3.2149 \(\int \frac{\left (a+b x+c x^2\right )^4}{(d+e x)^{10}} \, dx\)
Optimal. Leaf size=436 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{5 e^9 (d+e x)^5}-\frac{2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}+\frac{c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^9 (d+e x)^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{2 e^9 (d+e x)^8}-\frac{\left (a e^2-b d e+c d^2\right )^4}{9 e^9 (d+e x)^9}+\frac{2 c^3 (2 c d-b e)}{e^9 (d+e x)^2}-\frac{c^4}{e^9 (d+e x)} \]
[Out]
-(c*d^2 - b*d*e + a*e^2)^4/(9*e^9*(d + e*x)^9) + ((2*c*d - b*e)*(c*d^2 - b*d*e +
a*e^2)^3)/(2*e^9*(d + e*x)^8) - (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)))/(7*e^9*(d + e*x)^7) + (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)))/(3*e^9*(d + e*x)^6) -
(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))/(5*e^9*(d + e*x)^5) + (c*(2*
c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(e^9*(d + e*x)^4) - (2*c
^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(3*e^9*(d + e*x)^3) + (2*c^3*
(2*c*d - b*e))/(e^9*(d + e*x)^2) - c^4/(e^9*(d + e*x))
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Rubi [A] time = 1.68196, antiderivative size = 436, 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
\[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{5 e^9 (d+e x)^5}-\frac{2 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}+\frac{c (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^4}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{3 e^9 (d+e x)^6}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{7 e^9 (d+e x)^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{2 e^9 (d+e x)^8}-\frac{\left (a e^2-b d e+c d^2\right )^4}{9 e^9 (d+e x)^9}+\frac{2 c^3 (2 c d-b e)}{e^9 (d+e x)^2}-\frac{c^4}{e^9 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^4/(d + e*x)^10,x]
[Out]
-(c*d^2 - b*d*e + a*e^2)^4/(9*e^9*(d + e*x)^9) + ((2*c*d - b*e)*(c*d^2 - b*d*e +
a*e^2)^3)/(2*e^9*(d + e*x)^8) - (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)))/(7*e^9*(d + e*x)^7) + (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)))/(3*e^9*(d + e*x)^6) -
(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))/(5*e^9*(d + e*x)^5) + (c*(2*
c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e)))/(e^9*(d + e*x)^4) - (2*c
^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e)))/(3*e^9*(d + e*x)^3) + (2*c^3*
(2*c*d - b*e))/(e^9*(d + e*x)^2) - c^4/(e^9*(d + e*x))
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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)**10,x)
[Out]
Timed out
_______________________________________________________________________________________
Mathematica [A] time = 1.22842, size = 730, normalized size = 1.67 \[ -\frac{3 c^2 e^2 \left (2 a^2 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 a b e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+5 b^2 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )\right )+c e^3 \left (10 a^3 e^3 \left (d^2+9 d e x+36 e^2 x^2\right )+15 a^2 b e^2 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+12 a b^2 e \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+5 b^3 \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )\right )+e^4 \left (70 a^4 e^4+35 a^3 b e^3 (d+9 e x)+15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b^3 e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )\right )+5 c^3 e \left (2 a e \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )+7 b \left (d^7+9 d^6 e x+36 d^5 e^2 x^2+84 d^4 e^3 x^3+126 d^3 e^4 x^4+126 d^2 e^5 x^5+84 d e^6 x^6+36 e^7 x^7\right )\right )+70 c^4 \left (d^8+9 d^7 e x+36 d^6 e^2 x^2+84 d^5 e^3 x^3+126 d^4 e^4 x^4+126 d^3 e^5 x^5+84 d^2 e^6 x^6+36 d e^7 x^7+9 e^8 x^8\right )}{630 e^9 (d+e x)^9} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^4/(d + e*x)^10,x]
[Out]
-(70*c^4*(d^8 + 9*d^7*e*x + 36*d^6*e^2*x^2 + 84*d^5*e^3*x^3 + 126*d^4*e^4*x^4 +
126*d^3*e^5*x^5 + 84*d^2*e^6*x^6 + 36*d*e^7*x^7 + 9*e^8*x^8) + e^4*(70*a^4*e^4 +
35*a^3*b*e^3*(d + 9*e*x) + 15*a^2*b^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b^
3*e*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + b^4*(d^4 + 9*d^3*e*x + 36*d^
2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4)) + c*e^3*(10*a^3*e^3*(d^2 + 9*d*e*x + 36
*e^2*x^2) + 15*a^2*b*e^2*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 12*a*b^
2*e*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 5*b^3*(d^5
+ 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5)) +
3*c^2*e^2*(2*a^2*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4
*x^4) + 5*a*b*e*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x
^4 + 126*e^5*x^5) + 5*b^2*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 1
26*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6)) + 5*c^3*e*(2*a*e*(d^6 + 9*d^5*e*x
+ 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^5*x^5 + 84*e^6*x^6
) + 7*b*(d^7 + 9*d^6*e*x + 36*d^5*e^2*x^2 + 84*d^4*e^3*x^3 + 126*d^3*e^4*x^4 + 1
26*d^2*e^5*x^5 + 84*d*e^6*x^6 + 36*e^7*x^7)))/(630*e^9*(d + e*x)^9)
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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)^10,x)
[Out]
-1/7*(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)^7-2*c^3*(b*e
-2*c*d)/e^9/(e*x+d)^2-1/8*(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)^8-2/3*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)^3-1/5*(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)^5-1/6*(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)^6-c^4/e^9/(e*x+d)-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)^4-1/9*(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)^9
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Maxima [A] time = 0.86475, size = 1206, normalized size = 2.77 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4/(e*x + d)^10,x, algorithm="maxima")
[Out]
-1/630*(630*c^4*e^8*x^8 + 70*c^4*d^8 + 35*b*c^3*d^7*e + 35*a^3*b*d*e^7 + 70*a^4*
e^8 + 5*(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 + 1
2*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 5*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 5*(3*a^2*b^2 +
2*a^3*c)*d^2*e^6 + 1260*(2*c^4*d*e^7 + b*c^3*e^8)*x^7 + 420*(14*c^4*d^2*e^6 + 7*
b*c^3*d*e^7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 630*(14*c^4*d^3*e^5 + 7*b*c^3*d^2
*e^6 + (3*b^2*c^2 + 2*a*c^3)*d*e^7 + (b^3*c + 3*a*b*c^2)*e^8)*x^5 + 126*(70*c^4*
d^4*e^4 + 35*b*c^3*d^3*e^5 + 5*(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 + 84*(70*c^4*d^5*e^3 + 35*b
*c^3*d^4*e^4 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 36*(70*
c^4*d^6*e^2 + 35*b*c^3*d^5*e^3 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c
)*d*e^7 + 5*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 9*(70*c^4*d^7*e + 35*b*c^3*d^6*e^2
+ 35*a^3*b*e^8 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 5*(3
*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^18*x^9 + 9*d*e^17*x^8 + 36*d^2*e^16*x^7 + 84*d^
3*e^15*x^6 + 126*d^4*e^14*x^5 + 126*d^5*e^13*x^4 + 84*d^6*e^12*x^3 + 36*d^7*e^11
*x^2 + 9*d^8*e^10*x + d^9*e^9)
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Fricas [A] time = 0.207541, size = 1206, normalized size = 2.77 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4/(e*x + d)^10,x, algorithm="fricas")
[Out]
-1/630*(630*c^4*e^8*x^8 + 70*c^4*d^8 + 35*b*c^3*d^7*e + 35*a^3*b*d*e^7 + 70*a^4*
e^8 + 5*(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 + 1
2*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 5*(a*b^3 + 3*a^2*b*c)*d^3*e^5 + 5*(3*a^2*b^2 +
2*a^3*c)*d^2*e^6 + 1260*(2*c^4*d*e^7 + b*c^3*e^8)*x^7 + 420*(14*c^4*d^2*e^6 + 7*
b*c^3*d*e^7 + (3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 630*(14*c^4*d^3*e^5 + 7*b*c^3*d^2
*e^6 + (3*b^2*c^2 + 2*a*c^3)*d*e^7 + (b^3*c + 3*a*b*c^2)*e^8)*x^5 + 126*(70*c^4*
d^4*e^4 + 35*b*c^3*d^3*e^5 + 5*(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 + 84*(70*c^4*d^5*e^3 + 35*b
*c^3*d^4*e^4 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 36*(70*
c^4*d^6*e^2 + 35*b*c^3*d^5*e^3 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c
)*d*e^7 + 5*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 9*(70*c^4*d^7*e + 35*b*c^3*d^6*e^2
+ 35*a^3*b*e^8 + 5*(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 + 5*(a*b^3 + 3*a^2*b*c)*d^2*e^6 + 5*(3
*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^18*x^9 + 9*d*e^17*x^8 + 36*d^2*e^16*x^7 + 84*d^
3*e^15*x^6 + 126*d^4*e^14*x^5 + 126*d^5*e^13*x^4 + 84*d^6*e^12*x^3 + 36*d^7*e^11
*x^2 + 9*d^8*e^10*x + d^9*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)**10,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.204314, 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)^10,x, algorithm="giac")
[Out]
Done