3.2181 \(\int \frac{(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=173 \[ \frac{(2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{e^2 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^3 \log \left (a+b x+c x^2\right )}{2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.622919, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{(2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{e^2 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{(d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^3 \log \left (a+b x+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(2*c*d - b*e)*x)/(c*(b^2 - 4*a*c)) - ((d + e*x)^2*(b*d - 2*a*e + (2*c*d - b
*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + ((2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2
- 2*c*e*(b*d - 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*(b^2 - 4*a*c
)^(3/2)) + (e^3*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. \[ - \frac{e^{2} \left (b e - 2 c d\right ) \int \frac{1}{c}\, dx}{- 4 a c + b^{2}} + \frac{\left (d + e x\right )^{2} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} + \frac{e^{3} \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} + \frac{\left (b e - 2 c d\right ) \left (- 6 a c e^{2} + b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**2*(b*e - 2*c*d)*Integral(1/c, x)/(-4*a*c + b**2) + (d + e*x)**2*(2*a*e - b*d
+ x*(b*e - 2*c*d))/((-4*a*c + b**2)*(a + b*x + c*x**2)) + e**3*log(a + b*x + c*
x**2)/(2*c**2) + (b*e - 2*c*d)*(-6*a*c*e**2 + b**2*e**2 + 2*b*c*d*e - 2*c**2*d**
2)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(c**2*(-4*a*c + b**2)**(3/2))

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Mathematica [A]  time = 0.674243, size = 201, normalized size = 1.16 \[ \frac{\frac{2 \left (-2 c \left (a^2 e^3-3 a c d e (d+e x)+c^2 d^3 x\right )+b^2 e^2 (a e-3 c d x)-b c \left (3 a e^2 (d+e x)+c d^2 (d-3 e x)\right )+b^3 e^3 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{2 (b e-2 c d) \left (2 c e (b d-3 a e)+b^2 e^2-2 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+e^3 \log (a+x (b+c x))}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.029, size = 831, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((3*a*b*c*e^3-6*a*c^2*d*e^2-b^3*e^3+3*b^2*c*d*e^2-3*b*c^2*d^2*e+2*c^3*d^3)/c^2/(
4*a*c-b^2)*x+(2*a^2*c*e^3-a*b^2*e^3+3*a*b*c*d*e^2-6*a*c^2*d^2*e+b*c^2*d^3)/c^2/(
4*a*c-b^2))/(c*x^2+b*x+a)+2/c/(4*a*c-b^2)*ln(c*(4*a*c-b^2)*(c*x^2+b*x+a))*a*e^3-
1/2/c^2/(4*a*c-b^2)*ln(c*(4*a*c-b^2)*(c*x^2+b*x+a))*b^2*e^3-6/(64*a^3*c^5-48*a^2
*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x+b*c*(4*a*c-b^2)
)/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*a*b*e^3+12/(64*a^3*c^5
-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x+b*c*(4*a
*c-b^2))/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*a*d*e^2*c-6/(64
*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x+
b*c*(4*a*c-b^2))/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*b*c*d^2
*e+4/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c
-b^2)*x+b*c*(4*a*c-b^2))/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))
*c^2*d^3+1/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*
(4*a*c-b^2)*x+b*c*(4*a*c-b^2))/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^
(1/2))*b^3/c*e^3

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22313, 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)^3/(c*x^2 + b*x + a)^2,x, algorithm="fricas")

[Out]

[1/2*((4*a*c^3*d^3 - 6*a*b*c^2*d^2*e + 12*a^2*c^2*d*e^2 + (a*b^3 - 6*a^2*b*c)*e^
3 + (4*c^4*d^3 - 6*b*c^3*d^2*e + 12*a*c^3*d*e^2 + (b^3*c - 6*a*b*c^2)*e^3)*x^2 +
(4*b*c^3*d^3 - 6*b^2*c^2*d^2*e + 12*a*b*c^2*d*e^2 + (b^4 - 6*a*b^2*c)*e^3)*x)*l
og((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*b*c^2*d^3 - 12*a*c^2*d^2*e + 6*a*b*c*
d*e^2 - 2*(a*b^2 - 2*a^2*c)*e^3 + 2*(2*c^3*d^3 - 3*b*c^2*d^2*e + 3*(b^2*c - 2*a*
c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3)*x - ((b^2*c - 4*a*c^2)*e^3*x^2 + (b^3 - 4*a*b*
c)*e^3*x + (a*b^2 - 4*a^2*c)*e^3)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((a*b
^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^3*c^2 - 4*a*b*c^3)*x)*sqrt(b^2
- 4*a*c)), -1/2*(2*(4*a*c^3*d^3 - 6*a*b*c^2*d^2*e + 12*a^2*c^2*d*e^2 + (a*b^3 -
6*a^2*b*c)*e^3 + (4*c^4*d^3 - 6*b*c^3*d^2*e + 12*a*c^3*d*e^2 + (b^3*c - 6*a*b*c
^2)*e^3)*x^2 + (4*b*c^3*d^3 - 6*b^2*c^2*d^2*e + 12*a*b*c^2*d*e^2 + (b^4 - 6*a*b^
2*c)*e^3)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*b*c^2*d^
3 - 12*a*c^2*d^2*e + 6*a*b*c*d*e^2 - 2*(a*b^2 - 2*a^2*c)*e^3 + 2*(2*c^3*d^3 - 3*
b*c^2*d^2*e + 3*(b^2*c - 2*a*c^2)*d*e^2 - (b^3 - 3*a*b*c)*e^3)*x - ((b^2*c - 4*a
*c^2)*e^3*x^2 + (b^3 - 4*a*b*c)*e^3*x + (a*b^2 - 4*a^2*c)*e^3)*log(c*x^2 + b*x +
a))*sqrt(-b^2 + 4*a*c))/((a*b^2*c^2 - 4*a^2*c^3 + (b^2*c^3 - 4*a*c^4)*x^2 + (b^
3*c^2 - 4*a*b*c^3)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 19.0585, size = 1238, normalized size = 7.16 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2
- 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4
*c - b**6)))*log(x + (-16*a**2*c**3*(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(b
*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*
c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 8*a**2*c*e**3 + 8*a*b**2*c**2
*(e**3/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2
- 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**
4*c - b**6))) - a*b**2*e**3 - 6*a*b*c*d*e**2 - b**4*c*(e**3/(2*c**2) - sqrt(-(4*
a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)
/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 3*b**2*c*d*
*2*e - 2*b*c**2*d**3)/(6*a*b*c*e**3 - 12*a*c**2*d*e**2 - b**3*e**3 + 6*b*c**2*d*
*2*e - 4*c**3*d**3)) + (e**3/(2*c**2) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(
6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**
2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-16*a**2*c**3*(e**3/(2*c**2) + sqrt
(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*
d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 8*a**2
*c*e**3 + 8*a*b**2*c**2*(e**3/(2*c**2) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*
(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a*
*2*b**2*c**2 + 12*a*b**4*c - b**6))) - a*b**2*e**3 - 6*a*b*c*d*e**2 - b**4*c*(e*
*3/(2*c**2) + sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2
*b*c*d*e + 2*c**2*d**2)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6))) + 3*b**2*c*d**2*e - 2*b*c**2*d**3)/(6*a*b*c*e**3 - 12*a*c**2*d*e**2 -
b**3*e**3 + 6*b*c**2*d**2*e - 4*c**3*d**3)) + (2*a**2*c*e**3 - a*b**2*e**3 + 3*a
*b*c*d*e**2 - 6*a*c**2*d**2*e + b*c**2*d**3 + x*(3*a*b*c*e**3 - 6*a*c**2*d*e**2
- b**3*e**3 + 3*b**2*c*d*e**2 - 3*b*c**2*d**2*e + 2*c**3*d**3))/(4*a**2*c**3 - a
*b**2*c**2 + x**2*(4*a*c**4 - b**2*c**3) + x*(4*a*b*c**3 - b**3*c**2))

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GIAC/XCAS [A]  time = 0.20719, size = 319, normalized size = 1.84 \[ -\frac{{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 12 \, a c^{2} d e^{2} + b^{3} e^{3} - 6 \, a b c e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{e^{3}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac{b c^{2} d^{3} - 6 \, a c^{2} d^{2} e + 3 \, a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3} +{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} x}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(4*c^3*d^3 - 6*b*c^2*d^2*e + 12*a*c^2*d*e^2 + b^3*e^3 - 6*a*b*c*e^3)*arctan((2*
c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^2 - 4*a*c^3)*sqrt(-b^2 + 4*a*c)) + 1/2*e^3*
ln(c*x^2 + b*x + a)/c^2 - (b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - a*b^2*e^3
+ 2*a^2*c*e^3 + (2*c^3*d^3 - 3*b*c^2*d^2*e + 3*b^2*c*d*e^2 - 6*a*c^2*d*e^2 - b^
3*e^3 + 3*a*b*c*e^3)*x)/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^2)