Integrand size = 24, antiderivative size = 130 \[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=-\frac {\sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right )^2}} \left (\sqrt {613}+\frac {(6-x)^2}{x^2}\right ) x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {6-x}{\sqrt [4]{613} x}\right ),\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \]
-1/7356*x^2*(cos(2*arctan(1/613*(6-x)*613^(3/4)/x))^2)^(1/2)/cos(2*arctan( 1/613*(6-x)*613^(3/4)/x))*EllipticF(sin(2*arctan(1/613*(6-x)*613^(3/4)/x)) ,1/1226*(751538+111566*613^(1/2))^(1/2))*((6-x)^2/x^2+613^(1/2))*((613-182 *(1-6/x)^2+(-1+6/x)^4)/((6-x)^2/x^2+613^(1/2))^2)^(1/2)*613^(3/4)/(3*x^4+1 5*x^3-44*x^2-6*x+9)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 10.17 (sec) , antiderivative size = 826, normalized size of antiderivative = 6.35 \[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}{\left (x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}}\right ),\frac {\left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,3\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}{\left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,3\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}\right ) \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]}{x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]}} \left (x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]\right )^2 \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,3\right ]}{x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]}} \sqrt {\frac {x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]}{x-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]}}}{\sqrt {\left (9-6 x-44 x^2+15 x^3+3 x^4\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,1\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,3\right ]\right ) \left (\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,2\right ]-\text {Root}\left [3 \text {$\#$1}^4+15 \text {$\#$1}^3-44 \text {$\#$1}^2-6 \text {$\#$1}+9\&,4\right ]\right )}} \]
(-2*EllipticF[ArcSin[Sqrt[((x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))/((x - Root[9 - 6*#1 - 44* #1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3* #1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0]))]], ((Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])) /((Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])) ]*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 1, 0])/(x - Roo t[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])]*(x - Root[9 - 6*#1 - 4 4*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])^2*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1 ^4 & , 2, 0])]*Sqrt[(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 4, 0])/(x - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0])])/Sqrt[(9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4)*(Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1 ^4 & , 1, 0] - Root[9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 3, 0])*(Root [9 - 6*#1 - 44*#1^2 + 15*#1^3 + 3*#1^4 & , 2, 0] - Root[9 - 6*#1 - 44*#...
Time = 0.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.22, 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, 1409}
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.
| \(\displaystyle \int \frac {1}{\sqrt {3 x^4+15 x^3-44 x^2-6 x+9}} \, dx\) |
| \(\Big \downarrow \) 2504 |
| \(\displaystyle -1296 \int \frac {1}{108 \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}d\left (\frac {1}{x}-\frac {1}{6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -12 \int \frac {1}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}d\left (\frac {1}{x}-\frac {1}{6}\right )\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle -\frac {12 \sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613} \int \frac {1}{\sqrt {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}}d\left (\frac {1}{x}-\frac {1}{6}\right )}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
| \(\Big \downarrow \) 1409 |
| \(\displaystyle -\frac {\left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right ) \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {6 \left (\frac {1}{x}-\frac {1}{6}\right )}{\sqrt [4]{613}}\right ),\frac {613+91 \sqrt {613}}{1226}\right )}{\sqrt [4]{613} \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\) |
-(((Sqrt[613] + 36*(-1/6 + x^(-1))^2)*Sqrt[(613 - 6552*(-1/6 + x^(-1))^2 + 1296*(-1/6 + x^(-1))^4)/(Sqrt[613] + 36*(-1/6 + x^(-1))^2)^2]*EllipticF[2 *ArcTan[(6*(-1/6 + x^(-1)))/613^(1/4)], (613 + 91*Sqrt[613])/1226])/(613^( 1/4)*(1 + 6*(-1/6 + x^(-1)))^2*Sqrt[(613 - 6552*(-1/6 + x^(-1))^2 + 1296*( -1/6 + x^(-1))^4)/(1 + 6*(-1/6 + x^(-1)))^4]))
3.9.3.3.1 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] && GtQ[b^2 - 4*a*c, 0] && GtQ[c/a, 0] && LtQ[ b/a, 0]
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 = 0.99 (sec) , antiderivative size = 1180, normalized size of antiderivative = 9.08
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1180\) |
| elliptic | \(\text {Expression too large to display}\) | \(1180\) |
2/3*(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1)-RootOf(3*_Z^4+15*_Z^3-4 4*_Z^2-6*_Z+9,index=4))*((RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)-Ro otOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))*(x-RootOf(3*_Z^4+15*_Z^3-44*_ Z^2-6*_Z+9,index=1))/(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)-RootOf (3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2- 6*_Z+9,index=2)))^(1/2)*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2))^ 2*((RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+15*_Z^3-44 *_Z^2-6*_Z+9,index=1))*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3))/( RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3)-RootOf(3*_Z^4+15*_Z^3-44*_Z^ 2-6*_Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)))^(1/2 )*((RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootOf(3*_Z^4+15*_Z^3-44 *_Z^2-6*_Z+9,index=1))*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4))/( RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)-RootOf(3*_Z^4+15*_Z^3-44*_Z^ 2-6*_Z+9,index=1))/(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)))^(1/2 )/(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=4)-RootOf(3*_Z^4+15*_Z^3-44* _Z^2-6*_Z+9,index=2))/(RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2)-RootO f(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=1))*3^(1/2)/((x-RootOf(3*_Z^4+15*_Z^ 3-44*_Z^2-6*_Z+9,index=1))*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=2 ))*(x-RootOf(3*_Z^4+15*_Z^3-44*_Z^2-6*_Z+9,index=3))*(x-RootOf(3*_Z^4+15*_ Z^3-44*_Z^2-6*_Z+9,index=4)))^(1/2)*EllipticF(((RootOf(3*_Z^4+15*_Z^3-4...
\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}} \,d x } \]
\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int \frac {1}{\sqrt {3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9}}\, dx \]
\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}} \,d x } \]
\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int \frac {1}{\sqrt {3\,x^4+15\,x^3-44\,x^2-6\,x+9}} \,d x \]