3.2187 \(\int \frac{1}{(d+e x)^3 \left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=485 \[ \frac{(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^4}-\frac{e (2 c d-b e) \left (-c e (11 a e+b d)+3 b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac{e^3 \log (d+e x) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^4}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

[Out]

-(e*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e)))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d
*e + a*e^2)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(c^2*d^2 + 3*b^2*e^2 - c*e*(b*d +
11*a*e)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(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)^2
*(a + b*x + c*x^2)) + ((2*c*d - b*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d -
5*a*e) + 4*b^2*c*e^3*(b*d + 5*a*e) - 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e
^2))*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)^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[d + e*x])
/(c*d^2 - b*d*e + a*e^2)^4 - (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))
*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^4)

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Rubi [A]  time = 2.92136, antiderivative size = 485, 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 \[ \frac{(2 c d-b e) \left (-2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )+4 b^2 c e^3 (5 a e+b d)-4 c^3 d^2 e (b d-5 a e)-3 b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^4}-\frac{e (2 c d-b e) \left (-c e (11 a e+b d)+3 b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{e \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{2 \left (b^2-4 a c\right ) (d+e x)^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac{e^3 \log (d+e x) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right )}{\left (a e^2-b d e+c d^2\right )^4}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x)^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^3*(a + b*x + c*x^2)^2),x]

[Out]

-(e*(4*c^2*d^2 + 3*b^2*e^2 - 4*c*e*(b*d + 2*a*e)))/(2*(b^2 - 4*a*c)*(c*d^2 - b*d
*e + a*e^2)^2*(d + e*x)^2) - (e*(2*c*d - b*e)*(c^2*d^2 + 3*b^2*e^2 - c*e*(b*d +
11*a*e)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(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)^2
*(a + b*x + c*x^2)) + ((2*c*d - b*e)*(2*c^4*d^4 - 3*b^4*e^4 - 4*c^3*d^2*e*(b*d -
5*a*e) + 4*b^2*c*e^3*(b*d + 5*a*e) - 2*c^2*e^2*(b^2*d^2 + 10*a*b*d*e + 15*a^2*e
^2))*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)^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[d + e*x])
/(c*d^2 - b*d*e + a*e^2)^4 - (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(5*b*d + a*e))
*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^4)

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

[Out]

Timed out

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Mathematica [A]  time = 2.31467, size = 489, normalized size = 1.01 \[ \frac{(b e-2 c d) \left (2 c^2 e^2 \left (15 a^2 e^2+10 a b d e+b^2 d^2\right )-4 b^2 c e^3 (5 a e+b d)+4 c^3 d^2 e (b d-5 a e)+3 b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (a e-b d)+c d^2\right )^4}+\frac{2 c^2 \left (-a^2 e^3+3 a c d e (d-e x)+c^2 d^3 x\right )+b^2 c e \left (4 a e^2-3 c d (d-e x)\right )+b c^2 \left (3 a e^2 (e x-3 d)+c d^2 (d-3 e x)\right )+b^4 \left (-e^3\right )+b^3 c e^2 (3 d-e x)}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )^3}+\frac{e^3 \log (d+e x) \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^4}-\frac{e^3 \left (-2 c e (a e+5 b d)+3 b^2 e^2+10 c^2 d^2\right ) \log (a+x (b+c x))}{2 \left (e (a e-b d)+c d^2\right )^4}+\frac{2 e^3 (b e-2 c d)}{(d+e x) \left (e (a e-b d)+c d^2\right )^3}-\frac{e^3}{2 (d+e x)^2 \left (e (a e-b d)+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^3*(a + b*x + c*x^2)^2),x]

[Out]

-e^3/(2*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)^2) + (2*e^3*(-2*c*d + b*e))/((c*d
^2 + e*(-(b*d) + a*e))^3*(d + e*x)) + (-(b^4*e^3) + b^3*c*e^2*(3*d - e*x) + b^2*
c*e*(4*a*e^2 - 3*c*d*(d - e*x)) + 2*c^2*(-(a^2*e^3) + c^2*d^3*x + 3*a*c*d*e*(d -
e*x)) + b*c^2*(c*d^2*(d - 3*e*x) + 3*a*e^2*(-3*d + e*x)))/((b^2 - 4*a*c)*(-(c*d
^2) + e*(b*d - a*e))^3*(a + x*(b + c*x))) + ((-2*c*d + b*e)*(-2*c^4*d^4 + 3*b^4*
e^4 + 4*c^3*d^2*e*(b*d - 5*a*e) - 4*b^2*c*e^3*(b*d + 5*a*e) + 2*c^2*e^2*(b^2*d^2
+ 10*a*b*d*e + 15*a^2*e^2))*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))^4) + (e^3*(10*c^2*d^2 + 3*b^2*e^2 - 2*c*e*
(5*b*d + a*e))*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^4 - (e^3*(10*c^2*d^2 + 3
*b^2*e^2 - 2*c*e*(5*b*d + a*e))*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*
e))^4)

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Maple [B]  time = 0.036, size = 3346, normalized size = 6.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^3/(c*x^2+b*x+a)^2,x)

[Out]

6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^3*c^2*d^3*e^2-4/(a*e^2-b*d*e
+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^2*c^3*d^4*e+10/(a*e^2-b*d*e+c*d^2)^4/(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*c^2*d^2*e^3-10/(a*e^2
-b*d*e+c*d^2)^4/(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^
4*d^4*e-7/(a*e^2-b*d*e+c*d^2)^4*c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*b^
2*e^5-20/(a*e^2-b*d*e+c*d^2)^4*c^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*d
^2*e^3-5/(a*e^2-b*d*e+c*d^2)^4*c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^3*d
*e^4+5/(a*e^2-b*d*e+c*d^2)^4*c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^2*d
^2*e^3+30/(a*e^2-b*d*e+c*d^2)^4/(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^2*b*c^2*e^5-60/(a*e^2-b*d*e+c*d^2)^4/(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^3*d-20/(a*e^2-b*d*e+c*d^2)^4/(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^3*c*e^5+40/(a*e^2-b*d*e+c*
d^2)^4/(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^4*d^3*e^2
-10/(a*e^2-b*d*e+c*d^2)^4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arcta
n((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*c*d*e^4+2*e^4/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)*b+3*e^5/(a*e^2-b*d*e+c*d^2
)^4*ln(e*x+d)*b^2+3/2/(a*e^2-b*d*e+c*d^2)^4/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*
x+a))*b^4*e^5-4*e^3/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)*c*d-2*e^5/(a*e^2-b*d*e+c*d^2)^
4*ln(e*x+d)*a*c+10*e^3/(a*e^2-b*d*e+c*d^2)^4*ln(e*x+d)*c^2*d^2+2/(a*e^2-b*d*e+c*
d^2)^4/(c*x^2+b*x+a)*c^5/(4*a*c-b^2)*x*d^5-2/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)
/(4*a*c-b^2)*a^3*c^2*e^5-1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^4
*e^5+1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^5*d*e^4+3/(a*e^2-b*d*e+
c*d^2)^4/(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^5*e^5+4/(
a*e^2-b*d*e+c*d^2)^4/(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^5*d^5+10/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^2*c^2*d^2*e^3-1/
(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c/(4*a*c-b^2)*x*a*b^3*e^5+4/(a*e^2-b*d*e+c*d
^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b^2*c*e^5+4/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x
+a)/(4*a*c-b^2)*a^2*c^3*d^2*e^3+6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2
)*a*c^4*d^4*e-4/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b^4*c*d^2*e^3+1/
(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*b*c^4*d^5+4/(a*e^2-b*d*e+c*d^2)^
4*c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a^2*e^5-10*e^4/(a*e^2-b*d*e+c*d^
2)^4*ln(e*x+d)*b*c*d+6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*x*d^2
*a*b*e^3-1/2*e^3/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^2+1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+
b*x+a)*c/(4*a*c-b^2)*x*b^4*d*e^4-6/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^3/(4*a*
c-b^2)*x*a^2*d*e^4-14/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b*c^3*d^
3*e^2+3/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*a^2*b*e^5+60/(a*e^
2-b*d*e+c*d^2)^4/(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^2*d*e^4-1/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a*b^3*c*d*e^4-4/(
a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^2/(4*a*c-b^2)*x*b^3*d^2*e^3-60/(a*e^2-b*d*e
+c*d^2)^4/(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^3*d^
2*e^3+20/(a*e^2-b*d*e+c*d^2)^4*c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*b
*d*e^4-4/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)*c^4/(4*a*c-b^2)*x*d^3*a*e^2-7/(a*e^
2-b*d*e+c*d^2)^4/(c*x^2+b*x+a)/(4*a*c-b^2)*a^2*b*c^2*d*e^4+6/(a*e^2-b*d*e+c*d^2)
^4/(c*x^2+b*x+a)*c^3/(4*a*c-b^2)*x*b^2*d^3*e^2-5/(a*e^2-b*d*e+c*d^2)^4/(c*x^2+b*
x+a)*c^4/(4*a*c-b^2)*x*d^4*b*e

_______________________________________________________________________________________

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)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214021, 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)^3),x, algorithm="giac")

[Out]

Done