\(\int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx\) [92]

Optimal result

Integrand size = 24, antiderivative size = 126 \[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=-\frac {\left (\sqrt {613}+\left (-1+\frac {6}{x}\right )^2\right ) \sqrt {\frac {613-182 \left (1-\frac {6}{x}\right )^2+\left (-1+\frac {6}{x}\right )^4}{\left (\sqrt {613}+\left (-1+\frac {6}{x}\right )^2\right )^2}} x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {6-x}{\sqrt [4]{613} x}\right ),\frac {613+91 \sqrt {613}}{1226}\right )}{12 \sqrt [4]{613} \sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \] Output:

-1/7356*(613^(1/2)+(-1+6/x)^2)*((613-182*(1-6/x)^2+(-1+6/x)^4)/(613^(1/2)+ 
(-1+6/x)^2)^2)^(1/2)*x^2*InverseJacobiAM(2*arctan(1/613*(6-x)*613^(3/4)/x) 
,1/1226*(751538+111566*613^(1/2))^(1/2))*613^(3/4)/(3*x^4+15*x^3-44*x^2-6* 
x+9)^(1/2)
 

Mathematica [C] (warning: unable to verify)

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 = 6.56 \[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx =\text {Too large to display} \] Input:

Integrate[1/Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4],x]
 

Output:

(-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*#...
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.25, 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

\(\displaystyle -\frac {\left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right ) \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (36 \left (\frac {1}{x}-\frac {1}{6}\right )^2+\sqrt {613}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {6 \left (\frac {1}{x}-\frac {1}{6}\right )}{\sqrt [4]{613}}\right ),\frac {613+91 \sqrt {613}}{1226}\right )}{\sqrt [4]{613} \left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^2 \sqrt {\frac {1296 \left (\frac {1}{x}-\frac {1}{6}\right )^4-6552 \left (\frac {1}{x}-\frac {1}{6}\right )^2+613}{\left (6 \left (\frac {1}{x}-\frac {1}{6}\right )+1\right )^4}}}\)

Input:

Int[1/Sqrt[9 - 6*x - 44*x^2 + 15*x^3 + 3*x^4],x]
 

Output:

-(((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]))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1409

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]
 

rule 2504

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 
]
 

rule 7270

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]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1179\) vs. \(2(111)=222\).

Time = 0.89 (sec) , antiderivative size = 1180, normalized size of antiderivative = 9.37

method	result	size
default	\(\text {Expression too large to display}\)	\(1180\)
elliptic	\(\text {Expression too large to display}\)	\(1180\)

Input:

int(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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...
 

Fricas [F]

\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}} \,d x } \] Input:

integrate(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2),x, algorithm="fricas")
 

Output:

integral(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int \frac {1}{\sqrt {3 x^{4} + 15 x^{3} - 44 x^{2} - 6 x + 9}}\, dx \] Input:

integrate(1/(3*x**4+15*x**3-44*x**2-6*x+9)**(1/2),x)
 

Output:

Integral(1/sqrt(3*x**4 + 15*x**3 - 44*x**2 - 6*x + 9), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}} \,d x } \] Input:

integrate(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int { \frac {1}{\sqrt {3 \, x^{4} + 15 \, x^{3} - 44 \, x^{2} - 6 \, x + 9}} \,d x } \] Input:

integrate(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/sqrt(3*x^4 + 15*x^3 - 44*x^2 - 6*x + 9), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int \frac {1}{\sqrt {3\,x^4+15\,x^3-44\,x^2-6\,x+9}} \,d x \] Input:

int(1/(15*x^3 - 44*x^2 - 6*x + 3*x^4 + 9)^(1/2),x)
 

Output:

int(1/(15*x^3 - 44*x^2 - 6*x + 3*x^4 + 9)^(1/2), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {9-6 x-44 x^2+15 x^3+3 x^4}} \, dx=\int \frac {\sqrt {3 x^{4}+15 x^{3}-44 x^{2}-6 x +9}}{3 x^{4}+15 x^{3}-44 x^{2}-6 x +9}d x \] Input:

int(1/(3*x^4+15*x^3-44*x^2-6*x+9)^(1/2),x)
 

Output:

int(sqrt(3*x**4 + 15*x**3 - 44*x**2 - 6*x + 9)/(3*x**4 + 15*x**3 - 44*x**2 
 - 6*x + 9),x)