3.2185 \(\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=224 \[ \frac{(2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{e^3 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac{e^3 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.895128, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{(2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{e^3 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^2}+\frac{e^3 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + b*x + c*x^2)^2),x]

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.689707, size = 223, normalized size = 1. \[ \frac{(b e-2 c d) \left (2 c e (b d-3 a e)+b^2 e^2-2 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (a e-b d)+c d^2\right )^2}+\frac{2 c (a e+c d x)+b^2 (-e)+b c (d-e x)}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )}+\frac{e^3 \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^2}-\frac{e^3 \log (a+x (b+c x))}{2 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + b*x + c*x^2)^2),x]

[Out]

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

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Maple [B]  time = 0.034, size = 1384, normalized size = 6.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+b*x+a)^2,x)

[Out]

e^3*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^2-1/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)*c/(4*a
*c-b^2)*x*a*b*e^3+2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*a*d*e^
2+1/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*b^2*d*e^2-3/(a*e^2-b*d*e
+c*d^2)^2/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*b*d^2*e+2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2
+b*x+a)*c^3/(4*a*c-b^2)*x*d^3+2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)/(4*a*c-b^2)*
a^2*c*e^3-1/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^2*e^3-1/(a*e^2-b
*d*e+c*d^2)^2/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b*c*d*e^2+2/(a*e^2-b*d*e+c*d^2)^2/(c*x
^2+b*x+a)/(4*a*c-b^2)*a*c^2*d^2*e+1/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)/(4*a*c-b
^2)*b^3*d*e^2-2/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*c*d^2*e+1/(a
*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)/(4*a*c-b^2)*d^3*b*c^2-2/(a*e^2-b*d*e+c*d^2)^2*
c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*e^3+1/2/(a*e^2-b*d*e+c*d^2)^2/(4*a
*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^2*e^3-6/(a*e^2-b*d*e+c*d^2)^2/(64*a^3*c^
3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/
(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b*c*e^3+12/(a*e^2-b*d*e+c*d^
2)^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+
(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*c^2*e^2*d+1/(
a*e^2-b*d*e+c*d^2)^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*
c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))
*b^3*e^3-6/(a*e^2-b*d*e+c*d^2)^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2
)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c
-b^6)^(1/2))*b*c^2*d^2*e+4/(a*e^2-b*d*e+c*d^2)^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a
*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^
2*c^2+12*a*b^4*c-b^6)^(1/2))*c^3*d^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(1/((c*x^2 + b*x + a)^2*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 25.3282, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^2*(e*x + d)),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 - 4*(b^2*c - a*c^2)*d^2*e +
2*(b^3 - 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
- a*b*c*e^3 + (b^2*c + 2*a*c^2)*d*e^2)*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) - 2*((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(e*x + d))*sqrt(b
^2 - 4*a*c))/(((a*b^2*c^2 - 4*a^2*c^3)*d^4 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^3*e + (
a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d*e^3 + (a^3*
b^2 - 4*a^4*c)*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4 - 2*(b^3*c^2 - 4*a*b*c^3)*d^3*e +
(b^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^2 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a
^2*b^2*c - 4*a^3*c^2)*e^4)*x^2 + ((b^3*c^2 - 4*a*b*c^3)*d^4 - 2*(b^4*c - 4*a*b^2
*c^2)*d^3*e + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*e^2 - 2*(a*b^4 - 4*a^2*b^2*c)*
d*e^3 + (a^2*b^3 - 4*a^3*b*c)*e^4)*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 - 4*(b^2*c - a*c^2)*d^2*e + 2*(b
^3 - 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 - a*b
*c*e^3 + (b^2*c + 2*a*c^2)*d*e^2)*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) - 2*((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(e*x + d))*sqrt(-b^2 +
4*a*c))/(((a*b^2*c^2 - 4*a^2*c^3)*d^4 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^3*e + (a*b^
4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d*e^3 + (a^3*b^2
- 4*a^4*c)*e^4 + ((b^2*c^3 - 4*a*c^4)*d^4 - 2*(b^3*c^2 - 4*a*b*c^3)*d^3*e + (b^4
*c - 2*a*b^2*c^2 - 8*a^2*c^3)*d^2*e^2 - 2*(a*b^3*c - 4*a^2*b*c^2)*d*e^3 + (a^2*b
^2*c - 4*a^3*c^2)*e^4)*x^2 + ((b^3*c^2 - 4*a*b*c^3)*d^4 - 2*(b^4*c - 4*a*b^2*c^2
)*d^3*e + (b^5 - 2*a*b^3*c - 8*a^2*b*c^2)*d^2*e^2 - 2*(a*b^4 - 4*a^2*b^2*c)*d*e^
3 + (a^2*b^3 - 4*a^3*b*c)*e^4)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.210821, size = 635, normalized size = 2.83 \[ -\frac{e^{3}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} + \frac{e^{4}{\rm ln}\left ({\left | x e + d \right |}\right )}{c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}} - \frac{{\left (4 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 12 \, a c^{2} d e^{2} + b^{3} e^{3} - 6 \, a b c e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{b c^{2} d^{3} - 2 \, b^{2} c d^{2} e + 2 \, a c^{2} d^{2} e + b^{3} d e^{2} - a b c d e^{2} - a b^{2} e^{3} + 2 \, a^{2} c e^{3} +{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{2} c d e^{2} + 2 \, a c^{2} d e^{2} - a b c e^{3}\right )} x}{{\left (c d^{2} - b d e + a e^{2}\right )}^{2}{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^2*(e*x + d)),x, algorithm="giac")

[Out]

-1/2*e^3*ln(c*x^2 + b*x + a)/(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^
2 - 2*a*b*d*e^3 + a^2*e^4) + e^4*ln(abs(x*e + d))/(c^2*d^4*e - 2*b*c*d^3*e^2 + b
^2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5) - (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*d^4 - 4*a*c^3*d^4 - 2*b^3*c*d^3*e + 8*a*b*c^2*d^3*e + b^4*d^2*e^2 -
2*a*b^2*c*d^2*e^2 - 8*a^2*c^2*d^2*e^2 - 2*a*b^3*d*e^3 + 8*a^2*b*c*d*e^3 + a^2*b^
2*e^4 - 4*a^3*c*e^4)*sqrt(-b^2 + 4*a*c)) - (b*c^2*d^3 - 2*b^2*c*d^2*e + 2*a*c^2*
d^2*e + b^3*d*e^2 - 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 + b^2*c*d*e^2 + 2*a*c^2*d*e^2 - a*b*c*e^3)*x)/((c*d^2 - b*d*e + a*e^2)^2*
(c*x^2 + b*x + a)*(b^2 - 4*a*c))