Integrand size = 24, antiderivative size = 126 \[ \int \frac {1}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \, dx=-\frac {\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {\frac {517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4}{\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right )^2}} x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {4+3 x}{\sqrt [4]{517} x}\right ),\frac {517+19 \sqrt {517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \]
-1/4136*x^2*(cos(2*arctan(1/517*(4+3*x)*517^(3/4)/x))^2)^(1/2)/cos(2*arcta n(1/517*(4+3*x)*517^(3/4)/x))*EllipticF(sin(2*arctan(1/517*(4+3*x)*517^(3/ 4)/x)),1/1034*(534578+19646*517^(1/2))^(1/2))*((3+4/x)^2+517^(1/2))*((517- 38*(3+4/x)^2+(3+4/x)^4)/((3+4/x)^2+517^(1/2))^2)^(1/2)*517^(3/4)/(8*x^4-15 *x^3+8*x^2+24*x+8)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 10.62 (sec) , antiderivative size = 1148, normalized size of antiderivative = 9.11 \[ \int \frac {1}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \, dx =\text {Too large to display} \]
(-2*EllipticF[ArcSin[Sqrt[((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 24*#1 + 8* #1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8* #1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0]))]], ((Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0])) /((Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0])) ]*(x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])^2*Sqrt[((Root [8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0])*(x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8* #1^4 & , 3, 0]))/((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 2, 0 ])*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 3, 0]))]*(Root[8 + 24*#1 + 8*#1^2 - 15*#1^ 3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 4, 0 ])*Sqrt[((x - Root[8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0])*(Root[ 8 + 24*#1 + 8*#1^2 - 15*#1^3 + 8*#1^4 & , 1, 0] - Root[8 + 24*#1 + 8*#1...
Time = 0.42 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2504, 27, 7270, 1416}
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.
\(\displaystyle \int \frac {1}{\sqrt {8 x^4-15 x^3+8 x^2+24 x+8}} \, dx\) |
\(\Big \downarrow \) 2504 |
\(\displaystyle -1024 \int \frac {1}{128 \sqrt {2} \left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^2 \sqrt {\frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^4}}}d\left (\frac {3}{4}+\frac {1}{x}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -4 \sqrt {2} \int \frac {1}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^2 \sqrt {\frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^4}}}d\left (\frac {3}{4}+\frac {1}{x}\right )\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle -\frac {4 \sqrt {2} \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \int \frac {1}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle -\frac {\left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 \left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt [4]{517}}\right ),\frac {517+19 \sqrt {517}}{1034}\right )}{\sqrt {2} \sqrt [4]{517} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
-(((Sqrt[517] + 16*(3/4 + x^(-1))^2)*Sqrt[(517 - 608*(3/4 + x^(-1))^2 + 25 6*(3/4 + x^(-1))^4)/(Sqrt[517] + 16*(3/4 + x^(-1))^2)^2]*EllipticF[2*ArcTa n[(4*(3/4 + x^(-1)))/517^(1/4)], (517 + 19*Sqrt[517])/1034])/(Sqrt[2]*517^ (1/4)*(3 - 4*(3/4 + x^(-1)))^2*Sqrt[(517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4)/(3 - 4*(3/4 + x^(-1)))^4]))
3.8.100.3.1 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[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] , c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* a^2 Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a ^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0 ]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.16 (sec) , antiderivative size = 1180, normalized size of antiderivative = 9.37
method | result | size |
default | \(\text {Expression too large to display}\) | \(1180\) |
elliptic | \(\text {Expression too large to display}\) | \(1180\) |
1/2*(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1)-RootOf(8*_Z^4-15*_Z^3+8 *_Z^2+24*_Z+8,index=4))*((RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-Ro otOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z ^2+24*_Z+8,index=1))/(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf (8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1))/(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+2 4*_Z+8,index=2)))^(1/2)*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2))^ 2*((RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)-RootOf(8*_Z^4-15*_Z^3+8* _Z^2+24*_Z+8,index=1))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=3))/( RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=3)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2 +24*_Z+8,index=1))/(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)))^(1/2 )*((RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)-RootOf(8*_Z^4-15*_Z^3+8* _Z^2+24*_Z+8,index=1))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4))/( RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf(8*_Z^4-15*_Z^3+8*_Z^2 +24*_Z+8,index=1))/(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)))^(1/2 )/(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=4)-RootOf(8*_Z^4-15*_Z^3+8*_ Z^2+24*_Z+8,index=2))/(RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2)-RootO f(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=1))*2^(1/2)/((x-RootOf(8*_Z^4-15*_Z^ 3+8*_Z^2+24*_Z+8,index=1))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=2 ))*(x-RootOf(8*_Z^4-15*_Z^3+8*_Z^2+24*_Z+8,index=3))*(x-RootOf(8*_Z^4-15*_ Z^3+8*_Z^2+24*_Z+8,index=4)))^(1/2)*EllipticF(((RootOf(8*_Z^4-15*_Z^3+8...
\[ \int \frac {1}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \, dx=\int { \frac {1}{\sqrt {8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}} \,d x } \]
\[ \int \frac {1}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \, dx=\int \frac {1}{\sqrt {8 x^{4} - 15 x^{3} + 8 x^{2} + 24 x + 8}}\, dx \]
\[ \int \frac {1}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \, dx=\int { \frac {1}{\sqrt {8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}} \,d x } \]
\[ \int \frac {1}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \, dx=\int { \frac {1}{\sqrt {8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \, dx=\int \frac {1}{\sqrt {8\,x^4-15\,x^3+8\,x^2+24\,x+8}} \,d x \]