3.2196 \(\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=429 \[ -\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^5 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[Out]

-(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2)*(a + b*x + c*x^2)^2) - (3*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c
*e)*(6*c^2*d^2 - 2*b^2*e^2 - c*e*(3*b*d - 8*a*e)) - 2*c*(2*c*d - b*e)*(3*c^2*d^2
- b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^
2*(a + b*x + c*x^2)) - ((12*c^5*d^5 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^
5 - 10*c^4*d^3*e*(3*b*d - 4*a*e) + 20*c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2
))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e +
a*e^2)^3) + (e^5*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - (e^5*Log[a + b*x + c
*x^2])/(2*(c*d^2 - b*d*e + a*e^2)^3)

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Rubi [A]  time = 2.35853, antiderivative size = 429, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{-2 c x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (3 b d-8 a e)-2 b^2 e^2+6 c^2 d^2\right )+3 a c e (2 c d-b e)^2}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^5 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^5 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(2*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2)*(a + b*x + c*x^2)^2) - (3*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c
*e)*(6*c^2*d^2 - 2*b^2*e^2 - c*e*(3*b*d - 8*a*e)) - 2*c*(2*c*d - b*e)*(3*c^2*d^2
- b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/(2*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)^
2*(a + b*x + c*x^2)) - ((12*c^5*d^5 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^
5 - 10*c^4*d^3*e*(3*b*d - 4*a*e) + 20*c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2
))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e +
a*e^2)^3) + (e^5*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - (e^5*Log[a + b*x + c
*x^2])/(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)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.595, size = 429, normalized size = 1. \[ \frac{1}{2} \left (\frac{2 \left (-20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )+30 a^2 b c^2 e^5-10 a b^3 c e^5+10 c^4 d^3 e (3 b d-4 a e)+b^5 e^5-12 c^5 d^5\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2} \left (e (b d-a e)-c d^2\right )^3}+\frac{4 c^2 \left (4 a^2 e^3+7 a c d e^2 x+3 c^2 d^3 x\right )+b^2 c e \left (c d (2 e x-9 d)-15 a e^2\right )+2 b c^2 \left (7 a e^2 (d-e x)+3 c d^2 (d-3 e x)\right )+2 b^4 e^3+b^3 c e^2 (d+2 e x)}{\left (b^2-4 a c\right )^2 (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}+\frac{2 c (a e+c d x)+b^2 (-e)+b c (d-e x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^2 \left (e (b d-a e)-c d^2\right )}+\frac{2 e^5 \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}-\frac{e^5 \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + b*x + c*x^2)^3),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) + (2*b^4*e^3 + b^3*c*e^2*(d + 2*e*x) + 4*c^2*(4*a
^2*e^3 + 3*c^2*d^3*x + 7*a*c*d*e^2*x) + 2*b*c^2*(3*c*d^2*(d - 3*e*x) + 7*a*e^2*(
d - e*x)) + b^2*c*e*(-15*a*e^2 + c*d*(-9*d + 2*e*x)))/((b^2 - 4*a*c)^2*(c*d^2 +
e*(-(b*d) + a*e))^2*(a + x*(b + c*x))) + (2*(-12*c^5*d^5 + b^5*e^5 - 10*a*b^3*c*
e^5 + 30*a^2*b*c^2*e^5 + 10*c^4*d^3*e*(3*b*d - 4*a*e) - 20*c^3*d*e^2*(b^2*d^2 -
3*a*b*d*e + 3*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^
(5/2)*(-(c*d^2) + e*(b*d - a*e))^3) + (2*e^5*Log[d + e*x])/(c*d^2 + e*(-(b*d) +
a*e))^3 - (e^5*Log[a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^3)/2

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Maple [B]  time = 0.034, size = 5715, normalized size = 13.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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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)^3*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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)^3*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

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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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218215, 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)^3*(e*x + d)),x, algorithm="giac")

[Out]

Done