Integrand size = 24, antiderivative size = 330 \[ \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx=\frac {\sqrt {c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}-\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \sqrt {a x+b x^3+c x^5}} \]
x^(3/2)*(c*x^4+b*x^2+a)*c^(1/2)/a/(a^(1/2)+x^2*c^(1/2))/(c*x^5+b*x^3+a*x)^ (1/2)-(c*x^5+b*x^3+a*x)^(1/2)/a/x^(3/2)-c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^ (1/4)))^2)^(1/2)/cos(2*arctan(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))* x^(1/2)*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/(c*x^5+b*x ^3+a*x)^(1/2)+1/2*c^(1/4)*(cos(2*arctan(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))*x^(1/2)*((c*x^4+b*x^2+ a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/(c*x^5+b*x^3+a*x)^(1/2)
Result contains complex when optimal does not.
Time = 11.31 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx=\frac {-4 \left (a+b x^2+c x^4\right )+\frac {i \sqrt {2} \left (-b+\sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{4 a \sqrt {x} \sqrt {x \left (a+b x^2+c x^4\right )}} \]
(-4*(a + b*x^2 + c*x^4) + (I*Sqrt[2]*(-b + Sqrt[b^2 - 4*a*c])*x*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*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])] - Elliptic F[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])]))/Sqrt[c/(b + Sqrt[b^2 - 4*a*c])])/(4*a*Sqr t[x]*Sqrt[x*(a + b*x^2 + c*x^4)])
Time = 0.44 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1976, 27, 1961, 1459, 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.
| \(\displaystyle \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx\) |
| \(\Big \downarrow \) 1976 |
| \(\displaystyle \frac {\int \frac {c x^{5/2}}{\sqrt {c x^5+b x^3+a x}}dx}{a}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int \frac {x^{5/2}}{\sqrt {c x^5+b x^3+a x}}dx}{a}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}\) |
| \(\Big \downarrow \) 1961 |
| \(\displaystyle \frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \int \frac {x^2}{\sqrt {c x^4+b x^2+a}}dx}{a \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}\) |
| \(\Big \downarrow \) 1459 |
| \(\displaystyle \frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt {a} \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle \frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}}{\sqrt {c}}\right )}{a \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt {a x+b x^3+c x^5}}{a x^{3/2}}\) |
-(Sqrt[a*x + b*x^3 + c*x^5]/(a*x^(3/2))) + (c*Sqrt[x]*Sqrt[a + b*x^2 + c*x ^4]*(-((-((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] + 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*Sqrt[a*x + b*x^3 + c*x^5])
3.2.16.3.1 Defintions of rubi rules used
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[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[1/q Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c}, 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, 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[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)] , x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]/Sqrt[a *x^q + b*x^n + c*x^(2*n - q)]) Int[x^(m - q/2)/Sqrt[a + b*x^(n - q) + c*x ^(2*(n - q))], x], x] /; FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && ((EqQ[m, 1] && EqQ[n, 3] && EqQ[q, 2]) || ((EqQ[m + 1/2] || EqQ[m, 3/2] || EqQ[m, 1/2] || EqQ[m, 5/2]) && EqQ[n, 3] && EqQ[q, 1]))
Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ ), x_Symbol] :> Simp[x^(m - q + 1)*((a*x^q + b*x^n + c*x^(2*n - q))^(p + 1) /(a*(m + p*q + 1))), x] - Simp[1/(a*(m + p*q + 1)) Int[x^(m + n - q)*(b*( m + p*q + (n - q)*(p + 1) + 1) + c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c}, x] && Eq Q[r, 2*n - q] && PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGt Q[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && LtQ[m + p*q + 1, 0 ]
Time = 1.04 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(-\frac {c \,x^{4}+b \,x^{2}+a}{a \sqrt {x}\, \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}-\frac {c \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right ) \sqrt {x}}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}\) | \(258\) |
| default | \(\frac {\left (-\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, c \,x^{4}-\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b c \,x^{4}-c \sqrt {-\frac {2 \left (\sqrt {-4 a c +b^{2}}\, x^{2}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, x^{2}+b \,x^{2}+2 a}{a}}\, a x F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right )+c \sqrt {-\frac {2 \left (\sqrt {-4 a c +b^{2}}\, x^{2}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, x^{2}+b \,x^{2}+2 a}{a}}\, a x E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right )-\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b \,x^{2}-\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} x^{2}-\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a -\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b \right ) \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}{x^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}+a \right ) a \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\) | \(508\) |
-(c*x^4+b*x^2+a)/a/x^(1/2)/(x*(c*x^4+b*x^2+a))^(1/2)-1/2*c*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)/(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)))*x^(1/2)/(x*(c*x^4+b*x^2+a ))^(1/2)
\[ \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx=\int { \frac {1}{\sqrt {c x^{5} + b x^{3} + a x} x^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx=\int \frac {1}{x^{\frac {3}{2}} \sqrt {x \left (a + b x^{2} + c x^{4}\right )}}\, dx \]
\[ \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx=\int { \frac {1}{\sqrt {c x^{5} + b x^{3} + a x} x^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx=\int { \frac {1}{\sqrt {c x^{5} + b x^{3} + a x} x^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx=\int \frac {1}{x^{3/2}\,\sqrt {c\,x^5+b\,x^3+a\,x}} \,d x \]