\(\int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx\) [132]

Optimal result

Integrand size = 19, antiderivative size = 73 \[ \int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=-\frac {2 \sqrt {-7+x} \sqrt {-5+2 x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {19}{2}}}{\sqrt {2+3 x}}\right ),\frac {46}{19}\right )}{\sqrt {19} \sqrt {70+67 x-53 x^2+6 x^3}} \] Output:

-2/19*(-7+x)^(1/2)*(-5+2*x)^(1/2)*(2+3*x)^(1/2)*EllipticF(1/2*38^(1/2)/(2+ 
3*x)^(1/2),1/19*874^(1/2))*19^(1/2)/(6*x^3-53*x^2+67*x+70)^(1/2)
 

Mathematica [A] (verified)

Time = 10.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\frac {\sqrt {\frac {2}{23}} \sqrt {5-2 x} \sqrt {7-x} \sqrt {2+3 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {2}{19}} \sqrt {2+3 x}\right ),\frac {19}{46}\right )}{\sqrt {70+67 x-53 x^2+6 x^3}} \] Input:

Integrate[1/Sqrt[70 + 67*x - 53*x^2 + 6*x^3],x]
 

Output:

(Sqrt[2/23]*Sqrt[5 - 2*x]*Sqrt[7 - x]*Sqrt[2 + 3*x]*EllipticF[ArcSin[Sqrt[ 
2/19]*Sqrt[2 + 3*x]], 19/46])/Sqrt[70 + 67*x - 53*x^2 + 6*x^3]
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.36 (sec) , antiderivative size = 428, normalized size of antiderivative = 5.86, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2481, 2475, 27, 1172, 321}

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.

\(\displaystyle \int \frac {1}{\sqrt {6 x^3-53 x^2+67 x+70}} \, dx\)

\(\Big \downarrow \) 2481

\(\displaystyle \int \frac {1}{\sqrt {6 \left (x-\frac {53}{18}\right )^3-\frac {1603}{18} \left (x-\frac {53}{18}\right )-\frac {9490}{243}}}d\left (x-\frac {53}{18}\right )\)

\(\Big \downarrow \) 2475

\(\displaystyle \frac {3 \sqrt {2} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \int \frac {3 \sqrt {3}}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {9 \sqrt {6} \sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603} \int \frac {1}{\sqrt {18 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}} \sqrt {324 \left (x-\frac {53}{18}\right )^2+\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\left (18980+35397 i \sqrt {3}\right )^{2/3}+\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}-1603}}d\left (x-\frac {53}{18}\right )}{\sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \sqrt {-\frac {\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}-18 \left (x-\frac {53}{18}\right )}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \int \frac {1}{\sqrt {1-\frac {\sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}} \sqrt {\frac {\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt [3]{18980+35397 i \sqrt {3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{\sqrt {3} \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )\right )}+1}}d\frac {\sqrt {\frac {\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (-36 \left (x-\frac {53}{18}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1-\frac {1603}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )\right )}{1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}}}}{\sqrt {6}}}{\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 \left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \sqrt {-\frac {-18 \left (x-\frac {53}{18}\right )+\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}{\frac {1}{324} \sqrt {-\frac {1}{3} \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (324-\frac {519372}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}\right )-\frac {1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}}{\sqrt [3]{18980+35397 i \sqrt {3}}}}} \sqrt {-\frac {\left (18980+35397 i \sqrt {3}\right )^{2/3} \left (-324 \left (x-\frac {53}{18}\right )^2-\frac {18 \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right ) \left (x-\frac {53}{18}\right )}{\sqrt [3]{18980+35397 i \sqrt {3}}}-\left (18980+35397 i \sqrt {3}\right )^{2/3}-\frac {2569609}{\left (18980+35397 i \sqrt {3}\right )^{2/3}}+1603\right )}{\left (1603-\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {53}{18}-x\right ),\frac {2 \left (18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}\right )}{18980+35397 i \sqrt {3}-1603 \sqrt [3]{18980+35397 i \sqrt {3}}+\sqrt {-3 \left (18980+35397 i \sqrt {3}\right )^{2/3}} \left (1603+\left (18980+35397 i \sqrt {3}\right )^{2/3}\right )}\right )}{\sqrt {-\left (18980+35397 i \sqrt {3}\right )^{2/3}} \sqrt {2916 \left (x-\frac {53}{18}\right )^3-43281 \left (x-\frac {53}{18}\right )-18980}}\)

Input:

Int[1/Sqrt[70 + 67*x - 53*x^2 + 6*x^3],x]
 

Output:

(2*(1603 - (18980 + (35397*I)*Sqrt[3])^(2/3))*Sqrt[-(((1603 + (18980 + (35 
397*I)*Sqrt[3])^(2/3))/(18980 + (35397*I)*Sqrt[3])^(1/3) - 18*(-53/18 + x) 
)/((Sqrt[-1/3*(18980 + (35397*I)*Sqrt[3])^(2/3)]*(324 - 519372/(18980 + (3 
5397*I)*Sqrt[3])^(2/3)))/324 - (1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))/ 
(18980 + (35397*I)*Sqrt[3])^(1/3)))]*Sqrt[-(((18980 + (35397*I)*Sqrt[3])^( 
2/3)*(1603 - 2569609/(18980 + (35397*I)*Sqrt[3])^(2/3) - (18980 + (35397*I 
)*Sqrt[3])^(2/3) - (18*(1603 + (18980 + (35397*I)*Sqrt[3])^(2/3))*(-53/18 
+ x))/(18980 + (35397*I)*Sqrt[3])^(1/3) - 324*(-53/18 + x)^2))/(1603 - (18 
980 + (35397*I)*Sqrt[3])^(2/3))^2)]*EllipticF[ArcSin[53/18 - x], (2*(18980 
 + (35397*I)*Sqrt[3] - 1603*(18980 + (35397*I)*Sqrt[3])^(1/3)))/(18980 + ( 
35397*I)*Sqrt[3] - 1603*(18980 + (35397*I)*Sqrt[3])^(1/3) + Sqrt[-3*(18980 
 + (35397*I)*Sqrt[3])^(2/3)]*(1603 + (18980 + (35397*I)*Sqrt[3])^(2/3)))]) 
/(Sqrt[-(18980 + (35397*I)*Sqrt[3])^(2/3)]*Sqrt[-18980 - 43281*(-53/18 + x 
) + 2916*(-53/18 + x)^3])
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 2475

Int[((a_.) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9* 
a*d^2 + Sqrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Simp[(a + b*x + d*x^3)^p 
/(Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3 
)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r/1 
8^(1/3))*x + d^2*x^2, x]^p)   Int[Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/3) + 
d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d* 
(2^(1/3)*b*(d/(3^(1/3)*r)) - r/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; Free 
Q[{a, b, d, p}, x] && NeQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2481

Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] 
, c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c* 
d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, c/(3 
*d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
 

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77

method	result	size
default	\(\frac {\sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(56\)
elliptic	\(\frac {\sqrt {76+114 x}\, \sqrt {483-69 x}\, \sqrt {285-114 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {76+114 x}}{19}, \frac {\sqrt {874}}{46}\right )}{1311 \sqrt {6 x^{3}-53 x^{2}+67 x +70}}\)	\(56\)

Input:

int(1/(6*x^3-53*x^2+67*x+70)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/1311*(76+114*x)^(1/2)*(483-69*x)^(1/2)*(285-114*x)^(1/2)/(6*x^3-53*x^2+6 
7*x+70)^(1/2)*EllipticF(1/19*(76+114*x)^(1/2),1/46*874^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.15 \[ \int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\frac {1}{3} \, \sqrt {6} {\rm weierstrassPInverse}\left (\frac {1603}{27}, \frac {18980}{729}, x - \frac {53}{18}\right ) \] Input:

integrate(1/(6*x^3-53*x^2+67*x+70)^(1/2),x, algorithm="fricas")
 

Output:

1/3*sqrt(6)*weierstrassPInverse(1603/27, 18980/729, x - 53/18)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\int \frac {1}{\sqrt {6 x^{3} - 53 x^{2} + 67 x + 70}}\, dx \] Input:

integrate(1/(6*x**3-53*x**2+67*x+70)**(1/2),x)
 

Output:

Integral(1/sqrt(6*x**3 - 53*x**2 + 67*x + 70), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\int { \frac {1}{\sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70}} \,d x } \] Input:

integrate(1/(6*x^3-53*x^2+67*x+70)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/sqrt(6*x^3 - 53*x^2 + 67*x + 70), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\int { \frac {1}{\sqrt {6 \, x^{3} - 53 \, x^{2} + 67 \, x + 70}} \,d x } \] Input:

integrate(1/(6*x^3-53*x^2+67*x+70)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/sqrt(6*x^3 - 53*x^2 + 67*x + 70), x)
 

Mupad [B] (verification not implemented)

Time = 12.71 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=-\frac {46\,\sqrt {\frac {2\,x}{9}-\frac {5}{9}}\,\sqrt {\frac {3\,x}{23}+\frac {2}{23}}\,\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {21}{23}-\frac {3\,x}{23}}\right )\middle |\frac {46}{27}\right )}{3\,\sqrt {6\,x^3-53\,x^2+67\,x+70}} \] Input:

int(1/(67*x - 53*x^2 + 6*x^3 + 70)^(1/2),x)
 

Output:

-(46*((2*x)/9 - 5/9)^(1/2)*((3*x)/23 + 2/23)^(1/2)*(21/23 - (3*x)/23)^(1/2 
)*ellipticF(asin((21/23 - (3*x)/23)^(1/2)), 46/27))/(3*(67*x - 53*x^2 + 6* 
x^3 + 70)^(1/2))
 

Reduce [F]

\[ \int \frac {1}{\sqrt {70+67 x-53 x^2+6 x^3}} \, dx=\int \frac {\sqrt {6 x^{3}-53 x^{2}+67 x +70}}{6 x^{3}-53 x^{2}+67 x +70}d x \] Input:

int(1/(6*x^3-53*x^2+67*x+70)^(1/2),x)
 

Output:

int(sqrt(6*x**3 - 53*x**2 + 67*x + 70)/(6*x**3 - 53*x**2 + 67*x + 70),x)