Integrand size = 16, antiderivative size = 381 \[ \int \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {2 b \left (b^2-8 a c\right ) x \sqrt {a+b x^2+c x^4}}{35 c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {x \left (b^2+10 a c+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{35 c}+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}+\frac {2 \sqrt [4]{a} b \left (b^2-8 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{35 c^{7/4} \sqrt {a+b x^2+c x^4}}-\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{70 c^{5/4} \sqrt {a+b x^2+c x^4}} \] Output:
-2/35*b*(-8*a*c+b^2)*x*(c*x^4+b*x^2+a)^(1/2)/c^(3/2)/(a^(1/2)+c^(1/2)*x^2) +1/35*x*(3*b*c*x^2+10*a*c+b^2)*(c*x^4+b*x^2+a)^(1/2)/c+1/7*x*(c*x^4+b*x^2+ a)^(3/2)+2/35*a^(1/4)*b*(-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)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4)) ),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(7/4)/(c*x^4+b*x^2+a)^(1/2)-1/70*a^(1 /4)*(a^(1/2)*(-20*a*c+b^2)+2*b*(-8*a*c+b^2)/c^(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))/c^(5/4)/(c*x^4+b*x^2+a )^(1/2)
Result contains complex when optimal does not.
Time = 9.70 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.40 \[ \int \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {2 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (15 a^2 c+a \left (b^2+23 b c x^2+20 c^2 x^4\right )+x^2 \left (b^3+9 b^2 c x^2+13 b c^2 x^4+5 c^3 x^6\right )\right )-i b \left (b^2-8 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^4+9 a b^2 c-20 a^2 c^2+b^3 \sqrt {b^2-4 a c}-8 a b c \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{70 c^2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:
Integrate[(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(15*a^2*c + a*(b^2 + 23*b*c*x^2 + 2 0*c^2*x^4) + x^2*(b^3 + 9*b^2*c*x^2 + 13*b*c^2*x^4 + 5*c^3*x^6)) - I*b*(b^ 2 - 8*a*c)*(-b + Sqrt[b^2 - 4*a*c])*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*Sqrt[b^2 - 4*a*c])*S qrt[(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*ArcSi nh[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*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b* x^2 + c*x^4])
Time = 0.83 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {1404, 1490, 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.
| \(\displaystyle \int \left (a+b x^2+c x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1404 |
| \(\displaystyle \frac {3}{7} \int \left (b x^2+2 a\right ) \sqrt {c x^4+b x^2+a}dx+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {3}{7} \left (\frac {\int -\frac {2 b \left (b^2-8 a c\right ) x^2+a \left (b^2-20 a c\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}+\frac {x \sqrt {a+b x^2+c x^4} \left (10 a c+b^2+3 b c x^2\right )}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\int \frac {2 b \left (b^2-8 a c\right ) x^2+a \left (b^2-20 a c\right )}{\sqrt {c x^4+b x^2+a}}dx}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle \frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \sqrt {a} b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\sqrt {a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 b \left (b^2-8 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {3}{7} \left (\frac {x \left (10 a c+b^2+3 b c x^2\right ) \sqrt {a+b x^2+c x^4}}{15 c}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a} \left (b^2-20 a c\right )+\frac {2 b \left (b^2-8 a c\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 b \left (b^2-8 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}}{15 c}\right )+\frac {1}{7} x \left (a+b x^2+c x^4\right )^{3/2}\) |
Input:
Int[(a + b*x^2 + c*x^4)^(3/2),x]
Output:
(x*(a + b*x^2 + c*x^4)^(3/2))/7 + (3*((x*(b^2 + 10*a*c + 3*b*c*x^2)*Sqrt[a + b*x^2 + c*x^4])/(15*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]))/(15*c)))/7
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b *x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1)) Int[(2*a + b*x^2)*( a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* c, 0] && GtQ[p, 0] && IntegerQ[2*p]
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_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
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]
Time = 2.91 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.24
| method | result | size |
| default | \(\frac {c \,x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{7}+\frac {8 b \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35}+\frac {\left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (a^{2}-\frac {\left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) a}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {46 a b}{35}-\frac {2 \left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) b}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(471\) |
| elliptic | \(\frac {c \,x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{7}+\frac {8 b \,x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{35}+\frac {\left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (a^{2}-\frac {\left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) a}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {46 a b}{35}-\frac {2 \left (\frac {9 a c}{7}+\frac {3 b^{2}}{35}\right ) b}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(471\) |
| risch | \(\frac {x \left (5 c^{2} x^{4}+8 b c \,x^{2}+15 a c +b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{35 c}+\frac {-\frac {b \left (8 a c -b^{2}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}-\frac {b^{2} a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {5 c \,a^{2} \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}}{35 c}\) | \(571\) |
Input:
int((c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/7*c*x^5*(c*x^4+b*x^2+a)^(1/2)+8/35*b*x^3*(c*x^4+b*x^2+a)^(1/2)+1/3*(9/7* a*c+3/35*b^2)/c*x*(c*x^4+b*x^2+a)^(1/2)+1/4*(a^2-1/3*(9/7*a*c+3/35*b^2)/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 )*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))-1/2*(46/35*a*b-2/3*(9/7*a*c+3/35*b^2)/c*b )*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 = 396, normalized size of antiderivative = 1.04 \[ \int \left (a+b x^2+c x^4\right )^{3/2} \, dx=-\frac {2 \, \sqrt {\frac {1}{2}} {\left ({\left (b^{3} c - 8 \, a b c^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (b^{4} - 8 \, a b^{2} c\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (2 \, b^{3} c + 20 \, a c^{3} - {\left (16 \, a b + b^{2}\right )} c^{2}\right )} x \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (2 \, b^{4} - 20 \, a b c^{2} - {\left (16 \, a b^{2} - b^{3}\right )} c\right )} x\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 2 \, {\left (5 \, c^{4} x^{6} + 8 \, b c^{3} x^{4} - 2 \, b^{3} c + 16 \, a b c^{2} + {\left (b^{2} c^{2} + 15 \, a c^{3}\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2} + a}}{70 \, c^{3} x} \] Input:
integrate((c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas")
Output:
-1/70*(2*sqrt(1/2)*((b^3*c - 8*a*b*c^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (b^4 - 8*a*b^2*c)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e( arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt ((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) - sqrt(1/2)*((2*b^3*c + 20*a*c^3 - (16*a*b + b^2)*c^2)*x*sqrt((b^2 - 4*a*c)/c^2) - (2*b^4 - 20*a*b*c^2 - ( 16*a*b^2 - b^3)*c)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elli ptic_f(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b *c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) - 2*(5*c^4*x^6 + 8*b*c^3* x^4 - 2*b^3*c + 16*a*b*c^2 + (b^2*c^2 + 15*a*c^3)*x^2)*sqrt(c*x^4 + b*x^2 + a))/(c^3*x)
\[ \int \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int \left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((c*x**4+b*x**2+a)**(3/2),x)
Output:
Integral((a + b*x**2 + c*x**4)**(3/2), x)
\[ \int \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2), x)
\[ \int \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((c*x^4 + b*x^2 + a)^(3/2), x)
Timed out. \[ \int \left (a+b x^2+c x^4\right )^{3/2} \, dx=\int {\left (c\,x^4+b\,x^2+a\right )}^{3/2} \,d x \] Input:
int((a + b*x^2 + c*x^4)^(3/2),x)
Output:
int((a + b*x^2 + c*x^4)^(3/2), x)
\[ \int \left (a+b x^2+c x^4\right )^{3/2} \, dx=\frac {15 \sqrt {c \,x^{4}+b \,x^{2}+a}\, a c x +\sqrt {c \,x^{4}+b \,x^{2}+a}\, b^{2} x +8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b c \,x^{3}+5 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{2} x^{5}+20 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a^{2} c -\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a \,b^{2}+16 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a b c -2 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) b^{3}}{35 c} \] Input:
int((c*x^4+b*x^2+a)^(3/2),x)
Output:
(15*sqrt(a + b*x**2 + c*x**4)*a*c*x + sqrt(a + b*x**2 + c*x**4)*b**2*x + 8 *sqrt(a + b*x**2 + c*x**4)*b*c*x**3 + 5*sqrt(a + b*x**2 + c*x**4)*c**2*x** 5 + 20*int(sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a**2*c - int (sqrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a*b**2 + 16*int((sqrt( a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x)*a*b*c - 2*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x)*b**3)/(35*c)