3.194 \(\int \frac{\text{b1}+\text{c1} x}{a+2 b x+c x^2} \, dx\)

Optimal. Leaf size=65 \[ \frac{\text{c1} \log \left (a+2 b x+c x^2\right )}{2 c}-\frac{(\text{b1} c-b \text{c1}) \tanh ^{-1}\left (\frac{b+c x}{\sqrt{b^2-a c}}\right )}{c \sqrt{b^2-a c}} \]

[Out]

-(((b1*c - b*c1)*ArcTanh[(b + c*x)/Sqrt[b^2 - a*c]])/(c*Sqrt[b^2 - a*c])) + (c1*
Log[a + 2*b*x + c*x^2])/(2*c)

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Rubi [A]  time = 0.0886123, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{\text{c1} \log \left (a+2 b x+c x^2\right )}{2 c}-\frac{(\text{b1} c-b \text{c1}) \tanh ^{-1}\left (\frac{b+c x}{\sqrt{b^2-a c}}\right )}{c \sqrt{b^2-a c}} \]

Antiderivative was successfully verified.

[In]  Int[(b1 + c1*x)/(a + 2*b*x + c*x^2),x]

[Out]

-(((b1*c - b*c1)*ArcTanh[(b + c*x)/Sqrt[b^2 - a*c]])/(c*Sqrt[b^2 - a*c])) + (c1*
Log[a + 2*b*x + c*x^2])/(2*c)

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Rubi in Sympy [A]  time = 7.32027, size = 53, normalized size = 0.82 \[ \frac{c_{1} \log{\left (a + 2 b x + c x^{2} \right )}}{2 c} + \frac{\left (b c_{1} - b_{1} c\right ) \operatorname{atanh}{\left (\frac{b + c x}{\sqrt{- a c + b^{2}}} \right )}}{c \sqrt{- a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c1*x+b1)/(c*x**2+2*b*x+a),x)

[Out]

c1*log(a + 2*b*x + c*x**2)/(2*c) + (b*c1 - b1*c)*atanh((b + c*x)/sqrt(-a*c + b**
2))/(c*sqrt(-a*c + b**2))

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Mathematica [A]  time = 0.0754987, size = 66, normalized size = 1.02 \[ \frac{(\text{b1} c-b \text{c1}) \tan ^{-1}\left (\frac{b+c x}{\sqrt{a c-b^2}}\right )}{c \sqrt{a c-b^2}}+\frac{\text{c1} \log \left (a+2 b x+c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

[In]  Integrate[(b1 + c1*x)/(a + 2*b*x + c*x^2),x]

[Out]

((b1*c - b*c1)*ArcTan[(b + c*x)/Sqrt[-b^2 + a*c]])/(c*Sqrt[-b^2 + a*c]) + (c1*Lo
g[a + 2*b*x + c*x^2])/(2*c)

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Maple [A]  time = 0.007, size = 95, normalized size = 1.5 \[{\frac{{\it c1}\,\ln \left ( c{x}^{2}+2\,bx+a \right ) }{2\,c}}+{{\it b1}\arctan \left ({\frac{2\,cx+2\,b}{2}{\frac{1}{\sqrt{ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{ac-{b}^{2}}}}}-{\frac{b{\it c1}}{c}\arctan \left ({\frac{2\,cx+2\,b}{2}{\frac{1}{\sqrt{ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c1*x+b1)/(c*x^2+2*b*x+a),x)

[Out]

1/2*c1*ln(c*x^2+2*b*x+a)/c+1/(a*c-b^2)^(1/2)*arctan(1/2*(2*c*x+2*b)/(a*c-b^2)^(1
/2))*b1-1/(a*c-b^2)^(1/2)*arctan(1/2*(2*c*x+2*b)/(a*c-b^2)^(1/2))*c1*b/c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x + b1)/(c*x^2 + 2*b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217302, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b^{2} - a c} c_{1} \log \left (c x^{2} + 2 \, b x + a\right ) -{\left (b_{1} c - b c_{1}\right )} \log \left (\frac{2 \, b^{3} - 2 \, a b c + 2 \,{\left (b^{2} c - a c^{2}\right )} x +{\left (c^{2} x^{2} + 2 \, b c x + 2 \, b^{2} - a c\right )} \sqrt{b^{2} - a c}}{c x^{2} + 2 \, b x + a}\right )}{2 \, \sqrt{b^{2} - a c} c}, \frac{\sqrt{-b^{2} + a c} c_{1} \log \left (c x^{2} + 2 \, b x + a\right ) + 2 \,{\left (b_{1} c - b c_{1}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + a c}{\left (c x + b\right )}}{b^{2} - a c}\right )}{2 \, \sqrt{-b^{2} + a c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x + b1)/(c*x^2 + 2*b*x + a),x, algorithm="fricas")

[Out]

[1/2*(sqrt(b^2 - a*c)*c1*log(c*x^2 + 2*b*x + a) - (b1*c - b*c1)*log((2*b^3 - 2*a
*b*c + 2*(b^2*c - a*c^2)*x + (c^2*x^2 + 2*b*c*x + 2*b^2 - a*c)*sqrt(b^2 - a*c))/
(c*x^2 + 2*b*x + a)))/(sqrt(b^2 - a*c)*c), 1/2*(sqrt(-b^2 + a*c)*c1*log(c*x^2 +
2*b*x + a) + 2*(b1*c - b*c1)*arctan(-sqrt(-b^2 + a*c)*(c*x + b)/(b^2 - a*c)))/(s
qrt(-b^2 + a*c)*c)]

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Sympy [A]  time = 1.02035, size = 246, normalized size = 3.78 \[ \left (\frac{c_{1}}{2 c} - \frac{\sqrt{- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{c_{1}}{2 c} - \frac{\sqrt{- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) + a c_{1} + 2 b^{2} \left (\frac{c_{1}}{2 c} - \frac{\sqrt{- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) - b b_{1}}{b c_{1} - b_{1} c} \right )} + \left (\frac{c_{1}}{2 c} + \frac{\sqrt{- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{c_{1}}{2 c} + \frac{\sqrt{- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) + a c_{1} + 2 b^{2} \left (\frac{c_{1}}{2 c} + \frac{\sqrt{- a c + b^{2}} \left (b c_{1} - b_{1} c\right )}{2 c \left (a c - b^{2}\right )}\right ) - b b_{1}}{b c_{1} - b_{1} c} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x+b1)/(c*x**2+2*b*x+a),x)

[Out]

(c1/(2*c) - sqrt(-a*c + b**2)*(b*c1 - b1*c)/(2*c*(a*c - b**2)))*log(x + (-2*a*c*
(c1/(2*c) - sqrt(-a*c + b**2)*(b*c1 - b1*c)/(2*c*(a*c - b**2))) + a*c1 + 2*b**2*
(c1/(2*c) - sqrt(-a*c + b**2)*(b*c1 - b1*c)/(2*c*(a*c - b**2))) - b*b1)/(b*c1 -
b1*c)) + (c1/(2*c) + sqrt(-a*c + b**2)*(b*c1 - b1*c)/(2*c*(a*c - b**2)))*log(x +
(-2*a*c*(c1/(2*c) + sqrt(-a*c + b**2)*(b*c1 - b1*c)/(2*c*(a*c - b**2))) + a*c1
+ 2*b**2*(c1/(2*c) + sqrt(-a*c + b**2)*(b*c1 - b1*c)/(2*c*(a*c - b**2))) - b*b1)
/(b*c1 - b1*c))

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GIAC/XCAS [A]  time = 0.199813, size = 81, normalized size = 1.25 \[ \frac{c_{1}{\rm ln}\left (c x^{2} + 2 \, b x + a\right )}{2 \, c} + \frac{{\left (b_{1} c - b c_{1}\right )} \arctan \left (\frac{c x + b}{\sqrt{-b^{2} + a c}}\right )}{\sqrt{-b^{2} + a c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c1*x + b1)/(c*x^2 + 2*b*x + a),x, algorithm="giac")

[Out]

1/2*c1*ln(c*x^2 + 2*b*x + a)/c + (b1*c - b*c1)*arctan((c*x + b)/sqrt(-b^2 + a*c)
)/(sqrt(-b^2 + a*c)*c)