3.2208 \(\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=771 \[ \frac{\left (-140 a^3 b c^3 e^7+70 a^2 b^3 c^2 e^7+28 c^5 d^3 e^2 \left (10 a^2 e^2-15 a b d e+6 b^2 d^2\right )-70 c^4 d e^3 \left (-4 a^3 e^3+6 a^2 b d e^2-4 a b^2 d^2 e+b^3 d^3\right )-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} \left (a e^2-b d e+c d^2\right )^4}+\frac{-64 a^3 c^3 e^5+2 b^2 c^2 e \left (43 a^2 e^4+48 a c d^2 e^2+25 c^2 d^4\right )-2 c x (2 c d-b e) \left (c^2 e^2 \left (38 a^2 e^2-32 a b d e+7 b^2 d^2\right )+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+b^4 e^4+10 c^4 d^4\right )-4 b c^3 d \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (5 a e^2+17 c d^2\right )+2 b^6 e^5+b^5 c d e^4}{2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{-c x (2 c d-b e) \left (-2 c e (5 b d-11 a e)-3 b^2 e^2+10 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (5 b d-12 a e)-3 b^2 e^2+10 c^2 d^2\right )+5 a c e (2 c d-b e)^2}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}-\frac{e^7 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^4} \]

[Out]

-(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2)*(a + b*x + c*x^2)^3) - (5*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c
*e)*(10*c^2*d^2 - 3*b^2*e^2 - c*e*(5*b*d - 12*a*e)) - c*(2*c*d - b*e)*(10*c^2*d^
2 - 3*b^2*e^2 - 2*c*e*(5*b*d - 11*a*e))*x)/(6*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a
*e^2)^2*(a + b*x + c*x^2)^2) + (b^5*c*d*e^4 + 2*b^6*e^5 - 64*a^3*c^3*e^5 + b^4*c
*e^3*(c*d^2 - 23*a*e^2) - 2*b^3*c^2*d*e^2*(17*c*d^2 + 5*a*e^2) - 4*b*c^3*d*(5*c^
2*d^4 + 16*a*c*d^2*e^2 + 19*a^2*e^4) + 2*b^2*c^2*e*(25*c^2*d^4 + 48*a*c*d^2*e^2
+ 43*a^2*e^4) - 2*c*(2*c*d - b*e)*(10*c^4*d^4 + b^4*e^4 + b^2*c*e^3*(3*b*d - 11*
a*e) - 4*c^3*d^2*e*(5*b*d - 8*a*e) + c^2*e^2*(7*b^2*d^2 - 32*a*b*d*e + 38*a^2*e^
2))*x)/(2*(b^2 - 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)) + ((40*c^
7*d^7 + b^7*e^7 - 14*a*b^5*c*e^7 + 70*a^2*b^3*c^2*e^7 - 140*a^3*b*c^3*e^7 - 28*c
^6*d^5*e*(5*b*d - 6*a*e) + 28*c^5*d^3*e^2*(6*b^2*d^2 - 15*a*b*d*e + 10*a^2*e^2)
- 70*c^4*d*e^3*(b^3*d^3 - 4*a*b^2*d^2*e + 6*a^2*b*d*e^2 - 4*a^3*e^3))*ArcTanh[(b
+ 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(7/2)*(c*d^2 - b*d*e + a*e^2)^4) +
(e^7*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 - (e^7*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 = 14.476, antiderivative size = 771, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{\left (-140 a^3 b c^3 e^7+70 a^2 b^3 c^2 e^7+28 c^5 d^3 e^2 \left (10 a^2 e^2-15 a b d e+6 b^2 d^2\right )-70 c^4 d e^3 \left (-4 a^3 e^3+6 a^2 b d e^2-4 a b^2 d^2 e+b^3 d^3\right )-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2} \left (c d^2-e (b d-a e)\right )^4}+\frac{-64 a^3 c^3 e^5+2 b^2 c^2 e \left (43 a^2 e^4+48 a c d^2 e^2+25 c^2 d^4\right )-2 c x (2 c d-b e) \left (c^2 e^2 \left (38 a^2 e^2-32 a b d e+7 b^2 d^2\right )+b^2 c e^3 (3 b d-11 a e)-4 c^3 d^2 e (5 b d-8 a e)+b^4 e^4+10 c^4 d^4\right )-4 b c^3 d \left (19 a^2 e^4+16 a c d^2 e^2+5 c^2 d^4\right )+b^4 c e^3 \left (c d^2-23 a e^2\right )-2 b^3 c^2 d e^2 \left (5 a e^2+17 c d^2\right )+2 b^6 e^5+b^5 c d e^4}{2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^3}-\frac{-c x (2 c d-b e) \left (-2 c e (5 b d-11 a e)-3 b^2 e^2+10 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-c e (5 b d-12 a e)-3 b^2 e^2+10 c^2 d^2\right )+5 a c e (2 c d-b e)^2}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )^2}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3 \left (a e^2-b d e+c d^2\right )}-\frac{e^7 \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^4}+\frac{e^7 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + b*x + c*x^2)^4),x]

[Out]

-(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e +
a*e^2)*(a + b*x + c*x^2)^3) - (5*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c
*e)*(10*c^2*d^2 - 3*b^2*e^2 - c*e*(5*b*d - 12*a*e)) - c*(2*c*d - b*e)*(10*c^2*d^
2 - 3*b^2*e^2 - 2*c*e*(5*b*d - 11*a*e))*x)/(6*(b^2 - 4*a*c)^2*(c*d^2 - b*d*e + a
*e^2)^2*(a + b*x + c*x^2)^2) + (b^5*c*d*e^4 + 2*b^6*e^5 - 64*a^3*c^3*e^5 + b^4*c
*e^3*(c*d^2 - 23*a*e^2) - 2*b^3*c^2*d*e^2*(17*c*d^2 + 5*a*e^2) - 4*b*c^3*d*(5*c^
2*d^4 + 16*a*c*d^2*e^2 + 19*a^2*e^4) + 2*b^2*c^2*e*(25*c^2*d^4 + 48*a*c*d^2*e^2
+ 43*a^2*e^4) - 2*c*(2*c*d - b*e)*(10*c^4*d^4 + b^4*e^4 + b^2*c*e^3*(3*b*d - 11*
a*e) - 4*c^3*d^2*e*(5*b*d - 8*a*e) + c^2*e^2*(7*b^2*d^2 - 32*a*b*d*e + 38*a^2*e^
2))*x)/(2*(b^2 - 4*a*c)^3*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2)) + ((40*c^
7*d^7 + b^7*e^7 - 14*a*b^5*c*e^7 + 70*a^2*b^3*c^2*e^7 - 140*a^3*b*c^3*e^7 - 28*c
^6*d^5*e*(5*b*d - 6*a*e) + 28*c^5*d^3*e^2*(6*b^2*d^2 - 15*a*b*d*e + 10*a^2*e^2)
- 70*c^4*d*e^3*(b^3*d^3 - 4*a*b^2*d^2*e + 6*a^2*b*d*e^2 - 4*a^3*e^3))*ArcTanh[(b
+ 2*c*x)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(7/2)*(c*d^2 - e*(b*d - a*e))^4) +
(e^7*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^4 - (e^7*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)/(c*x**2+b*x+a)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 6.64657, size = 769, normalized size = 1. \[ \frac{1}{6} \left (\frac{4 c^2 \left (6 a^2 e^3+11 a c d e^2 x+5 c^2 d^3 x\right )+b^2 c e \left (c d (4 e x-15 d)-23 a e^2\right )+2 b c^2 \left (11 a e^2 (d-e x)+5 c d^2 (d-3 e x)\right )+3 b^4 e^3+b^3 c e^2 (2 d+3 e x)}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2 \left (e (a e-b d)+c d^2\right )^2}+\frac{6 \left (-140 a^3 b c^3 e^7+70 a^2 b^3 c^2 e^7+28 c^5 d^3 e^2 \left (10 a^2 e^2-15 a b d e+6 b^2 d^2\right )-70 c^4 d e^3 \left (-4 a^3 e^3+6 a^2 b d e^2-4 a b^2 d^2 e+b^3 d^3\right )-14 a b^5 c e^7-28 c^6 d^5 e (5 b d-6 a e)+b^7 e^7+40 c^7 d^7\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2} \left (e (a e-b d)+c d^2\right )^4}+\frac{3 \left (2 b^2 c^2 e \left (-43 a^2 e^4+2 a c d e^2 (5 e x-24 d)+c^2 d^3 (34 e x-25 d)\right )+4 b c^3 \left (19 a^2 e^4 (d-e x)+16 a c d^2 e^2 (d-3 e x)+5 c^2 d^4 (d-5 e x)\right )+8 c^3 \left (8 a^3 e^5+19 a^2 c d e^4 x+16 a c^2 d^3 e^2 x+5 c^3 d^5 x\right )-b^4 c e^3 \left (c d (d+2 e x)-23 a e^2\right )+2 b^3 c^2 e^2 \left (a e^2 (5 d+11 e x)+c d^2 (17 d-e x)\right )-2 b^6 e^5-b^5 c e^4 (d+2 e x)\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x)) \left (e (b d-a e)-c d^2\right )^3}+\frac{4 c (a e+c d x)-2 b^2 e+2 b c (d-e x)}{\left (b^2-4 a c\right ) (a+x (b+c x))^3 \left (e (b d-a e)-c d^2\right )}-\frac{3 e^7 \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^4}+\frac{6 e^7 \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + b*x + c*x^2)^4),x]

[Out]

((-2*b^2*e + 4*c*(a*e + c*d*x) + 2*b*c*(d - e*x))/((b^2 - 4*a*c)*(-(c*d^2) + e*(
b*d - a*e))*(a + x*(b + c*x))^3) + (3*b^4*e^3 + b^3*c*e^2*(2*d + 3*e*x) + 4*c^2*
(6*a^2*e^3 + 5*c^2*d^3*x + 11*a*c*d*e^2*x) + 2*b*c^2*(5*c*d^2*(d - 3*e*x) + 11*a
*e^2*(d - e*x)) + b^2*c*e*(-23*a*e^2 + c*d*(-15*d + 4*e*x)))/((b^2 - 4*a*c)^2*(c
*d^2 + e*(-(b*d) + a*e))^2*(a + x*(b + c*x))^2) + (3*(-2*b^6*e^5 - b^5*c*e^4*(d
+ 2*e*x) + 8*c^3*(8*a^3*e^5 + 5*c^3*d^5*x + 16*a*c^2*d^3*e^2*x + 19*a^2*c*d*e^4*
x) + 4*b*c^3*(5*c^2*d^4*(d - 5*e*x) + 16*a*c*d^2*e^2*(d - 3*e*x) + 19*a^2*e^4*(d
- e*x)) - b^4*c*e^3*(-23*a*e^2 + c*d*(d + 2*e*x)) + 2*b^3*c^2*e^2*(c*d^2*(17*d
- e*x) + a*e^2*(5*d + 11*e*x)) + 2*b^2*c^2*e*(-43*a^2*e^4 + 2*a*c*d*e^2*(-24*d +
5*e*x) + c^2*d^3*(-25*d + 34*e*x))))/((b^2 - 4*a*c)^3*(-(c*d^2) + e*(b*d - a*e)
)^3*(a + x*(b + c*x))) + (6*(40*c^7*d^7 + b^7*e^7 - 14*a*b^5*c*e^7 + 70*a^2*b^3*
c^2*e^7 - 140*a^3*b*c^3*e^7 - 28*c^6*d^5*e*(5*b*d - 6*a*e) + 28*c^5*d^3*e^2*(6*b
^2*d^2 - 15*a*b*d*e + 10*a^2*e^2) - 70*c^4*d*e^3*(b^3*d^3 - 4*a*b^2*d^2*e + 6*a^
2*b*d*e^2 - 4*a^3*e^3))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((-b^2 + 4*a*c)^
(7/2)*(c*d^2 + e*(-(b*d) + a*e))^4) + (6*e^7*Log[d + e*x])/(c*d^2 - b*d*e + a*e^
2)^4 - (3*e^7*Log[a + x*(b + c*x)])/(c*d^2 + e*(-(b*d) + a*e))^4)/6

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Maple [B]  time = 0.054, size = 16810, normalized size = 21.8 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+b*x+a)^4,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)^4*(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)^4*(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)**4,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.240563, 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)^4*(e*x + d)),x, algorithm="giac")

[Out]

Done