3.885 \(\int \frac{d+e x}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=66 \[ \frac{e \log \left (a+b x+c x^2\right )}{2 c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}} \]

[Out]

-(((2*c*d - b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4*a*c]))
+ (e*Log[a + b*x + c*x^2])/(2*c)

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Rubi [A]  time = 0.0775156, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{e \log \left (a+b x+c x^2\right )}{2 c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

-(((2*c*d - b*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4*a*c]))
+ (e*Log[a + b*x + c*x^2])/(2*c)

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Rubi in Sympy [A]  time = 13.837, size = 58, normalized size = 0.88 \[ \frac{e \log{\left (a + b x + c x^{2} \right )}}{2 c} + \frac{\left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)/(c*x**2+b*x+a),x)

[Out]

e*log(a + b*x + c*x**2)/(2*c) + (b*e - 2*c*d)*atanh((b + 2*c*x)/sqrt(-4*a*c + b*
*2))/(c*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.106636, size = 66, normalized size = 1. \[ \frac{e \log (a+x (b+c x))-\frac{2 (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

((-2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] +
e*Log[a + x*(b + c*x)])/(2*c)

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Maple [A]  time = 0.002, size = 93, normalized size = 1.4 \[{\frac{e\ln \left ( c{x}^{2}+bx+a \right ) }{2\,c}}+2\,{\frac{d}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{be}{c}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+b*x+a),x)

[Out]

1/2*e*ln(c*x^2+b*x+a)/c+2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*
d-1/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*e/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((e*x + d)/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.320129, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b^{2} - 4 \, a c} e \log \left (c x^{2} + b x + a\right ) -{\left (2 \, c d - b e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right )}{2 \, \sqrt{b^{2} - 4 \, a c} c}, \frac{\sqrt{-b^{2} + 4 \, a c} e \log \left (c x^{2} + b x + a\right ) + 2 \,{\left (2 \, c d - b e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(sqrt(b^2 - 4*a*c)*e*log(c*x^2 + b*x + a) - (2*c*d - b*e)*log((b^3 - 4*a*b*
c + 2*(b^2*c - 4*a*c^2)*x + (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c
))/(c*x^2 + b*x + a)))/(sqrt(b^2 - 4*a*c)*c), 1/2*(sqrt(-b^2 + 4*a*c)*e*log(c*x^
2 + b*x + a) + 2*(2*c*d - b*e)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a
*c)))/(sqrt(-b^2 + 4*a*c)*c)]

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Sympy [A]  time = 2.05298, size = 280, normalized size = 4.24 \[ \left (\frac{e}{2 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 4 a c \left (\frac{e}{2 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac{e}{2 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} + \left (\frac{e}{2 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- 4 a c \left (\frac{e}{2 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + b^{2} \left (\frac{e}{2 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)/(c*x**2+b*x+a),x)

[Out]

(e/(2*c) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b**2)))*log(x + (-4*a
*c*(e/(2*c) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b**2))) + 2*a*e +
b**2*(e/(2*c) - sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b**2))) - b*d)/(
b*e - 2*c*d)) + (e/(2*c) + sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b**2)
))*log(x + (-4*a*c*(e/(2*c) + sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c - b*
*2))) + 2*a*e + b**2*(e/(2*c) + sqrt(-4*a*c + b**2)*(b*e - 2*c*d)/(2*c*(4*a*c -
b**2))) - b*d)/(b*e - 2*c*d))

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GIAC/XCAS [A]  time = 0.270371, size = 88, normalized size = 1.33 \[ \frac{e{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c} + \frac{{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*e*ln(c*x^2 + b*x + a)/c + (2*c*d - b*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c
))/(sqrt(-b^2 + 4*a*c)*c)