3.1529 \(\int \frac{b+2 c x}{(d+e x) \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=130 \[ \frac{e \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac{(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2} \]

[Out]

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

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Rubi [A]  time = 0.350895, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{e \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac{(2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 48.6779, size = 114, normalized size = 0.88 \[ \frac{e \sqrt{- 4 a c + b^{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{a e^{2} - b d e + c d^{2}} - \frac{\left (\frac{b e}{2} - c d\right ) \log{\left (a + b x + c x^{2} \right )}}{a e^{2} - b d e + c d^{2}} + \frac{\left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{a e^{2} - b d e + c d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.207697, size = 116, normalized size = 0.89 \[ \frac{\sqrt{4 a c-b^2} (2 c d-b e) (2 \log (d+e x)-\log (a+x (b+c x)))+2 e \left (b^2-4 a c\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{2 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 233, normalized size = 1.8 \[{\frac{\ln \left ( ex+d \right ) be}{a{e}^{2}-bde+c{d}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ) cd}{a{e}^{2}-bde+c{d}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) be}{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}}+{\frac{c\ln \left ( c{x}^{2}+bx+a \right ) d}{a{e}^{2}-bde+c{d}^{2}}}+4\,{\frac{ace}{ \left ( a{e}^{2}-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{2}e}{a{e}^{2}-bde+c{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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((2*c*x + b)/((c*x^2 + b*x + a)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.353216, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{b^{2} - 4 \, a c} e \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) +{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}}, \frac{2 \, \sqrt{-b^{2} + 4 \, a c} e \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right ) +{\left (2 \, c d - b e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (2 \, c d - b e\right )} \log \left (e x + d\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.273724, size = 201, normalized size = 1.55 \[ \frac{{\left (2 \, c d - b e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}} - \frac{{\left (2 \, c d e - b e^{2}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2} + a e^{3}} - \frac{{\left (b^{2} e - 4 \, a c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*c*d - b*e)*ln(c*x^2 + b*x + a)/(c*d^2 - b*d*e + a*e^2) - (2*c*d*e - b*e^2
)*ln(abs(x*e + d))/(c*d^2*e - b*d*e^2 + a*e^3) - (b^2*e - 4*a*c*e)*arctan((2*c*x
+ b)/sqrt(-b^2 + 4*a*c))/((c*d^2 - b*d*e + a*e^2)*sqrt(-b^2 + 4*a*c))