3.2150 \(\int \frac{\left (a+b x+c x^2\right )^4}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=443 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{6 e^9 (d+e x)^6}-\frac{c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac{4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^9 (d+e x)^7}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^9 (d+e x)^8}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{9 e^9 (d+e x)^9}-\frac{\left (a e^2-b d e+c d^2\right )^4}{10 e^9 (d+e x)^{10}}+\frac{4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac{c^4}{2 e^9 (d+e x)^2} \]

[Out]

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

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Rubi [A]  time = 1.69163, antiderivative size = 443, 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 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{6 e^9 (d+e x)^6}-\frac{c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 e^9 (d+e x)^4}+\frac{4 c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{5 e^9 (d+e x)^5}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{7 e^9 (d+e x)^7}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{4 e^9 (d+e x)^8}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{9 e^9 (d+e x)^9}-\frac{\left (a e^2-b d e+c d^2\right )^4}{10 e^9 (d+e x)^{10}}+\frac{4 c^3 (2 c d-b e)}{3 e^9 (d+e x)^3}-\frac{c^4}{2 e^9 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^4/(d + e*x)^11,x]

[Out]

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

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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)**11,x)

[Out]

Timed out

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Mathematica [A]  time = 1.38959, size = 731, normalized size = 1.65 \[ -\frac{3 c^2 e^2 \left (2 a^2 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+3 b^2 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+2 c e^3 \left (7 a^3 e^3 \left (d^2+10 d e x+45 e^2 x^2\right )+9 a^2 b e^2 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+6 a b^2 e \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+2 b^3 \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )\right )+e^4 \left (126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+6 a b^3 e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )\right )+2 c^3 e \left (3 a e \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )+7 b \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )+14 c^4 \left (d^8+10 d^7 e x+45 d^6 e^2 x^2+120 d^5 e^3 x^3+210 d^4 e^4 x^4+252 d^3 e^5 x^5+210 d^2 e^6 x^6+120 d e^7 x^7+45 e^8 x^8\right )}{1260 e^9 (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^4/(d + e*x)^11,x]

[Out]

-(14*c^4*(d^8 + 10*d^7*e*x + 45*d^6*e^2*x^2 + 120*d^5*e^3*x^3 + 210*d^4*e^4*x^4
+ 252*d^3*e^5*x^5 + 210*d^2*e^6*x^6 + 120*d*e^7*x^7 + 45*e^8*x^8) + e^4*(126*a^4
*e^4 + 56*a^3*b*e^3*(d + 10*e*x) + 21*a^2*b^2*e^2*(d^2 + 10*d*e*x + 45*e^2*x^2)
+ 6*a*b^3*e*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + b^4*(d^4 + 10*d^3*
e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)) + 2*c*e^3*(7*a^3*e^3*(d^2 +
10*d*e*x + 45*e^2*x^2) + 9*a^2*b*e^2*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3
*x^3) + 6*a*b^2*e*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x
^4) + 2*b^3*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4
+ 252*e^5*x^5)) + 3*c^2*e^2*(2*a^2*e^2*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120
*d*e^3*x^3 + 210*e^4*x^4) + 4*a*b*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2
*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 3*b^2*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x
^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6)) + 2*c^3*e
*(3*a*e*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 +
252*d*e^5*x^5 + 210*e^6*x^6) + 7*b*(d^7 + 10*d^6*e*x + 45*d^5*e^2*x^2 + 120*d^4
*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120*e^7*x^7)))/(1
260*e^9*(d + e*x)^10)

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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)^11,x)

[Out]

-1/7*(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^3-80*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)^7-1/2*c^4/e^9/(e*x+d)^2-1/8*(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^4-12
0*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)^8-1/10*(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)^10-4/3*c^3*(b*e-2*c*d)/e
^9/(e*x+d)^3-4/5*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)^5-1/6*(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)^6-1/2*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)^4-1/9*(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)^9

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Maxima [A]  time = 0.860051, size = 1223, normalized size = 2.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^4/(e*x + d)^11,x, algorithm="maxima")

[Out]

-1/1260*(630*c^4*e^8*x^8 + 14*c^4*d^8 + 14*b*c^3*d^7*e + 56*a^3*b*d*e^7 + 126*a^
4*e^8 + 3*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 7*(3*a^2*b^2
+ 2*a^3*c)*d^2*e^6 + 1680*(c^4*d*e^7 + b*c^3*e^8)*x^7 + 210*(14*c^4*d^2*e^6 + 14
*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 252*(14*c^4*d^3*e^5 + 14*b*c^3
*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 4*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 210*
(14*c^4*d^4*e^4 + 14*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 4*(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 + 120*(14*c^4*d^5*e
^3 + 14*b*c^3*d^4*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*e^8)*x^3
+ 45*(14*c^4*d^6*e^2 + 14*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 4*(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 + 6*(a*b^3 +
3*a^2*b*c)*d*e^7 + 7*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 10*(14*c^4*d^7*e + 14*b*c^
3*d^6*e^2 + 56*a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^2*
e^6 + 7*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^19*x^10 + 10*d*e^18*x^9 + 45*d^2*e^17
*x^8 + 120*d^3*e^16*x^7 + 210*d^4*e^15*x^6 + 252*d^5*e^14*x^5 + 210*d^6*e^13*x^4
+ 120*d^7*e^12*x^3 + 45*d^8*e^11*x^2 + 10*d^9*e^10*x + d^10*e^9)

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Fricas [A]  time = 0.199988, size = 1223, normalized size = 2.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^4/(e*x + d)^11,x, algorithm="fricas")

[Out]

-1/1260*(630*c^4*e^8*x^8 + 14*c^4*d^8 + 14*b*c^3*d^7*e + 56*a^3*b*d*e^7 + 126*a^
4*e^8 + 3*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 7*(3*a^2*b^2
+ 2*a^3*c)*d^2*e^6 + 1680*(c^4*d*e^7 + b*c^3*e^8)*x^7 + 210*(14*c^4*d^2*e^6 + 14
*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 252*(14*c^4*d^3*e^5 + 14*b*c^3
*d^2*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 + 4*(b^3*c + 3*a*b*c^2)*e^8)*x^5 + 210*
(14*c^4*d^4*e^4 + 14*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 4*(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 + 120*(14*c^4*d^5*e
^3 + 14*b*c^3*d^4*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*e^8)*x^3
+ 45*(14*c^4*d^6*e^2 + 14*b*c^3*d^5*e^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 + 4*(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 + 6*(a*b^3 +
3*a^2*b*c)*d*e^7 + 7*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 10*(14*c^4*d^7*e + 14*b*c^
3*d^6*e^2 + 56*a^3*b*e^8 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 + 4*(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 + 6*(a*b^3 + 3*a^2*b*c)*d^2*
e^6 + 7*(3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^19*x^10 + 10*d*e^18*x^9 + 45*d^2*e^17
*x^8 + 120*d^3*e^16*x^7 + 210*d^4*e^15*x^6 + 252*d^5*e^14*x^5 + 210*d^6*e^13*x^4
+ 120*d^7*e^12*x^3 + 45*d^8*e^11*x^2 + 10*d^9*e^10*x + d^10*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)**11,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.205373, 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)^11,x, algorithm="giac")

[Out]

Done