Integrand size = 20, antiderivative size = 353 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\frac {8 b \sqrt {c} x \sqrt {a+b x^2+c x^4}}{3 \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (3 b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}-\frac {8 \sqrt [4]{a} b \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 \sqrt {a+b x^2+c x^4}}+\frac {\left (3 b^2+8 \sqrt {a} b \sqrt {c}+4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}} \] Output:
8*b*c^(1/2)*x*(c*x^4+b*x^2+a)^(1/2)/(3*a^(1/2)+3*c^(1/2)*x^2)-1/3*(-2*c*x^ 2+3*b)*(c*x^4+b*x^2+a)^(1/2)/x-1/3*(c*x^4+b*x^2+a)^(3/2)/x^3-8/3*a^(1/4)*b *c^(1/4)*(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))/(c*x^4+b*x^2+a)^(1/2)+1/6*(3*b^2+8*a^(1/2)*b*c^(1/2)+4*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)*InverseJaco biAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(1/4)/ c^(1/4)/(c*x^4+b*x^2+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.61 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\frac {2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (-a^2-5 a b x^2-4 b^2 x^4-3 b c x^6+c^2 x^8\right )+4 i b \left (-b+\sqrt {b^2-4 a c}\right ) x^3 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-b^2+4 a c+4 b \sqrt {b^2-4 a c}\right ) x^3 \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{6 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x^3 \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^4,x]
Output:
(2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(-a^2 - 5*a*b*x^2 - 4*b^2*x^4 - 3*b*c*x ^6 + c^2*x^8) + (4*I)*b*(-b + Sqrt[b^2 - 4*a*c])*x^3*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^2 + 4*a*c + 4*b*Sqrt[b^2 - 4*a*c])*x^3*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 + S qrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/( 6*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^3*Sqrt[a + b*x^2 + c*x^4])
Time = 0.76 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1437, 1594, 25, 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.
| \(\displaystyle \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx\) |
| \(\Big \downarrow \) 1437 |
| \(\displaystyle \int \frac {\left (2 c x^2+b\right ) \sqrt {c x^4+b x^2+a}}{x^2}dx-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}\) |
| \(\Big \downarrow \) 1594 |
| \(\displaystyle -\frac {1}{3} \int -\frac {3 b^2+8 c x^2 b+4 a c}{\sqrt {c x^4+b x^2+a}}dx-\frac {\left (3 b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \int \frac {3 b^2+8 c x^2 b+4 a c}{\sqrt {c x^4+b x^2+a}}dx-\frac {\left (3 b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {1}{3} \left (\left (8 \sqrt {a} b \sqrt {c}+4 a c+3 b^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-8 \sqrt {a} b \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx\right )-\frac {\left (3 b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (\left (8 \sqrt {a} b \sqrt {c}+4 a c+3 b^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-8 b \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx\right )-\frac {\left (3 b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {1}{3} \left (\frac {\left (8 \sqrt {a} b \sqrt {c}+4 a c+3 b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-8 b \sqrt {c} \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx\right )-\frac {\left (3 b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {1}{3} \left (\frac {\left (8 \sqrt {a} b \sqrt {c}+4 a c+3 b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-8 b \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )\right )-\frac {\left (3 b-2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{3 x}-\frac {\left (a+b x^2+c x^4\right )^{3/2}}{3 x^3}\) |
Input:
Int[(a + b*x^2 + c*x^4)^(3/2)/x^4,x]
Output:
-1/3*((3*b - 2*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/x - (a + b*x^2 + c*x^4)^(3/ 2)/(3*x^3) + (-8*b*Sqrt[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/(Sqr t[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])) + ((3*b^2 + 8*Sqrt[a ]*b*Sqrt[c] + 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqr t[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sq rt[a]*Sqrt[c]))/4])/(2*a^(1/4)*c^(1/4)*Sqrt[a + b*x^2 + c*x^4]))/3
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])
Time = 3.47 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.21
| method | result | size |
| default | \(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{3}}-\frac {4 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x}+\frac {c x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}+\frac {\left (\frac {4 a c}{3}+b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {4 b c a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(428\) |
| elliptic | \(-\frac {a \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x^{3}}-\frac {4 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 x}+\frac {c x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3}+\frac {\left (\frac {4 a c}{3}+b^{2}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {4 b c a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(428\) |
| risch | \(-\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (-c \,x^{4}+4 b \,x^{2}+a \right )}{3 x^{3}}+\frac {b^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {a c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{3 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {4 b c a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{3 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(543\) |
Input:
int((c*x^4+b*x^2+a)^(3/2)/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*a/x^3*(c*x^4+b*x^2+a)^(1/2)-4/3*b*(c*x^4+b*x^2+a)^(1/2)/x+1/3*c*x*(c* x^4+b*x^2+a)^(1/2)+1/4*(4/3*a*c+b^2)*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))-4/3*b*c *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)))
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^4,x, algorithm="fricas")
Output:
integral((c*x^4 + b*x^2 + a)^(3/2)/x^4, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{x^{4}}\, dx \] Input:
integrate((c*x**4+b*x**2+a)**(3/2)/x**4,x)
Output:
Integral((a + b*x**2 + c*x**4)**(3/2)/x**4, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^4,x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)/x^4, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{x^{4}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2)/x^4,x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2)/x^4, x)
Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{x^4} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(3/2)/x^4,x)
Output:
int((a + b*x^2 + c*x^4)^(3/2)/x^4, x)
\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{x^4} \, dx=\frac {-5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a +4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \,x^{2}+\sqrt {c \,x^{4}+b \,x^{2}+a}\, c \,x^{4}-12 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{8}+b \,x^{6}+a \,x^{4}}d x \right ) a^{2} x^{3}+3 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) b^{2} x^{3}}{3 x^{3}} \] Input:
int((c*x^4+b*x^2+a)^(3/2)/x^4,x)
Output:
( - 5*sqrt(a + b*x**2 + c*x**4)*a + 4*sqrt(a + b*x**2 + c*x**4)*b*x**2 + s qrt(a + b*x**2 + c*x**4)*c*x**4 - 12*int(sqrt(a + b*x**2 + c*x**4)/(a*x**4 + b*x**6 + c*x**8),x)*a**2*x**3 + 3*int(sqrt(a + b*x**2 + c*x**4)/(a + b* x**2 + c*x**4),x)*b**2*x**3)/(3*x**3)