3.1542 \(\int \frac{(b+2 c x) (d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{e (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 e (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{(d+e x)^2}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

-(d + e*x)^2/(2*(a + b*x + c*x^2)^2) - (e*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2
- 4*a*c)*(a + b*x + c*x^2)) + (2*e*(2*c*d - b*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 = 0.171976, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{e (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{2 e (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{(d+e x)^2}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(d + e*x)^2/(2*(a + b*x + c*x^2)^2) - (e*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2
- 4*a*c)*(a + b*x + c*x^2)) + (2*e*(2*c*d - b*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 = 40.1985, size = 102, normalized size = 0.91 \[ \frac{e \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{2 e \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{\left (d + e x\right )^{2}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.337455, size = 143, normalized size = 1.28 \[ \frac{1}{2} \left (\frac{e \left (4 c (c d x-2 a e)+b^2 e+2 b c (d-e x)\right )}{c \left (4 a c-b^2\right ) (a+x (b+c x))}-\frac{4 e (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{e^2 (a+b x)-c d (d+2 e x)}{c (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((e*(b^2*e + 4*c*(-2*a*e + c*d*x) + 2*b*c*(d - e*x)))/(c*(-b^2 + 4*a*c)*(a + x*(
b + c*x))) + (e^2*(a + b*x) - c*d*(d + 2*e*x))/(c*(a + x*(b + c*x))^2) - (4*e*(-
2*c*d + b*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 [B]  time = 0.015, size = 229, normalized size = 2. \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ( -{\frac{ \left ( be-2\,cd \right ) ce{x}^{3}}{4\,ac-{b}^{2}}}-{\frac{e \left ( 8\,ace+{b}^{2}e-6\,bcd \right ){x}^{2}}{8\,ac-2\,{b}^{2}}}-{\frac{e \left ( 3\,bea+2\,acd-2\,{b}^{2}d \right ) x}{4\,ac-{b}^{2}}}-{\frac{4\,{a}^{2}{e}^{2}-2\,abde+4\,ac{d}^{2}-{b}^{2}{d}^{2}}{8\,ac-2\,{b}^{2}}} \right ) }-2\,{\frac{b{e}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+4\,{\frac{dec}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-(b*e-2*c*d)*c*e/(4*a*c-b^2)*x^3-1/2*e*(8*a*c*e+b^2*e-6*b*c*d)/(4*a*c-b^2)*x^2-
e*(3*a*b*e+2*a*c*d-2*b^2*d)/(4*a*c-b^2)*x-1/2*(4*a^2*e^2-2*a*b*d*e+4*a*c*d^2-b^2
*d^2)/(4*a*c-b^2))/(c*x^2+b*x+a)^2-2*e^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*b+4*e/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c*d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.315303, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^2/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[1/2*(2*(2*a^2*c*d*e - a^2*b*e^2 + (2*c^3*d*e - b*c^2*e^2)*x^4 + 2*(2*b*c^2*d*e
- b^2*c*e^2)*x^3 + (2*(b^2*c + 2*a*c^2)*d*e - (b^3 + 2*a*b*c)*e^2)*x^2 + 2*(2*a*
b*c*d*e - a*b^2*e^2)*x)*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*a*b*d*e - 4*
a^2*e^2 + 2*(2*c^2*d*e - b*c*e^2)*x^3 + (b^2 - 4*a*c)*d^2 + (6*b*c*d*e - (b^2 +
8*a*c)*e^2)*x^2 - 2*(3*a*b*e^2 - 2*(b^2 - a*c)*d*e)*x)*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*(
2*a^2*c*d*e - a^2*b*e^2 + (2*c^3*d*e - b*c^2*e^2)*x^4 + 2*(2*b*c^2*d*e - b^2*c*e
^2)*x^3 + (2*(b^2*c + 2*a*c^2)*d*e - (b^3 + 2*a*b*c)*e^2)*x^2 + 2*(2*a*b*c*d*e -
a*b^2*e^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*a*b*d*
e - 4*a^2*e^2 + 2*(2*c^2*d*e - b*c*e^2)*x^3 + (b^2 - 4*a*c)*d^2 + (6*b*c*d*e - (
b^2 + 8*a*c)*e^2)*x^2 - 2*(3*a*b*e^2 - 2*(b^2 - a*c)*d*e)*x)*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 = 42.7666, size = 530, normalized size = 4.73 \[ e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log{\left (x + \frac{- 16 a^{2} c^{2} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + 8 a b^{2} c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - b^{4} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e^{2} - 2 b c d e}{2 b c e^{2} - 4 c^{2} d e} \right )} - e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) \log{\left (x + \frac{16 a^{2} c^{2} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) - 8 a b^{2} c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{4} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (b e - 2 c d\right ) + b^{2} e^{2} - 2 b c d e}{2 b c e^{2} - 4 c^{2} d e} \right )} - \frac{4 a^{2} e^{2} - 2 a b d e + 4 a c d^{2} - b^{2} d^{2} + x^{3} \left (2 b c e^{2} - 4 c^{2} d e\right ) + x^{2} \left (8 a c e^{2} + b^{2} e^{2} - 6 b c d e\right ) + x \left (6 a b e^{2} + 4 a c d e - 4 b^{2} d e\right )}{8 a^{3} c - 2 a^{2} b^{2} + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{3} \left (16 a b c^{2} - 4 b^{3} c\right ) + x^{2} \left (16 a^{2} c^{2} + 4 a b^{2} c - 2 b^{4}\right ) + x \left (16 a^{2} b c - 4 a b^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d)*log(x + (-16*a**2*c**2*e*sqrt(-1/(4*a
*c - b**2)**3)*(b*e - 2*c*d) + 8*a*b**2*c*e*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*
c*d) - b**4*e*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) + b**2*e**2 - 2*b*c*d*e)/
(2*b*c*e**2 - 4*c**2*d*e)) - e*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d)*log(x +
(16*a**2*c**2*e*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*d) - 8*a*b**2*c*e*sqrt(-1/
(4*a*c - b**2)**3)*(b*e - 2*c*d) + b**4*e*sqrt(-1/(4*a*c - b**2)**3)*(b*e - 2*c*
d) + b**2*e**2 - 2*b*c*d*e)/(2*b*c*e**2 - 4*c**2*d*e)) - (4*a**2*e**2 - 2*a*b*d*
e + 4*a*c*d**2 - b**2*d**2 + x**3*(2*b*c*e**2 - 4*c**2*d*e) + x**2*(8*a*c*e**2 +
b**2*e**2 - 6*b*c*d*e) + x*(6*a*b*e**2 + 4*a*c*d*e - 4*b**2*d*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 = 0.276053, size = 247, normalized size = 2.21 \[ -\frac{2 \,{\left (2 \, c d e - b e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{4 \, c^{2} d x^{3} e - 2 \, b c x^{3} e^{2} + 6 \, b c d x^{2} e - b^{2} x^{2} e^{2} - 8 \, a c x^{2} e^{2} + 4 \, b^{2} d x e - 4 \, a c d x e + b^{2} d^{2} - 4 \, a c d^{2} - 6 \, a b x e^{2} + 2 \, a b d e - 4 \, a^{2} e^{2}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^2/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

-2*(2*c*d*e - b*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt(
-b^2 + 4*a*c)) - 1/2*(4*c^2*d*x^3*e - 2*b*c*x^3*e^2 + 6*b*c*d*x^2*e - b^2*x^2*e^
2 - 8*a*c*x^2*e^2 + 4*b^2*d*x*e - 4*a*c*d*x*e + b^2*d^2 - 4*a*c*d^2 - 6*a*b*x*e^
2 + 2*a*b*d*e - 4*a^2*e^2)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c))