3.2375 \(\int \frac{f+g x}{(d+e x) \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=666 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (d g+5 e f)+b^2 d^2 (4 d g+5 e f)\right )-6 c^2 e^2 \left (-2 a^3 e^3 g+a^2 b e^2 (d g+5 e f)-6 a b^2 d^2 e g+2 b^3 d^3 g\right )+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (d g+5 e f))+b^5 \left (-e^4\right ) (e f-d g)+12 c^5 d^5 f\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{c x \left (3 c e (b d-2 a e) (2 a e g-b (d g+e f)+2 c d f)-(2 c d-b e) \left (c (2 a e (4 e f-d g)-3 b d (d g+e f))-2 b^2 e (e f-d g)+6 c^2 d^2 f\right )\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (c (2 a e (4 e f-d g)-3 b d (d g+e f))-2 b^2 e (e f-d g)+6 c^2 d^2 f\right )+3 a c e (2 c d-b e) (2 a e g-b (d g+e f)+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^4 (e f-d g) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

[Out]

-(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b
*(e*f + d*g))*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*c*d*f + 2*a*e*g - b*(e*f + d*g)) - (b*c*d - b^2*e + 2
*a*c*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g) + c*(2*a*e*(4*e*f - d*g) - 3*b*d*(e*f
+ d*g))) + c*(3*c*e*(b*d - 2*a*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g)) - (2*c*d
- b*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g) + c*(2*a*e*(4*e*f - d*g) - 3*b*d*(e*f
+ d*g))))*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*f - b^5*e^4*(e*f - d*g) + 10*a*b^3*c*e^4*(e*f - d*g) + 2*c^4*d^3*(2*
a*e*(10*e*f - d*g) - 3*b*d*(5*e*f + d*g)) - 6*c^2*e^2*(2*b^3*d^3*g - 6*a*b^2*d^2
*e*g - 2*a^3*e^3*g + a^2*b*e^2*(5*e*f + d*g)) + 4*c^3*d*e*(3*a^2*e^2*(5*e*f - 2*
d*g) - 3*a*b*d*e*(5*e*f + d*g) + b^2*d^2*(5*e*f + 4*d*g)))*ArcTanh[(b + 2*c*x)/S
qrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e + a*e^2)^3) + (e^4*(e*f -
d*g)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - (e^4*(e*f - d*g)*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 = 5.8403, antiderivative size = 664, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (4 c^3 d e \left (3 a^2 e^2 (5 e f-2 d g)-3 a b d e (d g+5 e f)+b^2 d^2 (4 d g+5 e f)\right )-6 c^2 e^2 \left (-2 a^3 e^3 g+a^2 b e^2 (d g+5 e f)-6 a b^2 d^2 e g+2 b^3 d^3 g\right )+10 a b^3 c e^4 (e f-d g)+2 c^4 d^3 (2 a e (10 e f-d g)-3 b d (d g+5 e f))+b^5 \left (-e^4\right ) (e f-d g)+12 c^5 d^5 f\right )}{\left (b^2-4 a c\right )^{5/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{c x \left (3 c e (b d-2 a e) (2 a e g-b (d g+e f)+2 c d f)-(2 c d-b e) \left (2 a c e (4 e f-d g)-2 b^2 e (e f-d g)-3 b c d (d g+e f)+6 c^2 d^2 f\right )\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (2 a c e (4 e f-d g)-2 b^2 e (e f-d g)-3 b c d (d g+e f)+6 c^2 d^2 f\right )+3 a c e (2 c d-b e) (2 a e g-b (d g+e f)+2 c d f)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac{c x (2 a e g-b (d g+e f)+2 c d f)+a b e g-2 a c d g+2 a c e f+b^2 (-e) f+b c d f}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2 \left (a e^2-b d e+c d^2\right )}-\frac{e^4 (e f-d g) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )^3}+\frac{e^4 (e f-d g) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + c*(2*c*d*f + 2*a*e*g - b
*(e*f + d*g))*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*c*d*f + 2*a*e*g - b*(e*f + d*g)) - (b*c*d - b^2*e + 2
*a*c*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g) + 2*a*c*e*(4*e*f - d*g) - 3*b*c*d*(e*
f + d*g)) + c*(3*c*e*(b*d - 2*a*e)*(2*c*d*f + 2*a*e*g - b*(e*f + d*g)) - (2*c*d
- b*e)*(6*c^2*d^2*f - 2*b^2*e*(e*f - d*g) + 2*a*c*e*(4*e*f - d*g) - 3*b*c*d*(e*f
+ d*g)))*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*f - b^5*e^4*(e*f - d*g) + 10*a*b^3*c*e^4*(e*f - d*g) + 2*c^4*d^3*(2*
a*e*(10*e*f - d*g) - 3*b*d*(5*e*f + d*g)) - 6*c^2*e^2*(2*b^3*d^3*g - 6*a*b^2*d^2
*e*g - 2*a^3*e^3*g + a^2*b*e^2*(5*e*f + d*g)) + 4*c^3*d*e*(3*a^2*e^2*(5*e*f - 2*
d*g) - 3*a*b*d*e*(5*e*f + d*g) + b^2*d^2*(5*e*f + 4*d*g)))*ArcTanh[(b + 2*c*x)/S
qrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*(c*d^2 - b*d*e + a*e^2)^3) + (e^4*(e*f -
d*g)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2)^3 - (e^4*(e*f - d*g)*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((g*x+f)/(e*x+d)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 6.89347, size = 857, normalized size = 1.29 \[ \frac{-e f b^2+c d f b+a e g b-c e f x b-c d g x b+2 a c e f-2 a c d g+2 c^2 d f x+2 a c e g x}{2 \left (4 a c-b^2\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^2}+\frac{\left (e^5 f b^5-d e^4 g b^5-10 a c e^5 f b^3+10 a c d e^4 g b^3+12 c^2 d^3 e^2 g b^3-20 c^3 d^3 e^2 f b^2-36 a c^2 d^2 e^3 g b^2-16 c^3 d^4 e g b^2+30 a^2 c^2 e^5 f b+60 a c^3 d^2 e^3 f b+30 c^4 d^4 e f b+6 c^4 d^5 g b+6 a^2 c^2 d e^4 g b+12 a c^3 d^3 e^2 g b-12 c^5 d^5 f-60 a^2 c^3 d e^4 f-40 a c^4 d^3 e^2 f-12 a^3 c^2 e^5 g+24 a^2 c^3 d^2 e^3 g+4 a c^4 d^4 e g\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (b^2-4 a c\right )^2 \sqrt{4 a c-b^2} \left (-c d^2+b e d-a e^2\right )^3}+\frac{\left (e^5 f-d e^4 g\right ) \log (d+e x)}{\left (c d^2-b e d+a e^2\right )^3}+\frac{\left (d e^4 g-e^5 f\right ) \log \left (c x^2+b x+a\right )}{2 \left (c d^2-b e d+a e^2\right )^3}+\frac{2 e^3 f b^4-2 d e^2 g b^4+c d e^2 f b^3+5 c d^2 e g b^3+2 c e^3 f x b^3-2 c d e^2 g x b^3-15 a c e^3 f b^2-9 c^2 d^2 e f b^2-3 c^2 d^3 g b^2+3 a c d e^2 g b^2+2 c^2 d e^2 f x b^2+10 c^2 d^2 e g x b^2+6 c^3 d^3 f b+14 a c^2 d e^2 f b+6 a^2 c e^3 g b-2 a c^2 d^2 e g b-14 a c^2 e^3 f x b-18 c^3 d^2 e f x b-6 c^3 d^3 g x b-10 a c^2 d e^2 g x b+16 a^2 c^2 e^3 f-16 a^2 c^2 d e^2 g+12 c^4 d^3 f x+28 a c^3 d e^2 f x+12 a^2 c^2 e^3 g x-4 a c^3 d^2 e g x}{2 \left (4 a c-b^2\right )^2 \left (c d^2-b e d+a e^2\right )^2 \left (c x^2+b x+a\right )} \]

Antiderivative was successfully verified.

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

[Out]

(b*c*d*f - b^2*e*f + 2*a*c*e*f - 2*a*c*d*g + a*b*e*g + 2*c^2*d*f*x - b*c*e*f*x -
b*c*d*g*x + 2*a*c*e*g*x)/(2*(-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c
*x^2)^2) + (6*b*c^3*d^3*f - 9*b^2*c^2*d^2*e*f + b^3*c*d*e^2*f + 14*a*b*c^2*d*e^2
*f + 2*b^4*e^3*f - 15*a*b^2*c*e^3*f + 16*a^2*c^2*e^3*f - 3*b^2*c^2*d^3*g + 5*b^3
*c*d^2*e*g - 2*a*b*c^2*d^2*e*g - 2*b^4*d*e^2*g + 3*a*b^2*c*d*e^2*g - 16*a^2*c^2*
d*e^2*g + 6*a^2*b*c*e^3*g + 12*c^4*d^3*f*x - 18*b*c^3*d^2*e*f*x + 2*b^2*c^2*d*e^
2*f*x + 28*a*c^3*d*e^2*f*x + 2*b^3*c*e^3*f*x - 14*a*b*c^2*e^3*f*x - 6*b*c^3*d^3*
g*x + 10*b^2*c^2*d^2*e*g*x - 4*a*c^3*d^2*e*g*x - 2*b^3*c*d*e^2*g*x - 10*a*b*c^2*
d*e^2*g*x + 12*a^2*c^2*e^3*g*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*f + 30*b*c^4*d^4*e*f - 20*b^2*c^3*d^3*e^2*f - 4
0*a*c^4*d^3*e^2*f + 60*a*b*c^3*d^2*e^3*f - 60*a^2*c^3*d*e^4*f + b^5*e^5*f - 10*a
*b^3*c*e^5*f + 30*a^2*b*c^2*e^5*f + 6*b*c^4*d^5*g - 16*b^2*c^3*d^4*e*g + 4*a*c^4
*d^4*e*g + 12*b^3*c^2*d^3*e^2*g + 12*a*b*c^3*d^3*e^2*g - 36*a*b^2*c^2*d^2*e^3*g
+ 24*a^2*c^3*d^2*e^3*g - b^5*d*e^4*g + 10*a*b^3*c*d*e^4*g + 6*a^2*b*c^2*d*e^4*g
- 12*a^3*c^2*e^5*g)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/((b^2 - 4*a*c)^2*Sqr
t[-b^2 + 4*a*c]*(-(c*d^2) + b*d*e - a*e^2)^3) + ((e^5*f - d*e^4*g)*Log[d + e*x])
/(c*d^2 - b*d*e + a*e^2)^3 + ((-(e^5*f) + d*e^4*g)*Log[a + b*x + c*x^2])/(2*(c*d
^2 - b*d*e + a*e^2)^3)

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Maple [B]  time = 0.069, size = 11600, normalized size = 17.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.478383, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)/((c*x^2 + b*x + a)^3*(e*x + d)),x, algorithm="giac")

[Out]

Done