3.2370 \(\int \frac{(d+e x)^4 (f+g x)}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=543 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (4 c^3 e \left (3 a^2 e^2 (4 d g+e f)-3 a b d e (3 d g+2 e f)+b^2 d^2 (2 d g+3 e f)\right )-30 a^2 b c^2 e^4 g+10 a b^3 c e^4 g-2 c^4 d^2 (3 b d (d g+4 e f)-4 a e (2 d g+3 e f))+b^5 \left (-e^4\right ) g+12 c^5 d^4 f\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{(d+e x) \left (x \left (-c^2 e \left (16 a^2 e^2 g+2 a b e (11 d g+3 e f)+b^2 (-d) (5 d g+6 e f)\right )+b^2 c e^2 g (15 a e+b d)-2 c^3 d (3 b d (d g+3 e f)-2 a e (4 d g+3 e f))-2 b^4 e^3 g+12 c^4 d^3 f\right )+2 b c \left (7 a^2 e^3 g+a c d e (7 d g+9 e f)+3 c^2 d^3 f\right )+b^3 e g \left (c d^2-2 a e^2\right )-b^2 c d \left (a e^2 g+3 c d (d g+2 e f)\right )-4 a c^2 e \left (a e (8 d g+3 e f)+3 c d^2 f\right )\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{(d+e x)^3 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{e^4 g \log \left (a+b x+c x^2\right )}{2 c^3} \]

[Out]

((d + e*x)^3*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c*(
b*e*f + b*d*g + 2*a*e*g))*x))/(2*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + ((d + e*
x)*(b^3*e*(c*d^2 - 2*a*e^2)*g - b^2*c*d*(a*e^2*g + 3*c*d*(2*e*f + d*g)) + 2*b*c*
(3*c^2*d^3*f + 7*a^2*e^3*g + a*c*d*e*(9*e*f + 7*d*g)) - 4*a*c^2*e*(3*c*d^2*f + a
*e*(3*e*f + 8*d*g)) + (12*c^4*d^3*f - 2*b^4*e^3*g + b^2*c*e^2*(b*d + 15*a*e)*g -
2*c^3*d*(3*b*d*(3*e*f + d*g) - 2*a*e*(3*e*f + 4*d*g)) - c^2*e*(16*a^2*e^2*g - b
^2*d*(6*e*f + 5*d*g) + 2*a*b*e*(3*e*f + 11*d*g)))*x))/(2*c^2*(b^2 - 4*a*c)^2*(a
+ b*x + c*x^2)) - ((12*c^5*d^4*f - b^5*e^4*g + 10*a*b^3*c*e^4*g - 30*a^2*b*c^2*e
^4*g - 2*c^4*d^2*(3*b*d*(4*e*f + d*g) - 4*a*e*(3*e*f + 2*d*g)) + 4*c^3*e*(b^2*d^
2*(3*e*f + 2*d*g) - 3*a*b*d*e*(2*e*f + 3*d*g) + 3*a^2*e^2*(e*f + 4*d*g)))*ArcTan
h[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(5/2)) + (e^4*g*Log[a + b*x
+ c*x^2])/(2*c^3)

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Rubi [A]  time = 2.72666, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (4 c^3 e \left (3 a^2 e^2 (4 d g+e f)-3 a b d e (3 d g+2 e f)+b^2 d^2 (2 d g+3 e f)\right )-30 a^2 b c^2 e^4 g+10 a b^3 c e^4 g-2 c^4 d^2 (3 b d (d g+4 e f)-4 a e (2 d g+3 e f))+b^5 \left (-e^4\right ) g+12 c^5 d^4 f\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac{(d+e x) \left (x \left (-c^2 e \left (16 a^2 e^2 g+2 a b e (11 d g+3 e f)+b^2 (-d) (5 d g+6 e f)\right )+b^2 c e^2 g (15 a e+b d)-2 c^3 d (3 b d (d g+3 e f)-2 a e (4 d g+3 e f))-2 b^4 e^3 g+12 c^4 d^3 f\right )+2 b c \left (7 a^2 e^3 g+a c d e (7 d g+9 e f)+3 c^2 d^3 f\right )+b^3 e g \left (c d^2-2 a e^2\right )-b^2 c d \left (a e^2 g+3 c d (d g+2 e f)\right )-4 a c^2 e \left (a e (8 d g+3 e f)+3 c d^2 f\right )\right )}{2 c^2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{(d+e x)^3 \left (-x \left (-c (2 a e g+b d g+b e f)+b^2 e g+2 c^2 d f\right )-b (a e g+c d f)+2 a c (d g+e f)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{e^4 g \log \left (a+b x+c x^2\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^4*(f + g*x))/(a + b*x + c*x^2)^3,x]

[Out]

((d + e*x)^3*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c*(
b*e*f + b*d*g + 2*a*e*g))*x))/(2*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + ((d + e*
x)*(b^3*e*(c*d^2 - 2*a*e^2)*g - b^2*c*d*(a*e^2*g + 3*c*d*(2*e*f + d*g)) + 2*b*c*
(3*c^2*d^3*f + 7*a^2*e^3*g + a*c*d*e*(9*e*f + 7*d*g)) - 4*a*c^2*e*(3*c*d^2*f + a
*e*(3*e*f + 8*d*g)) + (12*c^4*d^3*f - 2*b^4*e^3*g + b^2*c*e^2*(b*d + 15*a*e)*g -
2*c^3*d*(3*b*d*(3*e*f + d*g) - 2*a*e*(3*e*f + 4*d*g)) - c^2*e*(16*a^2*e^2*g - b
^2*d*(6*e*f + 5*d*g) + 2*a*b*e*(3*e*f + 11*d*g)))*x))/(2*c^2*(b^2 - 4*a*c)^2*(a
+ b*x + c*x^2)) - ((12*c^5*d^4*f - b^5*e^4*g + 10*a*b^3*c*e^4*g - 30*a^2*b*c^2*e
^4*g - 2*c^4*d^2*(3*b*d*(4*e*f + d*g) - 4*a*e*(3*e*f + 2*d*g)) + 4*c^3*e*(b^2*d^
2*(3*e*f + 2*d*g) - 3*a*b*d*e*(2*e*f + 3*d*g) + 3*a^2*e^2*(e*f + 4*d*g)))*ArcTan
h[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(5/2)) + (e^4*g*Log[a + b*x
+ c*x^2])/(2*c^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((e*x+d)**4*(g*x+f)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Mathematica [B]  time = 6.45031, size = 1298, normalized size = 2.39 \[ \frac{g \log \left (c x^2+b x+a\right ) e^4}{2 c^3}+\frac{\left (-e^4 g b^5+10 a c e^4 g b^3+12 c^3 d^2 e^2 f b^2+8 c^3 d^3 e g b^2-24 a c^3 d e^3 f b-24 c^4 d^3 e f b-6 c^4 d^4 g b-30 a^2 c^2 e^4 g b-36 a c^3 d^2 e^2 g b+12 c^5 d^4 f+12 a^2 c^3 e^4 f+24 a c^4 d^2 e^2 f+48 a^2 c^3 d e^3 g+16 a c^4 d^3 e g\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{c^3 \left (b^2-4 a c\right )^2 \sqrt{4 a c-b^2}}+\frac{-e^4 g b^6+c e^4 f b^5+4 c d e^3 g b^5+4 c e^4 g x b^5-4 c^2 d e^3 f b^4+11 a c e^4 g b^4-6 c^2 d^2 e^2 g b^4-2 c^2 e^4 f x b^4-8 c^2 d e^3 g x b^4-8 a c^2 e^4 f b^3+6 c^3 d^2 e^2 f b^3-32 a c^2 d e^3 g b^3+4 c^3 d^3 e g b^3-30 a c^2 e^4 g x b^3+20 a c^3 d e^3 f b^2-12 c^4 d^3 e f b^2-3 c^4 d^4 g b^2-39 a^2 c^2 e^4 g b^2+30 a c^3 d^2 e^2 g b^2+16 a c^3 e^4 f x b^2+12 c^4 d^2 e^2 f x b^2+64 a c^3 d e^3 g x b^2+8 c^4 d^3 e g x b^2+6 c^5 d^4 f b+22 a^2 c^3 e^4 f b+12 a c^4 d^2 e^2 f b+88 a^2 c^3 d e^3 g b+8 a c^4 d^3 e g b-24 a c^4 d e^3 f x b-24 c^5 d^3 e f x b-6 c^5 d^4 g x b+50 a^2 c^3 e^4 g x b-36 a c^4 d^2 e^2 g x b-64 a^2 c^4 d e^3 f+32 a^3 c^3 e^4 g-96 a^2 c^4 d^2 e^2 g+12 c^6 d^4 f x-20 a^2 c^4 e^4 f x+24 a c^5 d^2 e^2 f x-80 a^2 c^4 d e^3 g x+16 a c^5 d^3 e g x}{2 c^4 \left (4 a c-b^2\right )^2 \left (c x^2+b x+a\right )}+\frac{-e^4 g x b^5-a e^4 g b^4+c e^4 f x b^4+4 c d e^3 g x b^4+a c e^4 f b^3+4 a c d e^3 g b^3-4 c^2 d e^3 f x b^3+5 a c e^4 g x b^3-6 c^2 d^2 e^2 g x b^3-4 a c^2 d e^3 f b^2+4 a^2 c e^4 g b^2-6 a c^2 d^2 e^2 g b^2-4 a c^2 e^4 f x b^2+6 c^3 d^2 e^2 f x b^2-16 a c^2 d e^3 g x b^2+4 c^3 d^3 e g x b^2+c^4 d^4 f b-3 a^2 c^2 e^4 f b+6 a c^3 d^2 e^2 f b-12 a^2 c^2 d e^3 g b+4 a c^3 d^3 e g b+12 a c^3 d e^3 f x b-4 c^4 d^3 e f x b-c^4 d^4 g x b-5 a^2 c^2 e^4 g x b+18 a c^3 d^2 e^2 g x b+8 a^2 c^3 d e^3 f-8 a c^4 d^3 e f-2 a c^4 d^4 g-2 a^3 c^2 e^4 g+12 a^2 c^3 d^2 e^2 g+2 c^5 d^4 f x+2 a^2 c^3 e^4 f x-12 a c^4 d^2 e^2 f x+8 a^2 c^3 d e^3 g x-8 a c^4 d^3 e g x}{2 c^4 \left (4 a c-b^2\right ) \left (c x^2+b x+a\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^4*(f + g*x))/(a + b*x + c*x^2)^3,x]

[Out]

(b*c^4*d^4*f - 8*a*c^4*d^3*e*f + 6*a*b*c^3*d^2*e^2*f - 4*a*b^2*c^2*d*e^3*f + 8*a
^2*c^3*d*e^3*f + a*b^3*c*e^4*f - 3*a^2*b*c^2*e^4*f - 2*a*c^4*d^4*g + 4*a*b*c^3*d
^3*e*g - 6*a*b^2*c^2*d^2*e^2*g + 12*a^2*c^3*d^2*e^2*g + 4*a*b^3*c*d*e^3*g - 12*a
^2*b*c^2*d*e^3*g - a*b^4*e^4*g + 4*a^2*b^2*c*e^4*g - 2*a^3*c^2*e^4*g + 2*c^5*d^4
*f*x - 4*b*c^4*d^3*e*f*x + 6*b^2*c^3*d^2*e^2*f*x - 12*a*c^4*d^2*e^2*f*x - 4*b^3*
c^2*d*e^3*f*x + 12*a*b*c^3*d*e^3*f*x + b^4*c*e^4*f*x - 4*a*b^2*c^2*e^4*f*x + 2*a
^2*c^3*e^4*f*x - b*c^4*d^4*g*x + 4*b^2*c^3*d^3*e*g*x - 8*a*c^4*d^3*e*g*x - 6*b^3
*c^2*d^2*e^2*g*x + 18*a*b*c^3*d^2*e^2*g*x + 4*b^4*c*d*e^3*g*x - 16*a*b^2*c^2*d*e
^3*g*x + 8*a^2*c^3*d*e^3*g*x - b^5*e^4*g*x + 5*a*b^3*c*e^4*g*x - 5*a^2*b*c^2*e^4
*g*x)/(2*c^4*(-b^2 + 4*a*c)*(a + b*x + c*x^2)^2) + (6*b*c^5*d^4*f - 12*b^2*c^4*d
^3*e*f + 6*b^3*c^3*d^2*e^2*f + 12*a*b*c^4*d^2*e^2*f - 4*b^4*c^2*d*e^3*f + 20*a*b
^2*c^3*d*e^3*f - 64*a^2*c^4*d*e^3*f + b^5*c*e^4*f - 8*a*b^3*c^2*e^4*f + 22*a^2*b
*c^3*e^4*f - 3*b^2*c^4*d^4*g + 4*b^3*c^3*d^3*e*g + 8*a*b*c^4*d^3*e*g - 6*b^4*c^2
*d^2*e^2*g + 30*a*b^2*c^3*d^2*e^2*g - 96*a^2*c^4*d^2*e^2*g + 4*b^5*c*d*e^3*g - 3
2*a*b^3*c^2*d*e^3*g + 88*a^2*b*c^3*d*e^3*g - b^6*e^4*g + 11*a*b^4*c*e^4*g - 39*a
^2*b^2*c^2*e^4*g + 32*a^3*c^3*e^4*g + 12*c^6*d^4*f*x - 24*b*c^5*d^3*e*f*x + 12*b
^2*c^4*d^2*e^2*f*x + 24*a*c^5*d^2*e^2*f*x - 24*a*b*c^4*d*e^3*f*x - 2*b^4*c^2*e^4
*f*x + 16*a*b^2*c^3*e^4*f*x - 20*a^2*c^4*e^4*f*x - 6*b*c^5*d^4*g*x + 8*b^2*c^4*d
^3*e*g*x + 16*a*c^5*d^3*e*g*x - 36*a*b*c^4*d^2*e^2*g*x - 8*b^4*c^2*d*e^3*g*x + 6
4*a*b^2*c^3*d*e^3*g*x - 80*a^2*c^4*d*e^3*g*x + 4*b^5*c*e^4*g*x - 30*a*b^3*c^2*e^
4*g*x + 50*a^2*b*c^3*e^4*g*x)/(2*c^4*(-b^2 + 4*a*c)^2*(a + b*x + c*x^2)) + ((12*
c^5*d^4*f - 24*b*c^4*d^3*e*f + 12*b^2*c^3*d^2*e^2*f + 24*a*c^4*d^2*e^2*f - 24*a*
b*c^3*d*e^3*f + 12*a^2*c^3*e^4*f - 6*b*c^4*d^4*g + 8*b^2*c^3*d^3*e*g + 16*a*c^4*
d^3*e*g - 36*a*b*c^3*d^2*e^2*g + 48*a^2*c^3*d*e^3*g - b^5*e^4*g + 10*a*b^3*c*e^4
*g - 30*a^2*b*c^2*e^4*g)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(c^3*(b^2 - 4*a
*c)^2*Sqrt[-b^2 + 4*a*c]) + (e^4*g*Log[a + b*x + c*x^2])/(2*c^3)

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Maple [B]  time = 0.045, size = 4005, normalized size = 7.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4*(g*x+f)/(c*x^2+b*x+a)^3,x)

[Out]

((25*a^2*b*c^2*e^4*g-40*a^2*c^3*d*e^3*g-10*a^2*c^3*e^4*f-15*a*b^3*c*e^4*g+32*a*b
^2*c^2*d*e^3*g+8*a*b^2*c^2*e^4*f-18*a*b*c^3*d^2*e^2*g-12*a*b*c^3*d*e^3*f+8*a*c^4
*d^3*e*g+12*a*c^4*d^2*e^2*f+2*b^5*e^4*g-4*b^4*c*d*e^3*g-b^4*c*e^4*f+4*b^2*c^3*d^
3*e*g+6*b^2*c^3*d^2*e^2*f-3*b*c^4*d^4*g-12*b*c^4*d^3*e*f+6*c^5*d^4*f)/c^2/(16*a^
2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(32*a^3*c^3*e^4*g+11*a^2*b^2*c^2*e^4*g+8*a^2*b*c^3*
d*e^3*g+2*a^2*b*c^3*e^4*f-96*a^2*c^4*d^2*e^2*g-64*a^2*c^4*d*e^3*f-19*a*b^4*c*e^4
*g+32*a*b^3*c^2*d*e^3*g+8*a*b^3*c^2*e^4*f-6*a*b^2*c^3*d^2*e^2*g-4*a*b^2*c^3*d*e^
3*f+24*a*b*c^4*d^3*e*g+36*a*b*c^4*d^2*e^2*f+3*b^6*e^4*g-4*b^5*c*d*e^3*g-b^5*c*e^
4*f-6*b^4*c^2*d^2*e^2*g-4*b^4*c^2*d*e^3*f+12*b^3*c^3*d^3*e*g+18*b^3*c^3*d^2*e^2*
f-9*b^2*c^4*d^4*g-36*b^2*c^4*d^3*e*f+18*b*c^5*d^4*f)/(16*a^2*c^2-8*a*b^2*c+b^4)/
c^3*x^2+(31*a^3*b*c^2*e^4*g-24*a^3*c^3*d*e^3*g-6*a^3*c^3*e^4*f-22*a^2*b^3*c*e^4*
g+40*a^2*b^2*c^2*d*e^3*g+10*a^2*b^2*c^2*e^4*f-30*a^2*b*c^3*d^2*e^2*g-20*a^2*b*c^
3*d*e^3*f-8*a^2*c^4*d^3*e*g-12*a^2*c^4*d^2*e^2*f+3*a*b^5*e^4*g-4*a*b^4*c*d*e^3*g
-a*b^4*c*e^4*f-6*a*b^3*c^2*d^2*e^2*g-4*a*b^3*c^2*d*e^3*f+20*a*b^2*c^3*d^3*e*g+30
*a*b^2*c^3*d^2*e^2*f-5*a*b*c^4*d^4*g-20*a*b*c^4*d^3*e*f+10*a*c^5*d^4*f-b^3*c^3*d
^4*g-4*b^3*c^3*d^3*e*f+2*b^2*c^4*d^4*f)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x+1/2/c^3
*(24*a^4*c^2*e^4*g-21*a^3*b^2*c*e^4*g+40*a^3*b*c^2*d*e^3*g+10*a^3*b*c^2*e^4*f-48
*a^3*c^3*d^2*e^2*g-32*a^3*c^3*d*e^3*f+3*a^2*b^4*e^4*g-4*a^2*b^3*c*d*e^3*g-a^2*b^
3*c*e^4*f-6*a^2*b^2*c^2*d^2*e^2*g-4*a^2*b^2*c^2*d*e^3*f+24*a^2*b*c^3*d^3*e*g+36*
a^2*b*c^3*d^2*e^2*f-8*a^2*c^4*d^4*g-32*a^2*c^4*d^3*e*f-a*b^2*c^3*d^4*g-4*a*b^2*c
^3*d^3*e*f+10*a*b*c^4*d^4*f-b^3*c^3*d^4*f)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*
x+a)^2+8/(16*a^2*c^2-8*a*b^2*c+b^4)/c*ln(c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b
*x+a))*a^2*e^4*g-4/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*ln(c^2*(16*a^2*c^2-8*a*b^2*c+b
^4)*(c*x^2+b*x+a))*a*b^2*e^4*g+1/2/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*ln(c^2*(16*a^2
*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*b^4*e^4*g-30/(1024*a^5*c^9-1280*a^4*b^2*c^8+6
40*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^
2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*
b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*a^2*b*c*e^
4*g+48/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c
^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*
a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+2
0*a*b^8*c^5-b^10*c^4)^(1/2))*a^2*c^2*d*e^3*g+12/(1024*a^5*c^9-1280*a^4*b^2*c^8+6
40*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^
2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*
b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*a^2*c^2*e^
4*f+10/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c
^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*
a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+2
0*a*b^8*c^5-b^10*c^4)^(1/2))*a*b^3*e^4*g-36/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a
^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*
a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*
c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*a*b*c^2*d^2*e^
2*g-24/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c
^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*
a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+2
0*a*b^8*c^5-b^10*c^4)^(1/2))*a*b*c^2*d*e^3*f+16/(1024*a^5*c^9-1280*a^4*b^2*c^8+6
40*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^
2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*
b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*a*c^3*d^3*
e*g+24/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c
^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*
a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+2
0*a*b^8*c^5-b^10*c^4)^(1/2))*a*c^3*d^2*e^2*f+8/(1024*a^5*c^9-1280*a^4*b^2*c^8+64
0*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2
-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b
^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*b^2*c^2*d^3
*e*g+12/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*
c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8
*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+
20*a*b^8*c^5-b^10*c^4)^(1/2))*b^2*c^2*d^2*e^2*f-6/(1024*a^5*c^9-1280*a^4*b^2*c^8
+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*
c^2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^
4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*b*c^3*d^
4*g-24/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c
^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*
a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+2
0*a*b^8*c^5-b^10*c^4)^(1/2))*b*c^3*d^3*e*f+12/(1024*a^5*c^9-1280*a^4*b^2*c^8+640
*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-
8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^
2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*c^4*d^4*f-1/
(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10
*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x+c^2*(16*a^2*c^2-8*a*b^2*c
+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8
*c^5-b^10*c^4)^(1/2))/c*b^5*e^4*g

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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((e*x + d)^4*(g*x + f)/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.07824, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4*(g*x + f)/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[-1/2*(((12*(c^7*d^4 - 2*b*c^6*d^3*e - 2*a*b*c^5*d*e^3 + a^2*c^5*e^4 + (b^2*c^5
+ 2*a*c^6)*d^2*e^2)*f - (6*b*c^6*d^4 + 36*a*b*c^5*d^2*e^2 - 48*a^2*c^5*d*e^3 - 8
*(b^2*c^5 + 2*a*c^6)*d^3*e + (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^4)*g)*x^4
+ 2*(12*(b*c^6*d^4 - 2*b^2*c^5*d^3*e - 2*a*b^2*c^4*d*e^3 + a^2*b*c^4*e^4 + (b^3
*c^4 + 2*a*b*c^5)*d^2*e^2)*f - (6*b^2*c^5*d^4 + 36*a*b^2*c^4*d^2*e^2 - 48*a^2*b*
c^4*d*e^3 - 8*(b^3*c^4 + 2*a*b*c^5)*d^3*e + (b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c
^3)*e^4)*g)*x^3 + (12*((b^2*c^5 + 2*a*c^6)*d^4 - 2*(b^3*c^4 + 2*a*b*c^5)*d^3*e +
(b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c^3 + 2*a^2*b*c^4)*d*e^3
+ (a^2*b^2*c^3 + 2*a^3*c^4)*e^4)*f - (6*(b^3*c^4 + 2*a*b*c^5)*d^4 - 8*(b^4*c^3
+ 4*a*b^2*c^4 + 4*a^2*c^5)*d^3*e + 36*(a*b^3*c^3 + 2*a^2*b*c^4)*d^2*e^2 - 48*(a^
2*b^2*c^3 + 2*a^3*c^4)*d*e^3 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)
*e^4)*g)*x^2 + 12*(a^2*c^5*d^4 - 2*a^2*b*c^4*d^3*e - 2*a^3*b*c^3*d*e^3 + a^4*c^3
*e^4 + (a^2*b^2*c^3 + 2*a^3*c^4)*d^2*e^2)*f - (6*a^2*b*c^4*d^4 + 36*a^3*b*c^3*d^
2*e^2 - 48*a^4*c^3*d*e^3 - 8*(a^2*b^2*c^3 + 2*a^3*c^4)*d^3*e + (a^2*b^5 - 10*a^3
*b^3*c + 30*a^4*b*c^2)*e^4)*g + 2*(12*(a*b*c^5*d^4 - 2*a*b^2*c^4*d^3*e - 2*a^2*b
^2*c^3*d*e^3 + a^3*b*c^3*e^4 + (a*b^3*c^3 + 2*a^2*b*c^4)*d^2*e^2)*f - (6*a*b^2*c
^4*d^4 + 36*a^2*b^2*c^3*d^2*e^2 - 48*a^3*b*c^3*d*e^3 - 8*(a*b^3*c^3 + 2*a^2*b*c^
4)*d^3*e + (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*e^4)*g)*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)) - (2*((6*c^6*d^4 - 12*b*c^5*d^3*e - 12*a*b*c^4*d*e^3 + 6*(b
^2*c^4 + 2*a*c^5)*d^2*e^2 - (b^4*c^2 - 8*a*b^2*c^3 + 10*a^2*c^4)*e^4)*f - (3*b*c
^5*d^4 + 18*a*b*c^4*d^2*e^2 - 4*(b^2*c^4 + 2*a*c^5)*d^3*e + 4*(b^4*c^2 - 8*a*b^2
*c^3 + 10*a^2*c^4)*d*e^3 - (2*b^5*c - 15*a*b^3*c^2 + 25*a^2*b*c^3)*e^4)*g)*x^3 +
((18*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 18*(b^3*c^3 + 2*a*b*c^4)*d^2*e^2 - 4*(b^4*c
^2 + a*b^2*c^3 + 16*a^2*c^4)*d*e^3 - (b^5*c - 8*a*b^3*c^2 - 2*a^2*b*c^3)*e^4)*f
- (9*b^2*c^4*d^4 - 12*(b^3*c^3 + 2*a*b*c^4)*d^3*e + 6*(b^4*c^2 + a*b^2*c^3 + 16*
a^2*c^4)*d^2*e^2 + 4*(b^5*c - 8*a*b^3*c^2 - 2*a^2*b*c^3)*d*e^3 - (3*b^6 - 19*a*b
^4*c + 11*a^2*b^2*c^2 + 32*a^3*c^3)*e^4)*g)*x^2 + (36*a^2*b*c^3*d^2*e^2 - (b^3*c
^3 - 10*a*b*c^4)*d^4 - 4*(a*b^2*c^3 + 8*a^2*c^4)*d^3*e - 4*(a^2*b^2*c^2 + 8*a^3*
c^3)*d*e^3 - (a^2*b^3*c - 10*a^3*b*c^2)*e^4)*f + (24*a^2*b*c^3*d^3*e - (a*b^2*c^
3 + 8*a^2*c^4)*d^4 - 6*(a^2*b^2*c^2 + 8*a^3*c^3)*d^2*e^2 - 4*(a^2*b^3*c - 10*a^3
*b*c^2)*d*e^3 + 3*(a^2*b^4 - 7*a^3*b^2*c + 8*a^4*c^2)*e^4)*g + 2*((2*(b^2*c^4 +
5*a*c^5)*d^4 - 4*(b^3*c^3 + 5*a*b*c^4)*d^3*e + 6*(5*a*b^2*c^3 - 2*a^2*c^4)*d^2*e
^2 - 4*(a*b^3*c^2 + 5*a^2*b*c^3)*d*e^3 - (a*b^4*c - 10*a^2*b^2*c^2 + 6*a^3*c^3)*
e^4)*f - ((b^3*c^3 + 5*a*b*c^4)*d^4 - 4*(5*a*b^2*c^3 - 2*a^2*c^4)*d^3*e + 6*(a*b
^3*c^2 + 5*a^2*b*c^3)*d^2*e^2 + 4*(a*b^4*c - 10*a^2*b^2*c^2 + 6*a^3*c^3)*d*e^3 -
(3*a*b^5 - 22*a^2*b^3*c + 31*a^3*b*c^2)*e^4)*g)*x + ((b^4*c^2 - 8*a*b^2*c^3 + 1
6*a^2*c^4)*e^4*g*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4*g*x^3 + (b^6 -
6*a*b^4*c + 32*a^3*c^3)*e^4*g*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^4*
g*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^4*g)*log(c*x^2 + b*x + a))*sqrt(b^2
- 4*a*c))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 +
16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a
*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x)*sqr
t(b^2 - 4*a*c)), 1/2*(2*((12*(c^7*d^4 - 2*b*c^6*d^3*e - 2*a*b*c^5*d*e^3 + a^2*c^
5*e^4 + (b^2*c^5 + 2*a*c^6)*d^2*e^2)*f - (6*b*c^6*d^4 + 36*a*b*c^5*d^2*e^2 - 48*
a^2*c^5*d*e^3 - 8*(b^2*c^5 + 2*a*c^6)*d^3*e + (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b
*c^4)*e^4)*g)*x^4 + 2*(12*(b*c^6*d^4 - 2*b^2*c^5*d^3*e - 2*a*b^2*c^4*d*e^3 + a^2
*b*c^4*e^4 + (b^3*c^4 + 2*a*b*c^5)*d^2*e^2)*f - (6*b^2*c^5*d^4 + 36*a*b^2*c^4*d^
2*e^2 - 48*a^2*b*c^4*d*e^3 - 8*(b^3*c^4 + 2*a*b*c^5)*d^3*e + (b^6*c - 10*a*b^4*c
^2 + 30*a^2*b^2*c^3)*e^4)*g)*x^3 + (12*((b^2*c^5 + 2*a*c^6)*d^4 - 2*(b^3*c^4 + 2
*a*b*c^5)*d^3*e + (b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^2*e^2 - 2*(a*b^3*c^3 + 2
*a^2*b*c^4)*d*e^3 + (a^2*b^2*c^3 + 2*a^3*c^4)*e^4)*f - (6*(b^3*c^4 + 2*a*b*c^5)*
d^4 - 8*(b^4*c^3 + 4*a*b^2*c^4 + 4*a^2*c^5)*d^3*e + 36*(a*b^3*c^3 + 2*a^2*b*c^4)
*d^2*e^2 - 48*(a^2*b^2*c^3 + 2*a^3*c^4)*d*e^3 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^
2 + 60*a^3*b*c^3)*e^4)*g)*x^2 + 12*(a^2*c^5*d^4 - 2*a^2*b*c^4*d^3*e - 2*a^3*b*c^
3*d*e^3 + a^4*c^3*e^4 + (a^2*b^2*c^3 + 2*a^3*c^4)*d^2*e^2)*f - (6*a^2*b*c^4*d^4
+ 36*a^3*b*c^3*d^2*e^2 - 48*a^4*c^3*d*e^3 - 8*(a^2*b^2*c^3 + 2*a^3*c^4)*d^3*e +
(a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2)*e^4)*g + 2*(12*(a*b*c^5*d^4 - 2*a*b^2*c^
4*d^3*e - 2*a^2*b^2*c^3*d*e^3 + a^3*b*c^3*e^4 + (a*b^3*c^3 + 2*a^2*b*c^4)*d^2*e^
2)*f - (6*a*b^2*c^4*d^4 + 36*a^2*b^2*c^3*d^2*e^2 - 48*a^3*b*c^3*d*e^3 - 8*(a*b^3
*c^3 + 2*a^2*b*c^4)*d^3*e + (a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*e^4)*g)*x)*a
rctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (2*((6*c^6*d^4 - 12*b*c^5
*d^3*e - 12*a*b*c^4*d*e^3 + 6*(b^2*c^4 + 2*a*c^5)*d^2*e^2 - (b^4*c^2 - 8*a*b^2*c
^3 + 10*a^2*c^4)*e^4)*f - (3*b*c^5*d^4 + 18*a*b*c^4*d^2*e^2 - 4*(b^2*c^4 + 2*a*c
^5)*d^3*e + 4*(b^4*c^2 - 8*a*b^2*c^3 + 10*a^2*c^4)*d*e^3 - (2*b^5*c - 15*a*b^3*c
^2 + 25*a^2*b*c^3)*e^4)*g)*x^3 + ((18*b*c^5*d^4 - 36*b^2*c^4*d^3*e + 18*(b^3*c^3
+ 2*a*b*c^4)*d^2*e^2 - 4*(b^4*c^2 + a*b^2*c^3 + 16*a^2*c^4)*d*e^3 - (b^5*c - 8*
a*b^3*c^2 - 2*a^2*b*c^3)*e^4)*f - (9*b^2*c^4*d^4 - 12*(b^3*c^3 + 2*a*b*c^4)*d^3*
e + 6*(b^4*c^2 + a*b^2*c^3 + 16*a^2*c^4)*d^2*e^2 + 4*(b^5*c - 8*a*b^3*c^2 - 2*a^
2*b*c^3)*d*e^3 - (3*b^6 - 19*a*b^4*c + 11*a^2*b^2*c^2 + 32*a^3*c^3)*e^4)*g)*x^2
+ (36*a^2*b*c^3*d^2*e^2 - (b^3*c^3 - 10*a*b*c^4)*d^4 - 4*(a*b^2*c^3 + 8*a^2*c^4)
*d^3*e - 4*(a^2*b^2*c^2 + 8*a^3*c^3)*d*e^3 - (a^2*b^3*c - 10*a^3*b*c^2)*e^4)*f +
(24*a^2*b*c^3*d^3*e - (a*b^2*c^3 + 8*a^2*c^4)*d^4 - 6*(a^2*b^2*c^2 + 8*a^3*c^3)
*d^2*e^2 - 4*(a^2*b^3*c - 10*a^3*b*c^2)*d*e^3 + 3*(a^2*b^4 - 7*a^3*b^2*c + 8*a^4
*c^2)*e^4)*g + 2*((2*(b^2*c^4 + 5*a*c^5)*d^4 - 4*(b^3*c^3 + 5*a*b*c^4)*d^3*e + 6
*(5*a*b^2*c^3 - 2*a^2*c^4)*d^2*e^2 - 4*(a*b^3*c^2 + 5*a^2*b*c^3)*d*e^3 - (a*b^4*
c - 10*a^2*b^2*c^2 + 6*a^3*c^3)*e^4)*f - ((b^3*c^3 + 5*a*b*c^4)*d^4 - 4*(5*a*b^2
*c^3 - 2*a^2*c^4)*d^3*e + 6*(a*b^3*c^2 + 5*a^2*b*c^3)*d^2*e^2 + 4*(a*b^4*c - 10*
a^2*b^2*c^2 + 6*a^3*c^3)*d*e^3 - (3*a*b^5 - 22*a^2*b^3*c + 31*a^3*b*c^2)*e^4)*g)
*x + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*g*x^4 + 2*(b^5*c - 8*a*b^3*c^2 +
16*a^2*b*c^3)*e^4*g*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*e^4*g*x^2 + 2*(a*b^5 -
8*a^2*b^3*c + 16*a^3*b*c^2)*e^4*g*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^4*g
)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a
^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 1
6*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a
^2*b^3*c^4 + 16*a^3*b*c^5)*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((e*x+d)**4*(g*x+f)/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.434622, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4*(g*x + f)/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

Done