\(\int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx\) [2282]

Optimal result

Integrand size = 22, antiderivative size = 174 \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\frac {1}{4} \sqrt [3]{-1-2 x+6 x^2}+\frac {\sqrt [3]{\frac {7}{2}} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {2}{7}} \sqrt [3]{-1-2 x+6 x^2}}{\sqrt [6]{3}}\right )}{4\ 3^{5/6}}-\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log \left (7+\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}\right )+\frac {1}{24} \sqrt [3]{\frac {7}{6}} \log \left (7-\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}+6^{2/3} \sqrt [3]{7} \left (-1-2 x+6 x^2\right )^{2/3}\right ) \] Output:

1/4*(6*x^2-2*x-1)^(1/3)-1/24*7^(1/3)*2^(2/3)*arctan(-1/3*3^(1/2)+2/21*2^(1 
/3)*7^(2/3)*(6*x^2-2*x-1)^(1/3)*3^(5/6))*3^(1/6)-1/72*7^(1/3)*6^(2/3)*ln(7 
+6^(1/3)*7^(2/3)*(6*x^2-2*x-1)^(1/3))+1/144*7^(1/3)*6^(2/3)*ln(7-6^(1/3)*7 
^(2/3)*(6*x^2-2*x-1)^(1/3)+6^(2/3)*7^(1/3)*(6*x^2-2*x-1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\frac {1}{4} \sqrt [3]{-1-2 x+6 x^2}+\frac {\sqrt [3]{\frac {7}{2}} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {2}{7}} \sqrt [3]{-1-2 x+6 x^2}}{\sqrt [6]{3}}\right )}{4\ 3^{5/6}}-\frac {1}{12} \sqrt [3]{\frac {7}{6}} \log \left (7+\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}\right )+\frac {1}{24} \sqrt [3]{\frac {7}{6}} \log \left (7-\sqrt [3]{6} 7^{2/3} \sqrt [3]{-1-2 x+6 x^2}+6^{2/3} \sqrt [3]{7} \left (-1-2 x+6 x^2\right )^{2/3}\right ) \] Input:

Integrate[(-1 - 2*x + 6*x^2)^(1/3)/(-1 + 6*x),x]
 

Output:

(-1 - 2*x + 6*x^2)^(1/3)/4 + ((7/2)^(1/3)*ArcTan[1/Sqrt[3] - (2*(2/7)^(1/3 
)*(-1 - 2*x + 6*x^2)^(1/3))/3^(1/6)])/(4*3^(5/6)) - ((7/6)^(1/3)*Log[7 + 6 
^(1/3)*7^(2/3)*(-1 - 2*x + 6*x^2)^(1/3)])/12 + ((7/6)^(1/3)*Log[7 - 6^(1/3 
)*7^(2/3)*(-1 - 2*x + 6*x^2)^(1/3) + 6^(2/3)*7^(1/3)*(-1 - 2*x + 6*x^2)^(2 
/3)])/24
 

Rubi [A] (warning: unable to verify)

Time = 0.35 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.59, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1118, 27, 243, 60, 70, 16, 1082, 217}

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.

\(\displaystyle \int \frac {\sqrt [3]{6 x^2-2 x-1}}{6 x-1} \, dx\)

\(\Big \downarrow \) 1118

\(\displaystyle \frac {1}{6} \int \frac {\sqrt [3]{(6 x-1)^2-7}}{\sqrt [3]{6} (6 x-1)}d(6 x-1)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt [3]{(6 x-1)^2-7}}{6 x-1}d(6 x-1)}{6 \sqrt [3]{6}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\int \frac {\sqrt [3]{6 x-8}}{(6 x-1)^2}d(6 x-1)^2}{12 \sqrt [3]{6}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3 \sqrt [3]{6 x-8}-7 \int \frac {1}{(6 x-8)^{2/3} (6 x-1)^2}d(6 x-1)^2}{12 \sqrt [3]{6}}\)

\(\Big \downarrow \) 70

\(\displaystyle \frac {3 \sqrt [3]{6 x-8}-7 \left (\frac {3 \int \frac {1}{\sqrt [3]{6 x-8}+\sqrt [3]{7}}d\sqrt [3]{6 x-8}}{2\ 7^{2/3}}+\frac {3 \int \frac {1}{(6 x-1)^4-\sqrt [3]{7} \sqrt [3]{6 x-8}+7^{2/3}}d\sqrt [3]{6 x-8}}{2 \sqrt [3]{7}}-\frac {\log \left ((6 x-1)^2\right )}{2\ 7^{2/3}}\right )}{12 \sqrt [3]{6}}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {3 \sqrt [3]{6 x-8}-7 \left (\frac {3 \int \frac {1}{(6 x-1)^4-\sqrt [3]{7} \sqrt [3]{6 x-8}+7^{2/3}}d\sqrt [3]{6 x-8}}{2 \sqrt [3]{7}}-\frac {\log \left ((6 x-1)^2\right )}{2\ 7^{2/3}}+\frac {3 \log \left (\sqrt [3]{6 x-8}+\sqrt [3]{7}\right )}{2\ 7^{2/3}}\right )}{12 \sqrt [3]{6}}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {3 \sqrt [3]{6 x-8}-7 \left (\frac {3 \int \frac {1}{-(6 x-1)^4-3}d\left (1-\frac {2 \sqrt [3]{6 x-8}}{\sqrt [3]{7}}\right )}{7^{2/3}}-\frac {\log \left ((6 x-1)^2\right )}{2\ 7^{2/3}}+\frac {3 \log \left (\sqrt [3]{6 x-8}+\sqrt [3]{7}\right )}{2\ 7^{2/3}}\right )}{12 \sqrt [3]{6}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {3 \sqrt [3]{6 x-8}-7 \left (-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{6 x-8}}{\sqrt [3]{7}}}{\sqrt {3}}\right )}{7^{2/3}}-\frac {\log \left ((6 x-1)^2\right )}{2\ 7^{2/3}}+\frac {3 \log \left (\sqrt [3]{6 x-8}+\sqrt [3]{7}\right )}{2\ 7^{2/3}}\right )}{12 \sqrt [3]{6}}\)

Input:

Int[(-1 - 2*x + 6*x^2)^(1/3)/(-1 + 6*x),x]
 

Output:

(3*(-8 + 6*x)^(1/3) - 7*(-((Sqrt[3]*ArcTan[(1 - (2*(-8 + 6*x)^(1/3))/7^(1/ 
3))/Sqrt[3]])/7^(2/3)) - Log[(-1 + 6*x)^2]/(2*7^(2/3)) + (3*Log[7^(1/3) + 
(-8 + 6*x)^(1/3)])/(2*7^(2/3))))/(12*6^(1/3))
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 70

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2) 
, x] + (Simp[3/(2*b*q)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] + Simp[3/(2*b*q^2)   Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1118

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[1/e   Subst[Int[x^m*(a - b^2/(4*c) + (c*x^2)/e^2)^p, x], x, 
d + e*x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0]
 

Maple [A] (verified)

Time = 12.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {\left (6 x^{2}-2 x -1\right )^{\frac {1}{3}}}{4}-\frac {7^{\frac {1}{3}} 6^{\frac {2}{3}} \ln \left (6 \left (6 x^{2}-2 x -1\right )^{\frac {1}{3}}+7^{\frac {1}{3}} 6^{\frac {2}{3}}\right )}{72}+\frac {7^{\frac {1}{3}} 6^{\frac {2}{3}} \ln \left (6\right )}{144}+\frac {7^{\frac {1}{3}} 6^{\frac {2}{3}} \ln \left (6 \left (6 x^{2}-2 x -1\right )^{\frac {2}{3}}-7^{\frac {1}{3}} 6^{\frac {2}{3}} \left (6 x^{2}-2 x -1\right )^{\frac {1}{3}}+7^{\frac {2}{3}} 6^{\frac {1}{3}}\right )}{144}-\frac {7^{\frac {1}{3}} 3^{\frac {1}{6}} 2^{\frac {2}{3}} \arctan \left (\frac {2 \left (6 x^{2}-2 x -1\right )^{\frac {1}{3}} 7^{\frac {2}{3}} 6^{\frac {1}{3}} \sqrt {3}}{21}-\frac {\sqrt {3}}{3}\right )}{24}\)	\(149\)
trager	\(\text {Expression too large to display}\)	\(871\)
risch	\(\text {Expression too large to display}\)	\(2681\)

Input:

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

Output:

1/4*(6*x^2-2*x-1)^(1/3)-1/72*7^(1/3)*6^(2/3)*ln(6*(6*x^2-2*x-1)^(1/3)+7^(1 
/3)*6^(2/3))+1/144*7^(1/3)*6^(2/3)*ln(6)+1/144*7^(1/3)*6^(2/3)*ln(6*(6*x^2 
-2*x-1)^(2/3)-7^(1/3)*6^(2/3)*(6*x^2-2*x-1)^(1/3)+7^(2/3)*6^(1/3))-1/24*7^ 
(1/3)*3^(1/6)*2^(2/3)*arctan(2/21*(6*x^2-2*x-1)^(1/3)*7^(2/3)*6^(1/3)*3^(1 
/2)-1/3*3^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=-\frac {1}{12} \, \sqrt {3} \left (\frac {7}{6}\right )^{\frac {1}{3}} \arctan \left (\frac {4}{7} \, \sqrt {3} \left (\frac {7}{6}\right )^{\frac {2}{3}} {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{24} \, \left (\frac {7}{6}\right )^{\frac {1}{3}} \log \left (\left (\frac {7}{6}\right )^{\frac {2}{3}} - \left (\frac {7}{6}\right )^{\frac {1}{3}} {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} + {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {2}{3}}\right ) - \frac {1}{12} \, \left (\frac {7}{6}\right )^{\frac {1}{3}} \log \left (\left (\frac {7}{6}\right )^{\frac {1}{3}} + {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}\right ) + \frac {1}{4} \, {\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}} \] Input:

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

Output:

-1/12*sqrt(3)*(7/6)^(1/3)*arctan(4/7*sqrt(3)*(7/6)^(2/3)*(6*x^2 - 2*x - 1) 
^(1/3) - 1/3*sqrt(3)) + 1/24*(7/6)^(1/3)*log((7/6)^(2/3) - (7/6)^(1/3)*(6* 
x^2 - 2*x - 1)^(1/3) + (6*x^2 - 2*x - 1)^(2/3)) - 1/12*(7/6)^(1/3)*log((7/ 
6)^(1/3) + (6*x^2 - 2*x - 1)^(1/3)) + 1/4*(6*x^2 - 2*x - 1)^(1/3)
 

Sympy [F]

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

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

Output:

Integral((6*x**2 - 2*x - 1)**(1/3)/(6*x - 1), x)
 

Maxima [F]

\[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\int { \frac {{\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}}{6 \, x - 1} \,d x } \] Input:

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

Output:

integrate((6*x^2 - 2*x - 1)^(1/3)/(6*x - 1), x)
 

Giac [F]

\[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\int { \frac {{\left (6 \, x^{2} - 2 \, x - 1\right )}^{\frac {1}{3}}}{6 \, x - 1} \,d x } \] Input:

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

Output:

integrate((6*x^2 - 2*x - 1)^(1/3)/(6*x - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=\int \frac {{\left (6\,x^2-2\,x-1\right )}^{1/3}}{6\,x-1} \,d x \] Input:

int((6*x^2 - 2*x - 1)^(1/3)/(6*x - 1),x)
                                                                                    
                                                                                    
 

Output:

int((6*x^2 - 2*x - 1)^(1/3)/(6*x - 1), x)
 

Reduce [F]

\[ \int \frac {\sqrt [3]{-1-2 x+6 x^2}}{-1+6 x} \, dx=-\frac {3 \left (6 x^{2}-2 x -1\right )^{\frac {1}{3}}}{2}+42 \left (\int \frac {\left (6 x^{2}-2 x -1\right )^{\frac {1}{3}} x^{2}}{36 x^{3}-18 x^{2}-4 x +1}d x \right )-14 \left (\int \frac {\left (6 x^{2}-2 x -1\right )^{\frac {1}{3}} x}{36 x^{3}-18 x^{2}-4 x +1}d x \right ) \] Input:

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

Output:

( - 3*(6*x**2 - 2*x - 1)**(1/3) + 84*int(((6*x**2 - 2*x - 1)**(1/3)*x**2)/ 
(36*x**3 - 18*x**2 - 4*x + 1),x) - 28*int(((6*x**2 - 2*x - 1)**(1/3)*x)/(3 
6*x**3 - 18*x**2 - 4*x + 1),x))/2