3.2141 \(\int \frac{\left (a+b x+c x^2\right )^4}{(d+e x)^2} \, dx\)
Optimal. Leaf size=426 \[ \frac{(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^9}+\frac{2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac{c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}-\frac{2 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{2 x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}-\frac{\left (a e^2-b d e+c d^2\right )^4}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^9}-\frac{2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac{c^4 (d+e x)^7}{7 e^9} \]
[Out]
(2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*x)/e
^8 - (c*d^2 - b*d*e + a*e^2)^4/(e^9*(d + e*x)) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e
+ a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^2)/e^9 + ((70*c^
4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6
*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^3)/(3*e^9) - (c*(2*c*d -
b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^4)/e^9 + (2*c^2*(14*
c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^5)/(5*e^9) - (2*c^3*(2*c*d
- b*e)*(d + e*x)^6)/(3*e^9) + (c^4*(d + e*x)^7)/(7*e^9) - (4*(2*c*d - b*e)*(c*d^
2 - b*d*e + a*e^2)^3*Log[d + e*x])/e^9
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Rubi [A] time = 1.83947, antiderivative size = 426, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05
\[ \frac{(d+e x)^3 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{3 e^9}+\frac{2 c^2 (d+e x)^5 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9}-\frac{c (d+e x)^4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}-\frac{2 (d+e x)^2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{2 x \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^8}-\frac{\left (a e^2-b d e+c d^2\right )^4}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^9}-\frac{2 c^3 (d+e x)^6 (2 c d-b e)}{3 e^9}+\frac{c^4 (d+e x)^7}{7 e^9} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^4/(d + e*x)^2,x]
[Out]
(2*(c*d^2 - b*d*e + a*e^2)^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*x)/e
^8 - (c*d^2 - b*d*e + a*e^2)^4/(e^9*(d + e*x)) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e
+ a*e^2)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^2)/e^9 + ((70*c^
4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(5*b*d - 3*a*e) - 20*c^3*d^2*e*(7*b*d - 3*a*e) + 6
*c^2*e^2*(15*b^2*d^2 - 10*a*b*d*e + a^2*e^2))*(d + e*x)^3)/(3*e^9) - (c*(2*c*d -
b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))*(d + e*x)^4)/e^9 + (2*c^2*(14*
c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(d + e*x)^5)/(5*e^9) - (2*c^3*(2*c*d
- b*e)*(d + e*x)^6)/(3*e^9) + (c^4*(d + e*x)^7)/(7*e^9) - (4*(2*c*d - b*e)*(c*d^
2 - b*d*e + a*e^2)^3*Log[d + e*x])/e^9
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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((c*x**2+b*x+a)**4/(e*x+d)**2,x)
[Out]
Timed out
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Mathematica [A] time = 0.826646, size = 780, normalized size = 1.83 \[ \frac{21 c^2 e^2 \left (10 a^2 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )+b^2 \left (-30 d^6+150 d^5 e x+90 d^4 e^2 x^2-30 d^3 e^3 x^3+15 d^2 e^4 x^4-9 d e^5 x^5+6 e^6 x^6\right )\right )+35 c e^3 \left (12 a^3 e^3 \left (-d^2+d e x+e^2 x^2\right )+18 a^2 b e^2 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+12 a b^2 e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+b^3 \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )+35 e^4 \left (-3 a^4 e^4+12 a^3 b d e^3+18 a^2 b^2 e^2 \left (-d^2+d e x+e^2 x^2\right )+6 a b^3 e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+b^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+7 c^3 e \left (6 a e \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+b \left (60 d^7-360 d^6 e x-210 d^5 e^2 x^2+70 d^4 e^3 x^3-35 d^3 e^4 x^4+21 d^2 e^5 x^5-14 d e^6 x^6+10 e^7 x^7\right )\right )-420 (d+e x) (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^3+c^4 \left (-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right )}{105 e^9 (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^4/(d + e*x)^2,x]
[Out]
(c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^
4 - 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) + 35*e^4*(12*a^
3*b*d*e^3 - 3*a^4*e^4 + 18*a^2*b^2*e^2*(-d^2 + d*e*x + e^2*x^2) + 6*a*b^3*e*(2*d
^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + b^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^
2 - 2*d*e^3*x^3 + e^4*x^4)) + 35*c*e^3*(12*a^3*e^3*(-d^2 + d*e*x + e^2*x^2) + 18
*a^2*b*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 12*a*b^2*e*(-3*d^4 + 9*
d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^3*(12*d^5 - 48*d^4*e*x - 30
*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 21*c^2*e^2*(10*a^2*e
^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + 5*a*b*e*(12*d^
5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5) + b^
2*(-30*d^6 + 150*d^5*e*x + 90*d^4*e^2*x^2 - 30*d^3*e^3*x^3 + 15*d^2*e^4*x^4 - 9*
d*e^5*x^5 + 6*e^6*x^6)) + 7*c^3*e*(6*a*e*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2
- 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4 - 3*d*e^5*x^5 + 2*e^6*x^6) + b*(60*d^7 - 360*d^
6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*d^2*e^5*x^5 - 14*
d*e^6*x^6 + 10*e^7*x^7)) - 420*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3*(d + e
*x)*Log[d + e*x])/(105*e^9*(d + e*x))
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Maple [B] time = 0.023, size = 1159, normalized size = 2.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^4/(e*x+d)^2,x)
[Out]
12*b^3/e^4*ln(e*x+d)*a*d^2+4/e^2/(e*x+d)*d*a^3*b-6/e^3/(e*x+d)*d^2*a^2*b^2+4/e^4
/(e*x+d)*d^3*a*b^3-24/e^7*ln(e*x+d)*a*c^3*d^5+20/e^6*ln(e*x+d)*b^3*c*d^4-36/e^7*
ln(e*x+d)*b^2*c^2*d^5+28/e^8*ln(e*x+d)*b*c^3*d^6-4/e^3/(e*x+d)*a^3*c*d^2-6/e^5/(
e*x+d)*a^2*c^2*d^4-4/e^7/(e*x+d)*a*c^3*d^6+4/e^6/(e*x+d)*b^3*c*d^5-6/e^7/(e*x+d)
*b^2*c^2*d^6+3/e^2*x^4*a*b*c^2-8/3/e^3*x^3*b^3*c*d-8*b^3/e^3*d*a*x+18/e^4*a^2*c^
2*d^2*x+20/e^6*a*c^3*d^4*x-16/3/e^5*x^3*b*c^3*d^3-16/e^5*b^3*c*d^3*x-48/e^5*a*b*
c^2*d^3*x+36/e^4*ln(e*x+d)*a^2*b*c*d^2-48/e^5*ln(e*x+d)*a*b^2*c*d^3+60/e^6*ln(e*
x+d)*a*b*c^2*d^4+12/e^4/(e*x+d)*a^2*b*c*d^3-12/e^5/(e*x+d)*a*b^2*c*d^4+12/e^6/(e
*x+d)*a*b*c^2*d^5-8/e^3*x^3*a*b*c^2*d+4/e^8/(e*x+d)*b*c^3*d^7-12*b^2/e^3*ln(e*x+
d)*a^2*d+4/e^4*x^3*a*c^3*d^2+6/e^2*x^2*a^2*b*c-6/e^3*x^2*a^2*c^2*d-24/e^7*b*c^3*
d^5*x+30/e^6*b^2*c^2*d^4*x-8/e^3*ln(e*x+d)*a^3*c*d-24/e^5*ln(e*x+d)*a^2*c^2*d^3+
6/e^4*x^3*b^2*c^2*d^2-8/e^5*x^2*a*c^3*d^3-8/5/e^3*x^5*b*c^3*d-12/e^5*x^2*b^2*c^2
*d^3+10/e^6*x^2*b*c^3*d^4+6/e^4*x^2*b^3*c*d^2-2/e^3*x^4*a*c^3*d-3/e^3*x^4*b^2*c^
2*d+3/e^4*x^4*b*c^3*d^2+4/e^2*x^3*a*b^2*c-12/e^3*x^2*a*b^2*c*d+18/e^4*x^2*a*b*c^
2*d^2-24/e^3*a^2*b*c*d*x+36/e^4*a*b^2*c*d^2*x-1/e^5/(e*x+d)*b^4*d^4+1/7/e^2*c^4*
x^7+1/3*b^4/e^2*x^3-1/e/(e*x+d)*a^4+2*b^3/e^2*x^2*a-b^4/e^3*x^2*d+6*b^2/e^2*a^2*
x+3*b^4/e^4*d^2*x-4*b^4/e^5*ln(e*x+d)*d^3+4*b/e^2*ln(e*x+d)*a^3+5/3/e^6*x^3*c^4*
d^4-3/e^7*x^2*c^4*d^5+4/e^2*a^3*c*x+7/e^8*c^4*d^6*x+1/e^2*x^4*b^3*c+2/3/e^2*x^6*
b*c^3-1/3/e^3*x^6*c^4*d+4/5/e^2*x^5*a*c^3+6/5/e^2*x^5*b^2*c^2+3/5/e^4*x^5*c^4*d^
2-1/e^5*x^4*c^4*d^3+2/e^2*x^3*a^2*c^2-8/e^9*ln(e*x+d)*c^4*d^7-1/e^9/(e*x+d)*c^4*
d^8
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Maxima [A] time = 0.830069, size = 1089, normalized size = 2.56 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4/(e*x + d)^2,x, algorithm="maxima")
[Out]
-(c^4*d^8 - 4*b*c^3*d^7*e - 4*a^3*b*d*e^7 + a^4*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*d^
6*e^2 - 4*(b^3*c + 3*a*b*c^2)*d^5*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 -
4*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6)/(e^10*x + d*e^
9) + 1/105*(15*c^4*e^6*x^7 - 35*(c^4*d*e^5 - 2*b*c^3*e^6)*x^6 + 21*(3*c^4*d^2*e^
4 - 8*b*c^3*d*e^5 + 2*(3*b^2*c^2 + 2*a*c^3)*e^6)*x^5 - 105*(c^4*d^3*e^3 - 3*b*c^
3*d^2*e^4 + (3*b^2*c^2 + 2*a*c^3)*d*e^5 - (b^3*c + 3*a*b*c^2)*e^6)*x^4 + 35*(5*c
^4*d^4*e^2 - 16*b*c^3*d^3*e^3 + 6*(3*b^2*c^2 + 2*a*c^3)*d^2*e^4 - 8*(b^3*c + 3*a
*b*c^2)*d*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^6)*x^3 - 105*(3*c^4*d^5*e - 10*
b*c^3*d^4*e^2 + 4*(3*b^2*c^2 + 2*a*c^3)*d^3*e^3 - 6*(b^3*c + 3*a*b*c^2)*d^2*e^4
+ (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^5 - 2*(a*b^3 + 3*a^2*b*c)*e^6)*x^2 + 105*(7
*c^4*d^6 - 24*b*c^3*d^5*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^4*e^2 - 16*(b^3*c + 3*a*b
*c^2)*d^3*e^3 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^4 - 8*(a*b^3 + 3*a^2*b*c)
*d*e^5 + 2*(3*a^2*b^2 + 2*a^3*c)*e^6)*x)/e^8 - 4*(2*c^4*d^7 - 7*b*c^3*d^6*e - a^
3*b*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^3 + (b^4
+ 12*a*b^2*c + 6*a^2*c^2)*d^3*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^2*e^5 + (3*a^2*b^2
+ 2*a^3*c)*d*e^6)*log(e*x + d)/e^9
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Fricas [A] time = 0.216185, size = 1478, normalized size = 3.47 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4/(e*x + d)^2,x, algorithm="fricas")
[Out]
1/105*(15*c^4*e^8*x^8 - 105*c^4*d^8 + 420*b*c^3*d^7*e + 420*a^3*b*d*e^7 - 105*a^
4*e^8 - 210*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 420*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 10
5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 420*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 210
*(3*a^2*b^2 + 2*a^3*c)*d^2*e^6 - 10*(2*c^4*d*e^7 - 7*b*c^3*e^8)*x^7 + 14*(2*c^4*
d^2*e^6 - 7*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 21*(2*c^4*d^3*e^5 -
7*b*c^3*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - 5*(b^3*c + 3*a*b*c^2)*e^8)*x^
5 + 35*(2*c^4*d^4*e^4 - 7*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 5*(b
^3*c + 3*a*b*c^2)*d*e^7 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 - 70*(2*c^4*d^
5*e^3 - 7*b*c^3*d^4*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 5*(b^3*c + 3*a*b*c^2
)*d^2*e^6 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 - 3*(a*b^3 + 3*a^2*b*c)*e^8)*x^
3 + 210*(2*c^4*d^6*e^2 - 7*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 5*(
b^3*c + 3*a*b*c^2)*d^3*e^5 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - 3*(a*b^3 +
3*a^2*b*c)*d*e^7 + (3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 105*(7*c^4*d^7*e - 24*b*c^3
*d^6*e^2 + 10*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 16*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 3
*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 8*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 2*(3*a
^2*b^2 + 2*a^3*c)*d*e^7)*x - 420*(2*c^4*d^8 - 7*b*c^3*d^7*e - a^3*b*d*e^7 + 3*(3
*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 5*(b^3*c + 3*a*b*c^2)*d^5*e^3 + (b^4 + 12*a*b^2*c
+ 6*a^2*c^2)*d^4*e^4 - 3*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + (3*a^2*b^2 + 2*a^3*c)*d^2
*e^6 + (2*c^4*d^7*e - 7*b*c^3*d^6*e^2 - a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*
e^3 - 5*(b^3*c + 3*a*b*c^2)*d^4*e^4 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 3
*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + (3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)*log(e*x + d))/(e^
10*x + d*e^9)
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Sympy [A] time = 18.5737, size = 824, normalized size = 1.93 \[ \frac{c^{4} x^{7}}{7 e^{2}} - \frac{a^{4} e^{8} - 4 a^{3} b d e^{7} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} b^{2} d^{2} e^{6} - 12 a^{2} b c d^{3} e^{5} + 6 a^{2} c^{2} d^{4} e^{4} - 4 a b^{3} d^{3} e^{5} + 12 a b^{2} c d^{4} e^{4} - 12 a b c^{2} d^{5} e^{3} + 4 a c^{3} d^{6} e^{2} + b^{4} d^{4} e^{4} - 4 b^{3} c d^{5} e^{3} + 6 b^{2} c^{2} d^{6} e^{2} - 4 b c^{3} d^{7} e + c^{4} d^{8}}{d e^{9} + e^{10} x} + \frac{x^{6} \left (2 b c^{3} e - c^{4} d\right )}{3 e^{3}} + \frac{x^{5} \left (4 a c^{3} e^{2} + 6 b^{2} c^{2} e^{2} - 8 b c^{3} d e + 3 c^{4} d^{2}\right )}{5 e^{4}} + \frac{x^{4} \left (3 a b c^{2} e^{3} - 2 a c^{3} d e^{2} + b^{3} c e^{3} - 3 b^{2} c^{2} d e^{2} + 3 b c^{3} d^{2} e - c^{4} d^{3}\right )}{e^{5}} + \frac{x^{3} \left (6 a^{2} c^{2} e^{4} + 12 a b^{2} c e^{4} - 24 a b c^{2} d e^{3} + 12 a c^{3} d^{2} e^{2} + b^{4} e^{4} - 8 b^{3} c d e^{3} + 18 b^{2} c^{2} d^{2} e^{2} - 16 b c^{3} d^{3} e + 5 c^{4} d^{4}\right )}{3 e^{6}} + \frac{x^{2} \left (6 a^{2} b c e^{5} - 6 a^{2} c^{2} d e^{4} + 2 a b^{3} e^{5} - 12 a b^{2} c d e^{4} + 18 a b c^{2} d^{2} e^{3} - 8 a c^{3} d^{3} e^{2} - b^{4} d e^{4} + 6 b^{3} c d^{2} e^{3} - 12 b^{2} c^{2} d^{3} e^{2} + 10 b c^{3} d^{4} e - 3 c^{4} d^{5}\right )}{e^{7}} + \frac{x \left (4 a^{3} c e^{6} + 6 a^{2} b^{2} e^{6} - 24 a^{2} b c d e^{5} + 18 a^{2} c^{2} d^{2} e^{4} - 8 a b^{3} d e^{5} + 36 a b^{2} c d^{2} e^{4} - 48 a b c^{2} d^{3} e^{3} + 20 a c^{3} d^{4} e^{2} + 3 b^{4} d^{2} e^{4} - 16 b^{3} c d^{3} e^{3} + 30 b^{2} c^{2} d^{4} e^{2} - 24 b c^{3} d^{5} e + 7 c^{4} d^{6}\right )}{e^{8}} + \frac{4 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \log{\left (d + e x \right )}}{e^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**4/(e*x+d)**2,x)
[Out]
c**4*x**7/(7*e**2) - (a**4*e**8 - 4*a**3*b*d*e**7 + 4*a**3*c*d**2*e**6 + 6*a**2*
b**2*d**2*e**6 - 12*a**2*b*c*d**3*e**5 + 6*a**2*c**2*d**4*e**4 - 4*a*b**3*d**3*e
**5 + 12*a*b**2*c*d**4*e**4 - 12*a*b*c**2*d**5*e**3 + 4*a*c**3*d**6*e**2 + b**4*
d**4*e**4 - 4*b**3*c*d**5*e**3 + 6*b**2*c**2*d**6*e**2 - 4*b*c**3*d**7*e + c**4*
d**8)/(d*e**9 + e**10*x) + x**6*(2*b*c**3*e - c**4*d)/(3*e**3) + x**5*(4*a*c**3*
e**2 + 6*b**2*c**2*e**2 - 8*b*c**3*d*e + 3*c**4*d**2)/(5*e**4) + x**4*(3*a*b*c**
2*e**3 - 2*a*c**3*d*e**2 + b**3*c*e**3 - 3*b**2*c**2*d*e**2 + 3*b*c**3*d**2*e -
c**4*d**3)/e**5 + x**3*(6*a**2*c**2*e**4 + 12*a*b**2*c*e**4 - 24*a*b*c**2*d*e**3
+ 12*a*c**3*d**2*e**2 + b**4*e**4 - 8*b**3*c*d*e**3 + 18*b**2*c**2*d**2*e**2 -
16*b*c**3*d**3*e + 5*c**4*d**4)/(3*e**6) + x**2*(6*a**2*b*c*e**5 - 6*a**2*c**2*d
*e**4 + 2*a*b**3*e**5 - 12*a*b**2*c*d*e**4 + 18*a*b*c**2*d**2*e**3 - 8*a*c**3*d*
*3*e**2 - b**4*d*e**4 + 6*b**3*c*d**2*e**3 - 12*b**2*c**2*d**3*e**2 + 10*b*c**3*
d**4*e - 3*c**4*d**5)/e**7 + x*(4*a**3*c*e**6 + 6*a**2*b**2*e**6 - 24*a**2*b*c*d
*e**5 + 18*a**2*c**2*d**2*e**4 - 8*a*b**3*d*e**5 + 36*a*b**2*c*d**2*e**4 - 48*a*
b*c**2*d**3*e**3 + 20*a*c**3*d**4*e**2 + 3*b**4*d**2*e**4 - 16*b**3*c*d**3*e**3
+ 30*b**2*c**2*d**4*e**2 - 24*b*c**3*d**5*e + 7*c**4*d**6)/e**8 + 4*(b*e - 2*c*d
)*(a*e**2 - b*d*e + c*d**2)**3*log(d + e*x)/e**9
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GIAC/XCAS [A] time = 0.206781, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4/(e*x + d)^2,x, algorithm="giac")
[Out]
Done