Integrand size = 20, antiderivative size = 411 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a x^3}+\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{35 a^{3/2} x \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {3 \left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 x^5}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}+\frac {2 b \sqrt [4]{c} \left (b^2-8 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 a^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{c} \left (\sqrt {c} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {a}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 a^{5/4} \sqrt {a+b x^2+c x^4}} \] Output:
-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^(3/2)/x/(a^(1/2)+c^(1/2)*x^2)-3/35*(10*c*x^2+b)*(c*x^4+b *x^2+a)^(1/2)/x^5-1/7*(c*x^4+b*x^2+a)^(3/2)/x^7+2/35*b*c^(1/4)*(-8*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)*El lipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/ a^(7/4)/(c*x^4+b*x^2+a)^(1/2)-1/70*c^(1/4)*(c^(1/2)*(-20*a*c+b^2)+2*b*(-8* a*c+b^2)/a^(1/2))*(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)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2) /c^(1/2))^(1/2))/a^(5/4)/(c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.07 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.39 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\frac {-2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (5 a^4-2 b^3 x^8 \left (b+c x^2\right )+a^3 \left (13 b x^2+20 c x^4\right )+a b x^6 \left (-b^2+17 b c x^2+16 c^2 x^4\right )+3 a^2 \left (3 b^2 x^4+13 b c x^6+5 c^2 x^8\right )\right )-i b \left (b^2-8 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x^7 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^4+9 a b^2 c-20 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right ) x^7 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{70 a^2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x^7 \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^8,x]
Output:
(-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])]*Ellip ticE[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*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])]*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])
Time = 1.08 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1437, 1594, 1604, 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.
| \(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx\) |
| \(\Big \downarrow \) 1437 |
| \(\displaystyle \frac {3}{7} \int \frac {\left (2 c x^2+b\right ) \sqrt {c x^4+b x^2+a}}{x^6}dx-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 1594 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \int \frac {b^2-8 c x^2 b-20 a c}{x^4 \sqrt {c x^4+b x^2+a}}dx-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 1604 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \left (-\frac {\int \frac {c \left (b^2-20 a c\right ) x^2+2 b \left (b^2-8 a c\right )}{x^2 \sqrt {c x^4+b x^2+a}}dx}{3 a}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{3 a x^3}\right )-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 1604 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \left (-\frac {-\frac {\int -\frac {c \left (2 b \left (b^2-8 a c\right ) x^2+a \left (b^2-20 a c\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{a}-\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{a x}}{3 a}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{3 a x^3}\right )-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \left (-\frac {\frac {\int \frac {c \left (2 b \left (b^2-8 a c\right ) x^2+a \left (b^2-20 a c\right )\right )}{\sqrt {c x^4+b x^2+a}}dx}{a}-\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{a x}}{3 a}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{3 a x^3}\right )-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \left (-\frac {\frac {c \int \frac {2 b \left (b^2-8 a c\right ) x^2+a \left (b^2-20 a c\right )}{\sqrt {c x^4+b x^2+a}}dx}{a}-\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{a x}}{3 a}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{3 a x^3}\right )-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \left (-\frac {\frac {c \left (\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \sqrt {a} b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a}-\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{a x}}{3 a}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{3 a x^3}\right )-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \left (-\frac {\frac {c \left (\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a}-\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{a x}}{3 a}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{3 a x^3}\right )-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \left (-\frac {\frac {c \left (\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a}-\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{a x}}{3 a}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{3 a x^3}\right )-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {3}{7} \left (\frac {1}{5} \left (-\frac {\frac {c \left (\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 b \left (b^2-8 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{a}-\frac {2 b \left (b^2-8 a c\right ) \sqrt {a+b x^2+c x^4}}{a x}}{3 a}-\frac {\left (b^2-20 a c\right ) \sqrt {a+b x^2+c x^4}}{3 a x^3}\right )-\frac {\left (b+10 c x^2\right ) \sqrt {a+b x^2+c x^4}}{5 x^5}\right )-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{7 x^7}\) |
Input:
Int[(a + b*x^2 + c*x^4)^(3/2)/x^8,x]
Output:
-1/7*(a + b*x^2 + c*x^4)^(3/2)/x^7 + (3*(-1/5*((b + 10*c*x^2)*Sqrt[a + b*x ^2 + c*x^4])/x^5 + (-1/3*((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])/(a*x) + (c*((-2*b*(b^2 - 8 *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] + Sqrt[c]*x^2)^2] *EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c ^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c] + (a^(1/4)*(Sqrt[a]*(b^2 - 20*a* c) + (2*b*(b^2 - 8*a*c))/Sqrt[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^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/a)/ (3*a))/5))/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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])
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]
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]
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])
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])
Time = 4.32 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.20
| method | result | size |
| default | \(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{7 x^{7}}-\frac {8 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 x^{5}}-\frac {\left (15 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 a \,x^{3}}-\frac {2 b \left (8 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 a^{2} x}+\frac {\left (c^{2}-\frac {c \left (15 a c +b^{2}\right )}{35 a}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b c \left (8 a c -b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{35 a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(495\) |
| elliptic | \(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{7 x^{7}}-\frac {8 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 x^{5}}-\frac {\left (15 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 a \,x^{3}}-\frac {2 b \left (8 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 a^{2} x}+\frac {\left (c^{2}-\frac {c \left (15 a c +b^{2}\right )}{35 a}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b c \left (8 a c -b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{35 a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(495\) |
| risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (16 a b c \,x^{6}-2 b^{3} x^{6}+15 a^{2} c \,x^{4}+b^{2} x^{4} a +8 a^{2} b \,x^{2}+5 a^{3}\right )}{35 x^{7} a^{2}}+\frac {c \left (-\frac {b \left (8 a c -b^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}-\frac {b^{2} a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {5 c \,a^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{35 a^{2}}\) | \(599\) |
Input:
int((c*x^4+b*x^2+a)^(3/2)/x^8,x,method=_RETURNVERBOSE)
Output:
-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)))
Time = 0.08 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (a b^{3} - 8 \, a^{2} b c\right )} x^{7} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - {\left (b^{4} - 8 \, a b^{2} c\right )} x^{7}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} E(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + \sqrt {\frac {1}{2}} {\left ({\left (a^{2} b^{2} - 2 \, a b^{3} - 4 \, {\left (5 \, a^{3} - 4 \, a^{2} b\right )} c\right )} x^{7} \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + {\left (a b^{3} + 2 \, b^{4} - 4 \, {\left (5 \, a^{2} b + 4 \, a b^{2}\right )} c\right )} x^{7}\right )} \sqrt {a} \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}} F(\arcsin \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {a \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} - b}{a}}\right )\,|\,\frac {a b \sqrt {\frac {b^{2} - 4 \, a c}{a^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, {\left (2 \, {\left (a b^{3} - 8 \, a^{2} b c\right )} x^{6} - 8 \, a^{3} b x^{2} - {\left (a^{2} b^{2} + 15 \, a^{3} c\right )} x^{4} - 5 \, a^{4}\right )} \sqrt {c x^{4} + b x^{2} + a}}{70 \, a^{3} x^{7}} \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^8,x, algorithm="fricas")
Output:
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*sqrt((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*s qrt((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*sqrt((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)*sqrt(c*x^ 4 + b*x^2 + a))/(a^3*x^7)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{8}}\, dx \] Input:
integrate((c*x**4+b*x**2+a)**(3/2)/x**8,x)
Output:
Integral((a + b*x**2 + c*x**4)**(3/2)/x**8, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^8,x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)/x^8, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{8}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^8,x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)/x^8, x)
Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^8} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(3/2)/x^8,x)
Output:
int((a + b*x^2 + c*x^4)^(3/2)/x^8, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^8} \, dx=\frac {-\sqrt {c \,x^{4}+b \,x^{2}+a}\, a -7 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c \,x^{4}+8 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{10}+b \,x^{8}+a \,x^{6}}d x \right ) a b \,x^{7}-12 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{8}+b \,x^{6}+a \,x^{4}}d x \right ) a c \,x^{7}+7 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{8}+b \,x^{6}+a \,x^{4}}d x \right ) b^{2} x^{7}}{7 x^{7}} \] Input:
int((c*x^4+b*x^2+a)^(3/2)/x^8,x)
Output:
( - sqrt(a + b*x**2 + c*x**4)*a - 7*sqrt(a + b*x**2 + c*x**4)*c*x**4 + 8*i nt(sqrt(a + b*x**2 + c*x**4)/(a*x**6 + b*x**8 + c*x**10),x)*a*b*x**7 - 12* int(sqrt(a + b*x**2 + c*x**4)/(a*x**4 + b*x**6 + c*x**8),x)*a*c*x**7 + 7*i nt(sqrt(a + b*x**2 + c*x**4)/(a*x**4 + b*x**6 + c*x**8),x)*b**2*x**7)/(7*x **7)