3.2186 \(\int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=344 \[ -\frac{2 e \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 1.62712, antiderivative size = 344, 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 \[ -\frac{2 e \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}+\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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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)**2/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.34651, size = 339, normalized size = 0.99 \[ -\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (b d-a e)-c d^2\right )^3}+\frac{b c \left (3 a e^2-c d (d-2 e x)\right )-2 c^2 \left (a e (2 d-e x)+c d^2 x\right )+b^3 \left (-e^2\right )+b^2 c e (2 d-e x)}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}-\frac{e^3}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}-\frac{2 e^3 (b e-2 c d) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}+\frac{e^3 (b e-2 c d) \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.03, size = 2181, normalized size = 6.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*c^3*d^3*e+12/(a*e^2-b*d*e+c*
d^2)^3/(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^2*c*e^4+2
/(a*e^2-b*d*e+c*d^2)^3*c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^2*d*e^3+4/(
a*e^2-b*d*e+c*d^2)^3*c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*b*e^4+2/(a*e^
2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^4/(4*a*c-b^2)*x*d^4+1/(a*e^2-b*d*e+c*d^2)^3/(c*
x^2+b*x+a)/(4*a*c-b^2)*a*b^3*e^4-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^
2)*b^4*d*e^3+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*d^4*b*c^3-12/(a*e
^2-b*d*e+c*d^2)^3/(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))*e^
4*a^2*c^2-8/(a*e^2-b*d*e+c*d^2)^3*c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*
a*d*e^3+24/(a*e^2-b*d*e+c*d^2)^3/(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^3*e^2*d^2+4/(a*e^2-b*d*e+c*d^2)^3/(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^4*d^4-2*e^4/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*b-e
^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)-1/(a*e^2-b*d*e+c*d^2)^3/(4*a*c-b^2)*ln((4*a*c-b
^2)*(c*x^2+b*x+a))*b^3*e^4+4*e^3/(a*e^2-b*d*e+c*d^2)^3*ln(e*x+d)*c*d-24/(a*e^2-b
*d*e+c*d^2)^3/(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^
2*d*e^3-2/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*a^2*e^4-3/(a*e^2
-b*d*e+c*d^2)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b*c*e^4+4/(a*e^2-b*d*e+c*d^2)^3/(c
*x^2+b*x+a)/(4*a*c-b^2)*a^2*c^2*d*e^3+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c/(4
*a*c-b^2)*x*a*b^2*e^4-1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*b^3*
d*e^3-4/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*x*d^3*e*b+3/(a*e^2-b
*d*e+c*d^2)^3/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*b^2*d^2*e^2-6/(a*e^2-b*d*e+c*d^2)^
3/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b*c^2*d^2*e^2+1/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a
)/(4*a*c-b^2)*a*b^2*c*d*e^3+4/(a*e^2-b*d*e+c*d^2)^3/(64*a^3*c^3-48*a^2*b^2*c^2+1
2*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*c*d*e^3-8/(a*e^2-b*d*e+c*d^2)^3/(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)/(6
4*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b*c^3*d^3*e+3/(a*e^2-b*d*e+c*d^2
)^3/(c*x^2+b*x+a)/(4*a*c-b^2)*b^3*c*d^2*e^2-3/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a
)/(4*a*c-b^2)*b^2*c^2*d^3*e-2/(a*e^2-b*d*e+c*d^2)^3/(64*a^3*c^3-48*a^2*b^2*c^2+1
2*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^4*e^4

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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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 87.1261, 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)^2),x, algorithm="fricas")

[Out]

[((2*a*c^4*d^5 - 4*a*b*c^3*d^4*e + 12*a^2*c^3*d^3*e^2 + 2*(a*b^3*c - 6*a^2*b*c^2
)*d^2*e^3 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d*e^4 + (2*c^5*d^4*e - 4*b*c^4*d^3
*e^2 + 12*a*c^4*d^2*e^3 + 2*(b^3*c^2 - 6*a*b*c^3)*d*e^4 - (b^4*c - 6*a*b^2*c^2 +
6*a^2*c^3)*e^5)*x^3 + (2*c^5*d^5 - 2*b*c^4*d^4*e + 2*b^3*c^2*d^2*e^3 - 4*(b^2*c
^3 - 3*a*c^4)*d^3*e^2 + (b^4*c - 6*a*b^2*c^2 - 6*a^2*c^3)*d*e^4 - (b^5 - 6*a*b^3
*c + 6*a^2*b*c^2)*e^5)*x^2 + (2*b*c^4*d^5 + 8*a*b*c^3*d^3*e^2 - 2*(2*b^2*c^3 - a
*c^4)*d^4*e + 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^2*e^3 - (b^5 - 8*a*b^3*c + 1
8*a^2*b*c^2)*d*e^4 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e^5)*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)) - (b*c^3*d^5 + a^2*b*c*d*e^4 - (3*b^2*c^2 - 4*a*c^3)*d^4*e
+ 3*(b^3*c - 2*a*b*c^2)*d^3*e^2 - (b^4 - 2*a*b^2*c)*d^2*e^3 + (a^2*b^2 - 4*a^3*
c)*e^5 + 2*(c^4*d^4*e - 2*b*c^3*d^3*e^2 + 2*(b^2*c^2 - a*c^3)*d^2*e^3 - (b^3*c -
2*a*b*c^2)*d*e^4 + (a*b^2*c - 3*a^2*c^2)*e^5)*x^2 + (2*c^4*d^5 - 3*b*c^3*d^4*e
+ 4*a*c^3*d^3*e^2 + (3*b^3*c - 10*a*b*c^2)*d^2*e^3 - 2*(b^4 - 3*a*b^2*c - a^2*c^
2)*d*e^4 + (2*a*b^3 - 7*a^2*b*c)*e^5)*x + (2*(a*b^2*c - 4*a^2*c^2)*d^2*e^3 - (a*
b^3 - 4*a^2*b*c)*d*e^4 + (2*(b^2*c^2 - 4*a*c^3)*d*e^4 - (b^3*c - 4*a*b*c^2)*e^5)
*x^3 + (2*(b^2*c^2 - 4*a*c^3)*d^2*e^3 + (b^3*c - 4*a*b*c^2)*d*e^4 - (b^4 - 4*a*b
^2*c)*e^5)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d^2*e^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*
d*e^4 - (a*b^3 - 4*a^2*b*c)*e^5)*x)*log(c*x^2 + b*x + a) - 2*(2*(a*b^2*c - 4*a^2
*c^2)*d^2*e^3 - (a*b^3 - 4*a^2*b*c)*d*e^4 + (2*(b^2*c^2 - 4*a*c^3)*d*e^4 - (b^3*
c - 4*a*b*c^2)*e^5)*x^3 + (2*(b^2*c^2 - 4*a*c^3)*d^2*e^3 + (b^3*c - 4*a*b*c^2)*d
*e^4 - (b^4 - 4*a*b^2*c)*e^5)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d^2*e^3 - (b^4 - 6*a*
b^2*c + 8*a^2*c^2)*d*e^4 - (a*b^3 - 4*a^2*b*c)*e^5)*x)*log(e*x + d))*sqrt(b^2 -
4*a*c))/(((a*b^2*c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2 - 4*a^2*b*c^3)*d^6*e + 3*(a
*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5*e^2 - (a*b^5 + 2*a^2*b^3*c - 24*a^3*b*c^
2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^3*e^4 - 3*(a^3*b^3 - 4*a^4*
b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4 - 4*a*c^5)*d^6*e - 3*(b^3*c
^3 - 4*a*b*c^4)*d^5*e^2 + 3*(b^4*c^2 - 3*a*b^2*c^3 - 4*a^2*c^4)*d^4*e^3 - (b^5*c
+ 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)
*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b^2*c - 4*a^4*c^2)*e^7)*x^3
+ ((b^2*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*d^6*e + 3*(a*b^2*c^3 - 4*a^
2*c^4)*d^5*e^2 + (2*b^5*c - 11*a*b^3*c^2 + 12*a^2*b*c^3)*d^4*e^3 - (b^6 - a*b^4*
c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 - 4*a^2*b^3*c)*d^2*e^5 - (3*
a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3 - 4*a^4*b*c)*e^7)*x^2 + ((b
^3*c^3 - 4*a*b*c^4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 + 4*a^2*c^4)*d^6*e + 3*(b^5*
c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^4*e^3
+ (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 + 3*(a^3*b^2*c - 4*a^4*c^2)*d
^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4*a^5*c)*e^7)*x)*sqrt(b^2 -
4*a*c)), -(2*(2*a*c^4*d^5 - 4*a*b*c^3*d^4*e + 12*a^2*c^3*d^3*e^2 + 2*(a*b^3*c -
6*a^2*b*c^2)*d^2*e^3 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*d*e^4 + (2*c^5*d^4*e -
4*b*c^4*d^3*e^2 + 12*a*c^4*d^2*e^3 + 2*(b^3*c^2 - 6*a*b*c^3)*d*e^4 - (b^4*c - 6*
a*b^2*c^2 + 6*a^2*c^3)*e^5)*x^3 + (2*c^5*d^5 - 2*b*c^4*d^4*e + 2*b^3*c^2*d^2*e^3
- 4*(b^2*c^3 - 3*a*c^4)*d^3*e^2 + (b^4*c - 6*a*b^2*c^2 - 6*a^2*c^3)*d*e^4 - (b^
5 - 6*a*b^3*c + 6*a^2*b*c^2)*e^5)*x^2 + (2*b*c^4*d^5 + 8*a*b*c^3*d^3*e^2 - 2*(2*
b^2*c^3 - a*c^4)*d^4*e + 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*d^2*e^3 - (b^5 - 8*
a*b^3*c + 18*a^2*b*c^2)*d*e^4 - (a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e^5)*x)*arctan
(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (b*c^3*d^5 + a^2*b*c*d*e^4 - (
3*b^2*c^2 - 4*a*c^3)*d^4*e + 3*(b^3*c - 2*a*b*c^2)*d^3*e^2 - (b^4 - 2*a*b^2*c)*d
^2*e^3 + (a^2*b^2 - 4*a^3*c)*e^5 + 2*(c^4*d^4*e - 2*b*c^3*d^3*e^2 + 2*(b^2*c^2 -
a*c^3)*d^2*e^3 - (b^3*c - 2*a*b*c^2)*d*e^4 + (a*b^2*c - 3*a^2*c^2)*e^5)*x^2 + (
2*c^4*d^5 - 3*b*c^3*d^4*e + 4*a*c^3*d^3*e^2 + (3*b^3*c - 10*a*b*c^2)*d^2*e^3 - 2
*(b^4 - 3*a*b^2*c - a^2*c^2)*d*e^4 + (2*a*b^3 - 7*a^2*b*c)*e^5)*x + (2*(a*b^2*c
- 4*a^2*c^2)*d^2*e^3 - (a*b^3 - 4*a^2*b*c)*d*e^4 + (2*(b^2*c^2 - 4*a*c^3)*d*e^4
- (b^3*c - 4*a*b*c^2)*e^5)*x^3 + (2*(b^2*c^2 - 4*a*c^3)*d^2*e^3 + (b^3*c - 4*a*b
*c^2)*d*e^4 - (b^4 - 4*a*b^2*c)*e^5)*x^2 + (2*(b^3*c - 4*a*b*c^2)*d^2*e^3 - (b^4
- 6*a*b^2*c + 8*a^2*c^2)*d*e^4 - (a*b^3 - 4*a^2*b*c)*e^5)*x)*log(c*x^2 + b*x +
a) - 2*(2*(a*b^2*c - 4*a^2*c^2)*d^2*e^3 - (a*b^3 - 4*a^2*b*c)*d*e^4 + (2*(b^2*c^
2 - 4*a*c^3)*d*e^4 - (b^3*c - 4*a*b*c^2)*e^5)*x^3 + (2*(b^2*c^2 - 4*a*c^3)*d^2*e
^3 + (b^3*c - 4*a*b*c^2)*d*e^4 - (b^4 - 4*a*b^2*c)*e^5)*x^2 + (2*(b^3*c - 4*a*b*
c^2)*d^2*e^3 - (b^4 - 6*a*b^2*c + 8*a^2*c^2)*d*e^4 - (a*b^3 - 4*a^2*b*c)*e^5)*x)
*log(e*x + d))*sqrt(-b^2 + 4*a*c))/(((a*b^2*c^3 - 4*a^2*c^4)*d^7 - 3*(a*b^3*c^2
- 4*a^2*b*c^3)*d^6*e + 3*(a*b^4*c - 3*a^2*b^2*c^2 - 4*a^3*c^3)*d^5*e^2 - (a*b^5
+ 2*a^2*b^3*c - 24*a^3*b*c^2)*d^4*e^3 + 3*(a^2*b^4 - 3*a^3*b^2*c - 4*a^4*c^2)*d^
3*e^4 - 3*(a^3*b^3 - 4*a^4*b*c)*d^2*e^5 + (a^4*b^2 - 4*a^5*c)*d*e^6 + ((b^2*c^4
- 4*a*c^5)*d^6*e - 3*(b^3*c^3 - 4*a*b*c^4)*d^5*e^2 + 3*(b^4*c^2 - 3*a*b^2*c^3 -
4*a^2*c^4)*d^4*e^3 - (b^5*c + 2*a*b^3*c^2 - 24*a^2*b*c^3)*d^3*e^4 + 3*(a*b^4*c -
3*a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^5 - 3*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^6 + (a^3*b
^2*c - 4*a^4*c^2)*e^7)*x^3 + ((b^2*c^4 - 4*a*c^5)*d^7 - 2*(b^3*c^3 - 4*a*b*c^4)*
d^6*e + 3*(a*b^2*c^3 - 4*a^2*c^4)*d^5*e^2 + (2*b^5*c - 11*a*b^3*c^2 + 12*a^2*b*c
^3)*d^4*e^3 - (b^6 - a*b^4*c - 15*a^2*b^2*c^2 + 12*a^3*c^3)*d^3*e^4 + 3*(a*b^5 -
4*a^2*b^3*c)*d^2*e^5 - (3*a^2*b^4 - 13*a^3*b^2*c + 4*a^4*c^2)*d*e^6 + (a^3*b^3
- 4*a^4*b*c)*e^7)*x^2 + ((b^3*c^3 - 4*a*b*c^4)*d^7 - (3*b^4*c^2 - 13*a*b^2*c^3 +
4*a^2*c^4)*d^6*e + 3*(b^5*c - 4*a*b^3*c^2)*d^5*e^2 - (b^6 - a*b^4*c - 15*a^2*b^
2*c^2 + 12*a^3*c^3)*d^4*e^3 + (2*a*b^5 - 11*a^2*b^3*c + 12*a^3*b*c^2)*d^3*e^4 +
3*(a^3*b^2*c - 4*a^4*c^2)*d^2*e^5 - 2*(a^3*b^3 - 4*a^4*b*c)*d*e^6 + (a^4*b^2 - 4
*a^5*c)*e^7)*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)**2/(c*x**2+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212435, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done