} c\right )} x^{7} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + {\left (a b^{3} + 2 \, b^{4} - 4 \, {\left (5 \, a^{2} b + 4 \, a b^{2}\right )} c\right )} x^{7}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (2 \, {\left (a b^{3} - 8 \, a^{2} b c\right )} x^{6} - 8 \, a^{3} b x^{2} - {\left (a^{2} b^{2} + 15 \, a^{3} c\right )} x^{4} - 5 \, a^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{70 \, a^{3} x^{7}} \]

[In]

integrate((c*x^4+b*x^2+a)^(3/2)/x^8,x, algorithm="fricas")

[Out]

1/70*(2*sqrt(1/2)*((a*b^3 - 8*a^2*b*c)*x^7*sqrt((b^2 - 4*a*c)/a^2) - (b^4 - 8*a*b^2*c)*x^7)*sqrt(a)*sqrt((a*sq
rt((b^2 - 4*a*c)/a^2) - b)/a)*elliptic_e(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b
*sqrt((b^2 - 4*a*c)/a^2) + b^2 - 2*a*c)/(a*c)) + sqrt(1/2)*((a^2*b^2 - 2*a*b^3 - 4*(5*a^3 - 4*a^2*b)*c)*x^7*sq
rt((b^2 - 4*a*c)/a^2) + (a*b^3 + 2*b^4 - 4*(5*a^2*b + 4*a*b^2)*c)*x^7)*sqrt(a)*sqrt((a*sqrt((b^2 - 4*a*c)/a^2)
- b)/a)*elliptic_f(arcsin(sqrt(1/2)*x*sqrt((a*sqrt((b^2 - 4*a*c)/a^2) - b)/a)), 1/2*(a*b*sqrt((b^2 - 4*a*c)/a
^2) + b^2 - 2*a*c)/(a*c)) + 2*(2*(a*b^3 - 8*a^2*b*c)*x^6 - 8*a^3*b*x^2 - (a^2*b^2 + 15*a^3*c)*x^4 - 5*a^4)*sqr
t(c*x^4 + b*x^2 + a))/(a^3*x^7)

Sympy [F]

\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{8}}\, dx \]

[In]

integrate((c*x**4+b*x**2+a)**(3/2)/x**8,x)

[Out]

Integral((a + b*x**2 + c*x**4)**(3/2)/x**8, x)

Maxima [F]

\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \]

[In]

integrate((c*x^4+b*x^2+a)^(3/2)/x^8,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^8, x)

Giac [F]

\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \]

[In]

integrate((c*x^4+b*x^2+a)^(3/2)/x^8,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^8, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^8} \,d x \]

[In]

int((a + b*x^2 + c*x^4)^(3/2)/x^8,x)

[Out]

int((a + b*x^2 + c*x^4)^(3/2)/x^8, x)