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

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 = 0.0649665, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{35} (1-x)}{2 \sqrt{10 x^2-22 x+13}}\right )}{2 \sqrt{35}} \]

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])

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Rubi in Sympy [A]  time = 5.85201, size = 32, normalized size = 0.84 \[ \frac{\sqrt{35} \operatorname{atanh}{\left (\frac{\sqrt{35} \left (- 2 x + 2\right )}{4 \sqrt{10 x^{2} - 22 x + 13}} \right )}}{70} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2+x)/(5*x**2-18*x+17)/(10*x**2-22*x+13)**(1/2),x)

[Out]

sqrt(35)*atanh(sqrt(35)*(-2*x + 2)/(4*sqrt(10*x**2 - 22*x + 13)))/70

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Mathematica [C]  time = 0.597147, size = 410, normalized size = 10.79 \[ \frac{2 \log \left (i \left (5 x^2-18 x+17\right )\right )-\log \left ((1+2 i) \left ((1350-25 i) x^2+70 \sqrt{35} \sqrt{10 x^2-22 x+13} x-68 \sqrt{35} \sqrt{10 x^2-22 x+13}-(2844-118 i) x+(1566-127 i)\right )\right )-\log \left ((2+i) \left ((1350+25 i) x^2+70 \sqrt{35} \sqrt{10 x^2-22 x+13} x-68 \sqrt{35} \sqrt{10 x^2-22 x+13}-(2844+118 i) x+(1566+127 i)\right )\right )-2 i \tan ^{-1}\left (\frac{4 (5 x-(2-2 i)) \left (10 x^2-22 x+13\right )}{350 x^3+10 \left ((2+i) \sqrt{35} \sqrt{10 x^2-22 x+13}-(140+14 i)\right ) x^2+\left ((1841+308 i)-(62+21 i) \sqrt{35} \sqrt{10 x^2-22 x+13}\right ) x+(44+11 i) \sqrt{35} \sqrt{10 x^2-22 x+13}-(819+182 i)}\right )-2 i \tan ^{-1}\left (\frac{(7+14 i) \left ((85+30 i) x^3-(335+140 i) x^2+(419+218 i) x-(169+116 i)\right )}{700 x^3+20 i \left ((1+2 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(14+140 i)\right ) x^2+(4-2 i) \left ((29+4 i) \sqrt{35} \sqrt{10 x^2-22 x+13}+(798+245 i)\right ) x-(88-22 i) \sqrt{35} \sqrt{10 x^2-22 x+13}-(1638-364 i)}\right )}{8 \sqrt{35}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*I)*ArcTan[(4*((-2 + 2*I) + 5*x)*(13 - 22*x + 10*x^2))/((-819 - 182*I) + 350
*x^3 + (44 + 11*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2] + x*((1841 + 308*I) - (62 +
21*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2]) + 10*x^2*((-140 - 14*I) + (2 + I)*Sqrt
[35]*Sqrt[13 - 22*x + 10*x^2]))] - (2*I)*ArcTan[((7 + 14*I)*((-169 - 116*I) + (4
19 + 218*I)*x - (335 + 140*I)*x^2 + (85 + 30*I)*x^3))/((-1638 + 364*I) + 700*x^3
- (88 - 22*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2] + (20*I)*x^2*((14 + 140*I) + (1
+ 2*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2]) + (4 - 2*I)*x*((798 + 245*I) + (29 +
4*I)*Sqrt[35]*Sqrt[13 - 22*x + 10*x^2]))] + 2*Log[I*(17 - 18*x + 5*x^2)] - Log[(
1 + 2*I)*((1566 - 127*I) - (2844 - 118*I)*x + (1350 - 25*I)*x^2 - 68*Sqrt[35]*Sq
rt[13 - 22*x + 10*x^2] + 70*Sqrt[35]*x*Sqrt[13 - 22*x + 10*x^2])] - Log[(2 + I)*
((1566 + 127*I) - (2844 + 118*I)*x + (1350 + 25*I)*x^2 - 68*Sqrt[35]*Sqrt[13 - 2
2*x + 10*x^2] + 70*Sqrt[35]*x*Sqrt[13 - 22*x + 10*x^2])])/(8*Sqrt[35])

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Maple [B]  time = 0.016, size = 94, normalized size = 2.5 \[ -{\frac{\sqrt{35}}{70}\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9}{\it Artanh} \left ({\frac{2\,\sqrt{35}}{35}\sqrt{{\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9}} \right ){\frac{1}{\sqrt{{1 \left ({\frac{ \left ( -2+x \right ) ^{2}}{ \left ( 1-x \right ) ^{2}}}+9 \right ) \left ( 1+{\frac{-2+x}{1-x}} \right ) ^{-2}}}}} \left ( 1+{\frac{-2+x}{1-x}} \right ) ^{-1}} \]

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)*arctanh(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. \[ \int \frac{x - 2}{\sqrt{10 \, x^{2} - 22 \, x + 13}{\left (5 \, x^{2} - 18 \, x + 17\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x - 2)/(sqrt(10*x^2 - 22*x + 13)*(5*x^2 - 18*x + 17)),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 [A]  time = 0.220966, size = 112, normalized size = 2.95 \[ \frac{1}{280} \, \sqrt{35} \log \left (\frac{\sqrt{35}{\left (11225 \, x^{4} - 47220 \, x^{3} + 75534 \, x^{2} - 54372 \, x + 14849\right )} - 280 \,{\left (75 \, x^{3} - 233 \, x^{2} + 245 \, x - 87\right )} \sqrt{10 \, x^{2} - 22 \, x + 13}}{25 \, x^{4} - 180 \, x^{3} + 494 \, x^{2} - 612 \, x + 289}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/280*sqrt(35)*log((sqrt(35)*(11225*x^4 - 47220*x^3 + 75534*x^2 - 54372*x + 1484
9) - 280*(75*x^3 - 233*x^2 + 245*x - 87)*sqrt(10*x^2 - 22*x + 13))/(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. \[ \int \frac{x - 2}{\left (5 x^{2} - 18 x + 17\right ) \sqrt{10 x^{2} - 22 x + 13}}\, dx \]

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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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: RuntimeError