3.1543 \(\int \frac{(b+2 c x) (d+e x)}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=91 \[ -\frac{e (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 c e \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{d+e x}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

-(d + e*x)/(2*(a + b*x + c*x^2)^2) - (e*(b + 2*c*x))/(2*(b^2 - 4*a*c)*(a + b*x +
c*x^2)) + (2*c*e*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

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Rubi [A]  time = 0.110472, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{e (b+2 c x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 c e \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{d+e x}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(d + e*x))/(a + b*x + c*x^2)^3,x]

[Out]

-(d + e*x)/(2*(a + b*x + c*x^2)^2) - (e*(b + 2*c*x))/(2*(b^2 - 4*a*c)*(a + b*x +
c*x^2)) + (2*c*e*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

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Rubi in Sympy [A]  time = 22.446, size = 83, normalized size = 0.91 \[ \frac{2 c e \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{e \left (b + 2 c x\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{d + e x}{2 \left (a + b x + c x^{2}\right )^{2}} \]

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)**3,x)

[Out]

2*c*e*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(3/2) - e*(b + 2*c
*x)/(2*(-4*a*c + b**2)*(a + b*x + c*x**2)) - (d + e*x)/(2*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.241284, size = 93, normalized size = 1.02 \[ \frac{1}{2} \left (-\frac{e (b+2 c x)}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{4 c e \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}-\frac{d+e x}{(a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(d + e*x))/(a + b*x + c*x^2)^3,x]

[Out]

(-((d + e*x)/(a + x*(b + c*x))^2) - (e*(b + 2*c*x))/((b^2 - 4*a*c)*(a + x*(b + c
*x))) + (4*c*e*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2))/2

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Maple [A]  time = 0.013, size = 146, normalized size = 1.6 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ({\frac{{c}^{2}e{x}^{3}}{4\,ac-{b}^{2}}}+{\frac{3\,bce{x}^{2}}{8\,ac-2\,{b}^{2}}}-{\frac{e \left ( ac-{b}^{2} \right ) x}{4\,ac-{b}^{2}}}+{\frac{bea-4\,acd+{b}^{2}d}{8\,ac-2\,{b}^{2}}} \right ) }+2\,{\frac{ce}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)/(c*x^2+b*x+a)^3,x)

[Out]

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

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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)*(e*x + d)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.310143, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (c^{3} e x^{4} + 2 \, b c^{2} e x^{3} + 2 \, a b c e x + a^{2} c e +{\left (b^{2} c + 2 \, a c^{2}\right )} e x^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, c^{2} e x^{3} + 3 \, b c e x^{2} + a b e + 2 \,{\left (b^{2} - a c\right )} e x +{\left (b^{2} - 4 \, a c\right )} d\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{4 \,{\left (c^{3} e x^{4} + 2 \, b c^{2} e x^{3} + 2 \, a b c e x + a^{2} c e +{\left (b^{2} c + 2 \, a c^{2}\right )} e x^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c^{2} e x^{3} + 3 \, b c e x^{2} + a b e + 2 \,{\left (b^{2} - a c\right )} e x +{\left (b^{2} - 4 \, a c\right )} d\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[-1/2*(2*(c^3*e*x^4 + 2*b*c^2*e*x^3 + 2*a*b*c*e*x + a^2*c*e + (b^2*c + 2*a*c^2)*
e*x^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^2*e*x^3 + 3*b*c*e*x^2 + a*
b*e + 2*(b^2 - a*c)*e*x + (b^2 - 4*a*c)*d)*sqrt(b^2 - 4*a*c))/(((b^2*c^2 - 4*a*c
^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a
^2*c^2)*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt(b^2 - 4*a*c)), -1/2*(4*(c^3*e*x^4 +
2*b*c^2*e*x^3 + 2*a*b*c*e*x + a^2*c*e + (b^2*c + 2*a*c^2)*e*x^2)*arctan(-sqrt(-b
^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*c^2*e*x^3 + 3*b*c*e*x^2 + a*b*e + 2*
(b^2 - a*c)*e*x + (b^2 - 4*a*c)*d)*sqrt(-b^2 + 4*a*c))/(((b^2*c^2 - 4*a*c^3)*x^4
+ a^2*b^2 - 4*a^3*c + 2*(b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)
*x^2 + 2*(a*b^3 - 4*a^2*b*c)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 13.0465, size = 377, normalized size = 4.14 \[ - c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 16 a^{2} c^{3} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{2} c^{2} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - b^{4} c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b c e}{2 c^{2} e} \right )} + c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{16 a^{2} c^{3} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{2} c^{2} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{4} c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b c e}{2 c^{2} e} \right )} + \frac{a b e - 4 a c d + b^{2} d + 3 b c e x^{2} + 2 c^{2} e x^{3} + x \left (- 2 a c e + 2 b^{2} e\right )}{8 a^{3} c - 2 a^{2} b^{2} + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{3} \left (16 a b c^{2} - 4 b^{3} c\right ) + x^{2} \left (16 a^{2} c^{2} + 4 a b^{2} c - 2 b^{4}\right ) + x \left (16 a^{2} b c - 4 a b^{3}\right )} \]

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)**3,x)

[Out]

-c*e*sqrt(-1/(4*a*c - b**2)**3)*log(x + (-16*a**2*c**3*e*sqrt(-1/(4*a*c - b**2)*
*3) + 8*a*b**2*c**2*e*sqrt(-1/(4*a*c - b**2)**3) - b**4*c*e*sqrt(-1/(4*a*c - b**
2)**3) + b*c*e)/(2*c**2*e)) + c*e*sqrt(-1/(4*a*c - b**2)**3)*log(x + (16*a**2*c*
*3*e*sqrt(-1/(4*a*c - b**2)**3) - 8*a*b**2*c**2*e*sqrt(-1/(4*a*c - b**2)**3) + b
**4*c*e*sqrt(-1/(4*a*c - b**2)**3) + b*c*e)/(2*c**2*e)) + (a*b*e - 4*a*c*d + b**
2*d + 3*b*c*e*x**2 + 2*c**2*e*x**3 + x*(-2*a*c*e + 2*b**2*e))/(8*a**3*c - 2*a**2
*b**2 + x**4*(8*a*c**3 - 2*b**2*c**2) + x**3*(16*a*b*c**2 - 4*b**3*c) + x**2*(16
*a**2*c**2 + 4*a*b**2*c - 2*b**4) + x*(16*a**2*b*c - 4*a*b**3))

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GIAC/XCAS [A]  time = 0.27382, size = 165, normalized size = 1.81 \[ -\frac{2 \, c \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right ) e}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, c^{2} x^{3} e + 3 \, b c x^{2} e + 2 \, b^{2} x e - 2 \, a c x e + b^{2} d - 4 \, a c d + a b e}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}} \]

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)^3,x, algorithm="giac")

[Out]

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