3.2193 \(\int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=198 \[ -\frac{-x \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-3 b \left (a e^2+c d^2\right )+4 a c d e+2 b^2 d e}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

[Out]

-((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^
2) - (2*b^2*d*e + 4*a*c*d*e - 3*b*(c*d^2 + a*e^2) - (6*c^2*d^2 + b^2*e^2 - 2*c*e
*(3*b*d - a*e))*x)/((b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (2*(6*c^2*d^2 + b^2*e^2
- 2*c*e*(3*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5
/2)

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Rubi [A]  time = 0.484818, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{-x \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )-3 b \left (a e^2+c d^2\right )+4 a c d e+2 b^2 d e}{\left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^
2) - (2*b^2*d*e + 4*a*c*d*e - 3*b*(c*d^2 + a*e^2) - (6*c^2*d^2 + b^2*e^2 - 2*c*e
*(3*b*d - a*e))*x)/((b^2 - 4*a*c)^2*(a + b*x + c*x^2)) - (2*(6*c^2*d^2 + b^2*e^2
- 2*c*e*(3*b*d - a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5
/2)

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Rubi in Sympy [A]  time = 53.4912, size = 202, normalized size = 1.02 \[ \frac{\left (d + e x\right ) \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} + \frac{6 a b e^{2} - 8 a c d e - 4 b^{2} d e + 6 b c d^{2} + x \left (4 a c e^{2} + 2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{2 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{4 \left (a c e^{2} + \frac{b^{2} e^{2}}{2} - 3 b c d e + 3 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)*(2*a*e - b*d + x*(b*e - 2*c*d))/(2*(-4*a*c + b**2)*(a + b*x + c*x**2)*
*2) + (6*a*b*e**2 - 8*a*c*d*e - 4*b**2*d*e + 6*b*c*d**2 + x*(4*a*c*e**2 + 2*b**2
*e**2 - 12*b*c*d*e + 12*c**2*d**2))/(2*(-4*a*c + b**2)**2*(a + b*x + c*x**2)) -
4*(a*c*e**2 + b**2*e**2/2 - 3*b*c*d*e + 3*c**2*d**2)*atanh((b + 2*c*x)/sqrt(-4*a
*c + b**2))/(-4*a*c + b**2)**(5/2)

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Mathematica [A]  time = 0.544925, size = 203, normalized size = 1.03 \[ \frac{1}{2} \left (\frac{(b+2 c x) \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x}{c \left (4 a c-b^2\right ) (a+x (b+c x))^2}+\frac{4 \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(((6*c^2*d^2 + b^2*e^2 + 2*c*e*(-3*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^2*(
a + x*(b + c*x))) + (a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a
*c*e*(2*d + e*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))^2) + (4*(6*c^2*d^2 + b^2*e
^2 + 2*c*e*(-3*b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c
)^(5/2))/2

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Maple [B]  time = 0.016, size = 508, normalized size = 2.6 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ({\frac{c \left ( 2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2} \right ){x}^{3}}{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}}+{\frac{3\,b \left ( 2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2} \right ){x}^{2}}{32\,{a}^{2}{c}^{2}-16\,ac{b}^{2}+2\,{b}^{4}}}-{\frac{ \left ( 2\,{a}^{2}c{e}^{2}-5\,a{b}^{2}{e}^{2}+10\,abcde-10\,a{c}^{2}{d}^{2}+2\,{b}^{3}de-2\,c{b}^{2}{d}^{2} \right ) x}{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}}+{\frac{6\,{e}^{2}{a}^{2}b-16\,{a}^{2}cde-2\,a{b}^{2}de+10\,abc{d}^{2}-{b}^{3}{d}^{2}}{32\,{a}^{2}{c}^{2}-16\,ac{b}^{2}+2\,{b}^{4}}} \right ) }+4\,{\frac{ac{e}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{b}^{2}{e}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-12\,{\frac{bcde}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{c}^{2}{d}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*(2*a*c*e^2+b^2*e^2-6*b*c*d*e+6*c^2*d^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+3/2*b*
(2*a*c*e^2+b^2*e^2-6*b*c*d*e+6*c^2*d^2)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-(2*a^2*c*
e^2-5*a*b^2*e^2+10*a*b*c*d*e-10*a*c^2*d^2+2*b^3*d*e-2*b^2*c*d^2)/(16*a^2*c^2-8*a
*b^2*c+b^4)*x+1/2*(6*a^2*b*e^2-16*a^2*c*d*e-2*a*b^2*d*e+10*a*b*c*d^2-b^3*d^2)/(1
6*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^
2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c*e^2+2/(16*a^2*c^2-8*a*b^2*c+b^4
)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*e^2-12/(16*a^2*c^2-8
*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*c*d*e+12/(
16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*
c^2*d^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((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 = 0.235309, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(2*(6*a^2*c^2*d^2 - 6*a^2*b*c*d*e + (6*c^4*d^2 - 6*b*c^3*d*e + (b^2*c^2 + 2
*a*c^3)*e^2)*x^4 + 2*(6*b*c^3*d^2 - 6*b^2*c^2*d*e + (b^3*c + 2*a*b*c^2)*e^2)*x^3
+ (a^2*b^2 + 2*a^3*c)*e^2 + (6*(b^2*c^2 + 2*a*c^3)*d^2 - 6*(b^3*c + 2*a*b*c^2)*
d*e + (b^4 + 4*a*b^2*c + 4*a^2*c^2)*e^2)*x^2 + 2*(6*a*b*c^2*d^2 - 6*a*b^2*c*d*e
+ (a*b^3 + 2*a^2*b*c)*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)) + (6*a^2*b
*e^2 + 2*(6*c^3*d^2 - 6*b*c^2*d*e + (b^2*c + 2*a*c^2)*e^2)*x^3 - (b^3 - 10*a*b*c
)*d^2 - 2*(a*b^2 + 8*a^2*c)*d*e + 3*(6*b*c^2*d^2 - 6*b^2*c*d*e + (b^3 + 2*a*b*c)
*e^2)*x^2 + 2*(2*(b^2*c + 5*a*c^2)*d^2 - 2*(b^3 + 5*a*b*c)*d*e + (5*a*b^2 - 2*a^
2*c)*e^2)*x)*sqrt(b^2 - 4*a*c))/((a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2
- 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (
b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)*sq
rt(b^2 - 4*a*c)), 1/2*(4*(6*a^2*c^2*d^2 - 6*a^2*b*c*d*e + (6*c^4*d^2 - 6*b*c^3*d
*e + (b^2*c^2 + 2*a*c^3)*e^2)*x^4 + 2*(6*b*c^3*d^2 - 6*b^2*c^2*d*e + (b^3*c + 2*
a*b*c^2)*e^2)*x^3 + (a^2*b^2 + 2*a^3*c)*e^2 + (6*(b^2*c^2 + 2*a*c^3)*d^2 - 6*(b^
3*c + 2*a*b*c^2)*d*e + (b^4 + 4*a*b^2*c + 4*a^2*c^2)*e^2)*x^2 + 2*(6*a*b*c^2*d^2
- 6*a*b^2*c*d*e + (a*b^3 + 2*a^2*b*c)*e^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x
+ b)/(b^2 - 4*a*c)) + (6*a^2*b*e^2 + 2*(6*c^3*d^2 - 6*b*c^2*d*e + (b^2*c + 2*a*
c^2)*e^2)*x^3 - (b^3 - 10*a*b*c)*d^2 - 2*(a*b^2 + 8*a^2*c)*d*e + 3*(6*b*c^2*d^2
- 6*b^2*c*d*e + (b^3 + 2*a*b*c)*e^2)*x^2 + 2*(2*(b^2*c + 5*a*c^2)*d^2 - 2*(b^3 +
5*a*b*c)*d*e + (5*a*b^2 - 2*a^2*c)*e^2)*x)*sqrt(-b^2 + 4*a*c))/((a^2*b^4 - 8*a^
3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^4 + 2*(b^5*c - 8*a
*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8
*a^2*b^3*c + 16*a^3*b*c^2)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 14.0238, size = 1052, normalized size = 5.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)*l
og(x + (-64*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c
*d*e + 6*c**2*d**2) + 48*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 +
b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) - 12*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5)*(
2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + 2*a*b*c*e**2 + b**6*sqrt(-1/
(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + b**3*e**
2 - 6*b**2*c*d*e + 6*b*c**2*d**2)/(4*a*c**2*e**2 + 2*b**2*c*e**2 - 12*b*c**2*d*e
+ 12*c**3*d**2)) + sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d
*e + 6*c**2*d**2)*log(x + (64*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 +
b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) - 48*a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)
**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + 12*a*b**4*c*sqrt(-1/(4
*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2) + 2*a*b*c*e*
*2 - b**6*sqrt(-1/(4*a*c - b**2)**5)*(2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**
2*d**2) + b**3*e**2 - 6*b**2*c*d*e + 6*b*c**2*d**2)/(4*a*c**2*e**2 + 2*b**2*c*e*
*2 - 12*b*c**2*d*e + 12*c**3*d**2)) + (6*a**2*b*e**2 - 16*a**2*c*d*e - 2*a*b**2*
d*e + 10*a*b*c*d**2 - b**3*d**2 + x**3*(4*a*c**2*e**2 + 2*b**2*c*e**2 - 12*b*c**
2*d*e + 12*c**3*d**2) + x**2*(6*a*b*c*e**2 + 3*b**3*e**2 - 18*b**2*c*d*e + 18*b*
c**2*d**2) + x*(-4*a**2*c*e**2 + 10*a*b**2*e**2 - 20*a*b*c*d*e + 20*a*c**2*d**2
- 4*b**3*d*e + 4*b**2*c*d**2))/(32*a**4*c**2 - 16*a**3*b**2*c + 2*a**2*b**4 + x*
*4*(32*a**2*c**4 - 16*a*b**2*c**3 + 2*b**4*c**2) + x**3*(64*a**2*b*c**3 - 32*a*b
**3*c**2 + 4*b**5*c) + x**2*(64*a**3*c**3 - 12*a*b**4*c + 2*b**6) + x*(64*a**3*b
*c**2 - 32*a**2*b**3*c + 4*a*b**5))

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GIAC/XCAS [A]  time = 0.206811, size = 414, normalized size = 2.09 \[ \frac{2 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{3} d^{2} x^{3} - 12 \, b c^{2} d x^{3} e + 18 \, b c^{2} d^{2} x^{2} + 2 \, b^{2} c x^{3} e^{2} + 4 \, a c^{2} x^{3} e^{2} - 18 \, b^{2} c d x^{2} e + 4 \, b^{2} c d^{2} x + 20 \, a c^{2} d^{2} x + 3 \, b^{3} x^{2} e^{2} + 6 \, a b c x^{2} e^{2} - 4 \, b^{3} d x e - 20 \, a b c d x e - b^{3} d^{2} + 10 \, a b c d^{2} + 10 \, a b^{2} x e^{2} - 4 \, a^{2} c x e^{2} - 2 \, a b^{2} d e - 16 \, a^{2} c d e + 6 \, a^{2} b e^{2}}{2 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(6*c^2*d^2 - 6*b*c*d*e + b^2*e^2 + 2*a*c*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4
*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*sqrt(-b^2 + 4*a*c)) + 1/2*(12*c^3*d^2*x^3
- 12*b*c^2*d*x^3*e + 18*b*c^2*d^2*x^2 + 2*b^2*c*x^3*e^2 + 4*a*c^2*x^3*e^2 - 18*
b^2*c*d*x^2*e + 4*b^2*c*d^2*x + 20*a*c^2*d^2*x + 3*b^3*x^2*e^2 + 6*a*b*c*x^2*e^2
- 4*b^3*d*x*e - 20*a*b*c*d*x*e - b^3*d^2 + 10*a*b*c*d^2 + 10*a*b^2*x*e^2 - 4*a^
2*c*x*e^2 - 2*a*b^2*d*e - 16*a^2*c*d*e + 6*a^2*b*e^2)/((b^4 - 8*a*b^2*c + 16*a^2
*c^2)*(c*x^2 + b*x + a)^2)