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

Optimal. Leaf size=38 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {35} (1-x)}{2 \sqrt {10 x^2-22 x+13}}\right )}{2 \sqrt {35}} \]

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1029, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {35} (1-x)}{2 \sqrt {10 x^2-22 x+13}}\right )}{2 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 206

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

Rule 1029

Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symb
ol] :> Dist[-2*g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - (b*d - a*e)*x^2, x], x], x, S
imp[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] && NeQ[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]

Rubi steps

\begin {align*} \int \frac {-2+x}{\left (17-18 x+5 x^2\right ) \sqrt {13-22 x+10 x^2}} \, dx &=8 \operatorname {Subst}\left (\int \frac {1}{64-140 x^2} \, dx,x,\frac {2-2 x}{\sqrt {13-22 x+10 x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {35} (1-x)}{2 \sqrt {13-22 x+10 x^2}}\right )}{2 \sqrt {35}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 76, normalized size = 2.00 \[ \frac {i \left (\tan ^{-1}\left (\frac {(2-18 i)-(1-18 i) x}{\sqrt {35} \sqrt {10 x^2-22 x+13}}\right )+i \tanh ^{-1}\left (\frac {(18-i) x-(18-2 i)}{\sqrt {35} \sqrt {10 x^2-22 x+13}}\right )\right )}{4 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/4)*(ArcTan[((2 - 18*I) - (1 - 18*I)*x)/(Sqrt[35]*Sqrt[13 - 22*x + 10*x^2])] + I*ArcTanh[((-18 + 2*I) + (18
- I)*x)/(Sqrt[35]*Sqrt[13 - 22*x + 10*x^2])]))/Sqrt[35]

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IntegrateAlgebraic [B]  time = 0.50, size = 91, normalized size = 2.39 \[ -\frac {\tanh ^{-1}\left (\frac {-50 x^2+\left (5 \sqrt {10} x-9 \sqrt {10}\right ) \sqrt {10 x^2-22 x+13}+145 x-135}{-2 \sqrt {35} \sqrt {10 x^2-22 x+13}+10 \sqrt {14} x-20 \sqrt {14}}\right )}{2 \sqrt {35}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.68, size = 81, normalized size = 2.13 \[ \frac {1}{280} \, \sqrt {35} \log \left (\frac {11225 \, x^{4} - 47220 \, x^{3} - 8 \, \sqrt {35} {\left (75 \, x^{3} - 233 \, x^{2} + 245 \, x - 87\right )} \sqrt {10 \, x^{2} - 22 \, x + 13} + 75534 \, x^{2} - 54372 \, x + 14849}{25 \, x^{4} - 180 \, x^{3} + 494 \, x^{2} - 612 \, x + 289}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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giac [B]  time = 1.84, size = 231, normalized size = 6.08 \[ \frac {1}{140} \, \sqrt {35} \log \left ({\left | 21875000000 \, \sqrt {14} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 82031250000 \, {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} - 91875000000 \, \sqrt {35} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 172812500000 \, \sqrt {10} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} + 240625000000 \, \sqrt {14} + 913281250000 \right |}\right ) - \frac {1}{140} \, \sqrt {35} \log \left ({\left | -21875000000 \, \sqrt {14} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 82031250000 \, {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )}^{2} + 91875000000 \, \sqrt {35} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 172812500000 \, \sqrt {10} {\left (\sqrt {10} x - \sqrt {10 \, x^{2} - 22 \, x + 13}\right )} - 240625000000 \, \sqrt {14} + 913281250000 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)) + 240625000000*sqrt(14) + 913281250000)) - 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)) - 240625000000*sqrt(14) + 913281250000))

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maple [C]  time = 0.36, size = 82, normalized size = 2.16

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x - 2}{\sqrt {10 \, x^{2} - 22 \, x + 13} {\left (5 \, x^{2} - 18 \, x + 17\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x-2}{\left (5\,x^2-18\,x+17\right )\,\sqrt {10\,x^2-22\,x+13}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x - 2}{\left (5 x^{2} - 18 x + 17\right ) \sqrt {10 x^{2} - 22 x + 13}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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