3.2151 \(\int \frac{\left (a+b x+c x^2\right )^4}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=440 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{7 e^9 (d+e x)^7}-\frac{2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9 (d+e x)^8}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9 (d+e x)^9}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^{10}}-\frac{\left (a e^2-b d e+c d^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac{c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac{c^4}{3 e^9 (d+e x)^3} \]

[Out]

-(c*d^2 - b*d*e + a*e^2)^4/(11*e^9*(d + e*x)^11) + (2*(2*c*d - b*e)*(c*d^2 - b*d
*e + a*e^2)^3)/(5*e^9*(d + e*x)^10) - (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)))/(9*e^9*(d + e*x)^9) + ((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)))/(2*e^9*(d + e*x)^8)
- (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))/(7*e^9*(d + e*x)^7) + (2*
c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(3*e^9*(d + e*x)^6)
- (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(5*e^9*(d + e*x)^5) +
(c^3*(2*c*d - b*e))/(e^9*(d + e*x)^4) - c^4/(3*e^9*(d + e*x)^3)

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Rubi [A]  time = 1.67934, antiderivative size = 440, 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 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{7 e^9 (d+e x)^7}-\frac{2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{2 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^9 (d+e x)^8}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{9 e^9 (d+e x)^9}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{5 e^9 (d+e x)^{10}}-\frac{\left (a e^2-b d e+c d^2\right )^4}{11 e^9 (d+e x)^{11}}+\frac{c^3 (2 c d-b e)}{e^9 (d+e x)^4}-\frac{c^4}{3 e^9 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^4/(d + e*x)^12,x]

[Out]

-(c*d^2 - b*d*e + a*e^2)^4/(11*e^9*(d + e*x)^11) + (2*(2*c*d - b*e)*(c*d^2 - b*d
*e + a*e^2)^3)/(5*e^9*(d + e*x)^10) - (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)))/(9*e^9*(d + e*x)^9) + ((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)))/(2*e^9*(d + e*x)^8)
- (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))/(7*e^9*(d + e*x)^7) + (2*
c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(3*e^9*(d + e*x)^6)
- (2*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(5*e^9*(d + e*x)^5) +
(c^3*(2*c*d - b*e))/(e^9*(d + e*x)^4) - c^4/(3*e^9*(d + e*x)^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((c*x**2+b*x+a)**4/(e*x+d)**12,x)

[Out]

Timed out

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Mathematica [A]  time = 1.05169, size = 731, normalized size = 1.66 \[ -\frac{6 c^2 e^2 \left (3 a^2 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+3 b^2 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )\right )+c e^3 \left (56 a^3 e^3 \left (d^2+11 d e x+55 e^2 x^2\right )+63 a^2 b e^2 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+36 a b^2 e \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+10 b^3 \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )\right )+3 e^4 \left (210 a^4 e^4+84 a^3 b e^3 (d+11 e x)+28 a^2 b^2 e^2 \left (d^2+11 d e x+55 e^2 x^2\right )+7 a b^3 e \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+b^4 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )\right )+3 c^3 e \left (4 a e \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )+7 b \left (d^7+11 d^6 e x+55 d^5 e^2 x^2+165 d^4 e^3 x^3+330 d^3 e^4 x^4+462 d^2 e^5 x^5+462 d e^6 x^6+330 e^7 x^7\right )\right )+14 c^4 \left (d^8+11 d^7 e x+55 d^6 e^2 x^2+165 d^5 e^3 x^3+330 d^4 e^4 x^4+462 d^3 e^5 x^5+462 d^2 e^6 x^6+330 d e^7 x^7+165 e^8 x^8\right )}{6930 e^9 (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^4/(d + e*x)^12,x]

[Out]

-(14*c^4*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4
+ 462*d^3*e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8) + 3*e^4*(210*
a^4*e^4 + 84*a^3*b*e^3*(d + 11*e*x) + 28*a^2*b^2*e^2*(d^2 + 11*d*e*x + 55*e^2*x^
2) + 7*a*b^3*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + b^4*(d^4 + 11*d
^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + c*e^3*(56*a^3*e^3*(d^2
+ 11*d*e*x + 55*e^2*x^2) + 63*a^2*b*e^2*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*
e^3*x^3) + 36*a*b^2*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e
^4*x^4) + 10*b^3*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^
4*x^4 + 462*e^5*x^5)) + 6*c^2*e^2*(3*a^2*e^2*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2
+ 165*d*e^3*x^3 + 330*e^4*x^4) + 5*a*b*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 16
5*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5) + 3*b^2*(d^6 + 11*d^5*e*x + 55*d^4*
e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6)) + 3*
c^3*e*(4*a*e*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*
x^4 + 462*d*e^5*x^5 + 462*e^6*x^6) + 7*b*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 16
5*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7)
))/(6930*e^9*(d + e*x)^11)

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Maple [B]  time = 0.013, size = 914, normalized size = 2.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^4/(e*x+d)^12,x)

[Out]

-1/7*(6*a^2*c^2*e^4+12*a*b^2*c*e^4-60*a*b*c^2*d*e^3+60*a*c^3*d^2*e^2+b^4*e^4-20*
b^3*c*d*e^3+90*b^2*c^2*d^2*e^2-140*b*c^3*d^3*e+70*c^4*d^4)/e^9/(e*x+d)^7-1/8*(12
*a^2*b*c*e^5-24*a^2*c^2*d*e^4+4*a*b^3*e^5-48*a*b^2*c*d*e^4+120*a*b*c^2*d^2*e^3-8
0*a*c^3*d^3*e^2-4*b^4*d*e^4+40*b^3*c*d^2*e^3-120*b^2*c^2*d^3*e^2+140*b*c^3*d^4*e
-56*c^4*d^5)/e^9/(e*x+d)^8-1/11*(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)/e^9/(e*x+d)^11-1/10*(4*a^3*b*e^7-8*a^3*c*d*e^6-12*a^2*b
^2*d*e^6+36*a^2*b*c*d^2*e^5-24*a^2*c^2*d^3*e^4+12*a*b^3*d^2*e^5-48*a*b^2*c*d^3*e
^4+60*a*b*c^2*d^4*e^3-24*a*c^3*d^5*e^2-4*b^4*d^3*e^4+20*b^3*c*d^4*e^3-36*b^2*c^2
*d^5*e^2+28*b*c^3*d^6*e-8*c^4*d^7)/e^9/(e*x+d)^10-1/3*c^4/e^9/(e*x+d)^3-2/5*c^2*
(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^2)/e^9/(e*x+d)^5-2/3*c*(3*a*b*c*e^3-6*a
*c^2*d*e^2+b^3*e^3-9*b^2*c*d*e^2+21*b*c^2*d^2*e-14*c^3*d^3)/e^9/(e*x+d)^6-c^3*(b
*e-2*c*d)/e^9/(e*x+d)^4-1/9*(4*a^3*c*e^6+6*a^2*b^2*e^6-36*a^2*b*c*d*e^5+36*a^2*c
^2*d^2*e^4-12*a*b^3*d*e^5+72*a*b^2*c*d^2*e^4-120*a*b*c^2*d^3*e^3+60*a*c^3*d^4*e^
2+6*b^4*d^2*e^4-40*b^3*c*d^3*e^3+90*b^2*c^2*d^4*e^2-84*b*c^3*d^5*e+28*c^4*d^6)/e
^9/(e*x+d)^9

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Maxima [A]  time = 0.858791, size = 1247, normalized size = 2.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^4/(e*x + d)^12,x, algorithm="maxima")

[Out]

-1/6930*(2310*c^4*e^8*x^8 + 14*c^4*d^8 + 21*b*c^3*d^7*e + 252*a^3*b*d*e^7 + 630*
a^4*e^8 + 6*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 10*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 3*(
b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 21*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 28*(3*a
^2*b^2 + 2*a^3*c)*d^2*e^6 + 2310*(2*c^4*d*e^7 + 3*b*c^3*e^8)*x^7 + 462*(14*c^4*d
^2*e^6 + 21*b*c^3*d*e^7 + 6*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 462*(14*c^4*d^3*e^5
+ 21*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 10*(b^3*c + 3*a*b*c^2)*e^8
)*x^5 + 330*(14*c^4*d^4*e^4 + 21*b*c^3*d^3*e^5 + 6*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6
+ 10*(b^3*c + 3*a*b*c^2)*d*e^7 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 16
5*(14*c^4*d^5*e^3 + 21*b*c^3*d^4*e^4 + 6*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(b^3
*c + 3*a*b*c^2)*d^2*e^6 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 21*(a*b^3 + 3
*a^2*b*c)*e^8)*x^3 + 55*(14*c^4*d^6*e^2 + 21*b*c^3*d^5*e^3 + 6*(3*b^2*c^2 + 2*a*
c^3)*d^4*e^4 + 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)
*d^2*e^6 + 21*(a*b^3 + 3*a^2*b*c)*d*e^7 + 28*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 11
*(14*c^4*d^7*e + 21*b*c^3*d^6*e^2 + 252*a^3*b*e^8 + 6*(3*b^2*c^2 + 2*a*c^3)*d^5*
e^3 + 10*(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
+ 21*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 28*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^20*x^11
+ 11*d*e^19*x^10 + 55*d^2*e^18*x^9 + 165*d^3*e^17*x^8 + 330*d^4*e^16*x^7 + 462*
d^5*e^15*x^6 + 462*d^6*e^14*x^5 + 330*d^7*e^13*x^4 + 165*d^8*e^12*x^3 + 55*d^9*e
^11*x^2 + 11*d^10*e^10*x + d^11*e^9)

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Fricas [A]  time = 0.200781, size = 1247, normalized size = 2.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^4/(e*x + d)^12,x, algorithm="fricas")

[Out]

-1/6930*(2310*c^4*e^8*x^8 + 14*c^4*d^8 + 21*b*c^3*d^7*e + 252*a^3*b*d*e^7 + 630*
a^4*e^8 + 6*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 10*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 3*(
b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 21*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 28*(3*a
^2*b^2 + 2*a^3*c)*d^2*e^6 + 2310*(2*c^4*d*e^7 + 3*b*c^3*e^8)*x^7 + 462*(14*c^4*d
^2*e^6 + 21*b*c^3*d*e^7 + 6*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 462*(14*c^4*d^3*e^5
+ 21*b*c^3*d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 10*(b^3*c + 3*a*b*c^2)*e^8
)*x^5 + 330*(14*c^4*d^4*e^4 + 21*b*c^3*d^3*e^5 + 6*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6
+ 10*(b^3*c + 3*a*b*c^2)*d*e^7 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 16
5*(14*c^4*d^5*e^3 + 21*b*c^3*d^4*e^4 + 6*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 10*(b^3
*c + 3*a*b*c^2)*d^2*e^6 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7 + 21*(a*b^3 + 3
*a^2*b*c)*e^8)*x^3 + 55*(14*c^4*d^6*e^2 + 21*b*c^3*d^5*e^3 + 6*(3*b^2*c^2 + 2*a*
c^3)*d^4*e^4 + 10*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 3*(b^4 + 12*a*b^2*c + 6*a^2*c^2)
*d^2*e^6 + 21*(a*b^3 + 3*a^2*b*c)*d*e^7 + 28*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 11
*(14*c^4*d^7*e + 21*b*c^3*d^6*e^2 + 252*a^3*b*e^8 + 6*(3*b^2*c^2 + 2*a*c^3)*d^5*
e^3 + 10*(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
+ 21*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 28*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^20*x^11
+ 11*d*e^19*x^10 + 55*d^2*e^18*x^9 + 165*d^3*e^17*x^8 + 330*d^4*e^16*x^7 + 462*
d^5*e^15*x^6 + 462*d^6*e^14*x^5 + 330*d^7*e^13*x^4 + 165*d^8*e^12*x^3 + 55*d^9*e
^11*x^2 + 11*d^10*e^10*x + d^11*e^9)

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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((c*x**2+b*x+a)**4/(e*x+d)**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.203977, 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)^12,x, algorithm="giac")

[Out]

Done