\(\int \frac {-2+x}{(17-18 x+5 x^2) \sqrt {13-22 x+10 x^2}} \, dx\) [249]

Optimal result

Integrand size = 30, antiderivative size = 38 \[ \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {35} (1-x)}{2 \sqrt {13-22 x+10 x^2}}\right )}{2 \sqrt {35}} \] Output:

1/70*arctanh(1/2*(1-x)*35^(1/2)/(10*x^2-22*x+13)^(1/2))*35^(1/2)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(38)=76\).

Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.24 \[ \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {-135+145 x-50 x^2+\sqrt {10} (-9+5 x) \sqrt {13-22 x+10 x^2}}{-20 \sqrt {14}+10 \sqrt {14} x-2 \sqrt {35} \sqrt {13-22 x+10 x^2}}\right )}{2 \sqrt {35}} \] Input:

Integrate[(-2 + x)/((17 - 18*x + 5*x^2)*Sqrt[13 - 22*x + 10*x^2]),x]
 

Output:

-1/2*ArcTanh[(-135 + 145*x - 50*x^2 + Sqrt[10]*(-9 + 5*x)*Sqrt[13 - 22*x + 
 10*x^2])/(-20*Sqrt[14] + 10*Sqrt[14]*x - 2*Sqrt[35]*Sqrt[13 - 22*x + 10*x 
^2])]/Sqrt[35]
 

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1362, 219}

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.

\(\displaystyle \int \frac {x-2}{\left (5 x^2-18 x+17\right ) \sqrt {10 x^2-22 x+13}} \, dx\)

\(\Big \downarrow \) 1362

\(\displaystyle 8 \int \frac {1}{64-\frac {560 (1-x)^2}{10 x^2-22 x+13}}d\frac {2 (1-x)}{\sqrt {10 x^2-22 x+13}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\text {arctanh}\left (\frac {\sqrt {35} (1-x)}{2 \sqrt {10 x^2-22 x+13}}\right )}{2 \sqrt {35}}\)

Input:

Int[(-2 + x)/((17 - 18*x + 5*x^2)*Sqrt[13 - 22*x + 10*x^2]),x]
 

Output:

ArcTanh[(Sqrt[35]*(1 - x))/(2*Sqrt[13 - 22*x + 10*x^2])]/(2*Sqrt[35])
 

Defintions of rubi rules used

rule 219

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

rule 1362

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + 
(e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[-2*g*(g*b - 2*a*h)   Subst[I 
nt[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, Simp[ 
g*b - 2*a*h - (b*h - 2*g*c)*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ[{a, b 
, c, d, e, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[e^2 - 4*d*f, 0] && Ne 
Q[b*d - a*e, 0] && EqQ[h^2*(b*d - a*e) - 2*g*h*(c*d - a*f) + g^2*(c*e - b*f 
), 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.16

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (-\frac {75 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x^{2}-158 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +140 \sqrt {10 x^{2}-22 x +13}\, x +87 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )-140 \sqrt {10 x^{2}-22 x +13}}{5 x^{2}-18 x +17}\right )}{140}\)	\(82\)
default	\(-\frac {\sqrt {\frac {\left (-2+x \right )^{2}}{\left (1-x \right )^{2}}+9}\, \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \sqrt {\frac {\left (-2+x \right )^{2}}{\left (1-x \right )^{2}}+9}\, \sqrt {35}}{35}\right )}{70 \sqrt {\frac {\frac {\left (-2+x \right )^{2}}{\left (1-x \right )^{2}}+9}{\left (\frac {-2+x}{1-x}+1\right )^{2}}}\, \left (\frac {-2+x}{1-x}+1\right )}\)	\(94\)

Input:

int((-2+x)/(5*x^2-18*x+17)/(10*x^2-22*x+13)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/140*RootOf(_Z^2-35)*ln(-(75*RootOf(_Z^2-35)*x^2-158*RootOf(_Z^2-35)*x+1 
40*(10*x^2-22*x+13)^(1/2)*x+87*RootOf(_Z^2-35)-140*(10*x^2-22*x+13)^(1/2)) 
/(5*x^2-18*x+17))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (26) = 52\).

Time = 0.07 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.13 \[ \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx=\frac {1}{280} \, \sqrt {35} \log \left (\frac {11225 \, x^{4} - 47220 \, x^{3} - 8 \, \sqrt {35} {\left (75 \, x^{3} - 233 \, x^{2} + 245 \, x - 87\right )} \sqrt {10 \, x^{2} - 22 \, x + 13} + 75534 \, x^{2} - 54372 \, x + 14849}{25 \, x^{4} - 180 \, x^{3} + 494 \, x^{2} - 612 \, x + 289}\right ) \] Input:

integrate((-2+x)/(5*x^2-18*x+17)/(10*x^2-22*x+13)^(1/2),x, algorithm="fric 
as")
 

Output:

1/280*sqrt(35)*log((11225*x^4 - 47220*x^3 - 8*sqrt(35)*(75*x^3 - 233*x^2 + 
 245*x - 87)*sqrt(10*x^2 - 22*x + 13) + 75534*x^2 - 54372*x + 14849)/(25*x 
^4 - 180*x^3 + 494*x^2 - 612*x + 289))
 

Sympy [F]

\[ \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx=\int \frac {x - 2}{\left (5 x^{2} - 18 x + 17\right ) \sqrt {10 x^{2} - 22 x + 13}}\, dx \] Input:

integrate((-2+x)/(5*x**2-18*x+17)/(10*x**2-22*x+13)**(1/2),x)
                                                                                    
                                                                                    
 

Output:

Integral((x - 2)/((5*x**2 - 18*x + 17)*sqrt(10*x**2 - 22*x + 13)), x)
 

Maxima [F]

\[ \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx=\int { \frac {x - 2}{\sqrt {10 \, x^{2} - 22 \, x + 13} {\left (5 \, x^{2} - 18 \, x + 17\right )}} \,d x } \] Input:

integrate((-2+x)/(5*x^2-18*x+17)/(10*x^2-22*x+13)^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate((x - 2)/(sqrt(10*x^2 - 22*x + 13)*(5*x^2 - 18*x + 17)), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (26) = 52\).

Time = 0.24 (sec) , antiderivative size = 231, normalized size of antiderivative = 6.08 \[ \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx=\frac {1}{140} \, \sqrt {35} \log \left ({\left | 21875000000 \, \sqrt {14} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 82031250000 \, {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} - 91875000000 \, \sqrt {35} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 172812500000 \, \sqrt {10} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} + 240625000000 \, \sqrt {14} + 913281250000 \right |}\right ) - \frac {1}{140} \, \sqrt {35} \log \left ({\left | -21875000000 \, \sqrt {14} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 82031250000 \, {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 91875000000 \, \sqrt {35} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 172812500000 \, \sqrt {10} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 240625000000 \, \sqrt {14} + 913281250000 \right |}\right ) \] Input:

integrate((-2+x)/(5*x^2-18*x+17)/(10*x^2-22*x+13)^(1/2),x, algorithm="giac 
")
 

Output:

1/140*sqrt(35)*log(abs(21875000000*sqrt(14)*(sqrt(10)*x - sqrt(10*x^2 - 22 
*x + 13))^2 + 82031250000*(sqrt(10)*x - sqrt(10*x^2 - 22*x + 13))^2 - 9187 
5000000*sqrt(35)*(sqrt(10)*x - sqrt(10*x^2 - 22*x + 13)) - 172812500000*sq 
rt(10)*(sqrt(10)*x - sqrt(10*x^2 - 22*x + 13)) + 240625000000*sqrt(14) + 9 
13281250000)) - 1/140*sqrt(35)*log(abs(-21875000000*sqrt(14)*(sqrt(10)*x - 
 sqrt(10*x^2 - 22*x + 13))^2 + 82031250000*(sqrt(10)*x - sqrt(10*x^2 - 22* 
x + 13))^2 + 91875000000*sqrt(35)*(sqrt(10)*x - sqrt(10*x^2 - 22*x + 13)) 
- 172812500000*sqrt(10)*(sqrt(10)*x - sqrt(10*x^2 - 22*x + 13)) - 24062500 
0000*sqrt(14) + 913281250000))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx=\int \frac {x-2}{\left (5\,x^2-18\,x+17\right )\,\sqrt {10\,x^2-22\,x+13}} \,d x \] Input:

int((x - 2)/((5*x^2 - 18*x + 17)*(10*x^2 - 22*x + 13)^(1/2)),x)
 

Output:

int((x - 2)/((5*x^2 - 18*x + 17)*(10*x^2 - 22*x + 13)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx=\int \frac {x}{5 \sqrt {10 x^{2}-22 x +13}\, x^{2}-18 \sqrt {10 x^{2}-22 x +13}\, x +17 \sqrt {10 x^{2}-22 x +13}}d x -2 \left (\int \frac {1}{5 \sqrt {10 x^{2}-22 x +13}\, x^{2}-18 \sqrt {10 x^{2}-22 x +13}\, x +17 \sqrt {10 x^{2}-22 x +13}}d x \right ) \] Input:

int((-2+x)/(5*x^2-18*x+17)/(10*x^2-22*x+13)^(1/2),x)
 

Output:

int(x/(5*sqrt(10*x**2 - 22*x + 13)*x**2 - 18*sqrt(10*x**2 - 22*x + 13)*x + 
 17*sqrt(10*x**2 - 22*x + 13)),x) - 2*int(1/(5*sqrt(10*x**2 - 22*x + 13)*x 
**2 - 18*sqrt(10*x**2 - 22*x + 13)*x + 17*sqrt(10*x**2 - 22*x + 13)),x)