3.2173 \(\int \frac{(d+e x)^2}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=101 \[ -\frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{e^2 x}{c} \]

[Out]

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

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Rubi [A]  time = 0.284547, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^2}+\frac{e^2 x}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{2} \int \frac{1}{c}\, dx - \frac{e \left (b e - 2 c d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} + \frac{\left (- b e \left (b e - 2 c d\right ) + 2 c \left (a e^{2} - c d^{2}\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**2*Integral(1/c, x) - e*(b*e - 2*c*d)*log(a + b*x + c*x**2)/(2*c**2) + (-b*e*(
b*e - 2*c*d) + 2*c*(a*e**2 - c*d**2))*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(c*
*2*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.127274, size = 101, normalized size = 1. \[ \frac{\frac{2 \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+e (2 c d-b e) \log (a+x (b+c x))+2 c e^2 x}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.005, size = 207, normalized size = 2.1 \[{\frac{{e}^{2}x}{c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) b{e}^{2}}{2\,{c}^{2}}}+{\frac{d\ln \left ( c{x}^{2}+bx+a \right ) e}{c}}-2\,{\frac{a{e}^{2}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}{e}^{2}}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{bde}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^2+b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221662, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, c^{2} d^{2} - 2 \, b c d e +{\left (b^{2} - 2 \, a c\right )} e^{2}\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (2 \, c e^{2} x +{\left (2 \, c d e - b e^{2}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, \frac{2 \,{\left (2 \, c^{2} d^{2} - 2 \, b c d e +{\left (b^{2} - 2 \, a c\right )} e^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c e^{2} x +{\left (2 \, c d e - b e^{2}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.31535, size = 588, normalized size = 5.82 \[ \left (- \frac{e \left (b e - 2 c d\right )}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- a b e^{2} - 4 a c^{2} \left (- \frac{e \left (b e - 2 c d\right )}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + 4 a c d e + b^{2} c \left (- \frac{e \left (b e - 2 c d\right )}{2 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) - b c d^{2}}{2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}} \right )} + \left (- \frac{e \left (b e - 2 c d\right )}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x + \frac{- a b e^{2} - 4 a c^{2} \left (- \frac{e \left (b e - 2 c d\right )}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) + 4 a c d e + b^{2} c \left (- \frac{e \left (b e - 2 c d\right )}{2 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{2} \left (4 a c - b^{2}\right )}\right ) - b c d^{2}}{2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}} \right )} + \frac{e^{2} x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.205922, size = 135, normalized size = 1.34 \[ \frac{x e^{2}}{c} + \frac{{\left (2 \, c d e - b e^{2}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac{{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, a c e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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