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

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

[Out]

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

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Rubi [A]  time = 0.0238798, antiderivative size = 38, normalized size of antiderivative = 1., 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 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]

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])

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*}

Mathematica [C]  time = 0.0391635, size = 76, normalized size = 2. \[ \frac{i \left (\tan ^{-1}\left (\frac{(2-18 i)-(1-18 i) x}{\sqrt{35} \sqrt{10 x^2-22 x+13}}\right )+i \tanh ^{-1}\left (\frac{(18-i) x-(18-2 i)}{\sqrt{35} \sqrt{10 x^2-22 x+13}}\right )\right )}{4 \sqrt{35}} \]

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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Maple [B]  time = 0.01, size = 94, normalized size = 2.5 \begin{align*} -{\frac{\sqrt{35}}{70}\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9}{\it Artanh} \left ({\frac{2\,\sqrt{35}}{35}\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9}} \right ){\frac{1}{\sqrt{{ \left ({\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9 \right ) \left ( 1+{\frac{-2+x}{1-x}} \right ) ^{-2}}}}} \left ( 1+{\frac{-2+x}{1-x}} \right ) ^{-1}} \end{align*}

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)

[Out]

-1/70/(((-2+x)^2/(1-x)^2+9)/(1+(-2+x)/(1-x))^2)^(1/2)/(1+(-2+x)/(1-x))*((-2+x)^2/(1-x)^2+9)^(1/2)*35^(1/2)*arc
tanh(2/35*((-2+x)^2/(1-x)^2+9)^(1/2)*35^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - 2}{\sqrt{10 \, x^{2} - 22 \, x + 13}{\left (5 \, x^{2} - 18 \, x + 17\right )}}\,{d x} \end{align*}

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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Fricas [B]  time = 2.10608, size = 254, normalized size = 6.68 \begin{align*} \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 ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x - 2}{\left (5 x^{2} - 18 x + 17\right ) \sqrt{10 x^{2} - 22 x + 13}}\, dx \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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]

Exception raised: TypeError