3.2145 \(\int \frac{\left (a+b x+c x^2\right )^4}{(d+e x)^6} \, dx\)
Optimal. Leaf size=414 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac{4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac{c^3 x^2 (3 c d-2 b e)}{e^7}+\frac{c^4 x^3}{3 e^6} \]
[Out]
(c^2*(21*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(6*b*d - a*e))*x)/e^8 - (c^3*(3*c*d - 2*b*e
)*x^2)/e^7 + (c^4*x^3)/(3*e^6) - (c*d^2 - b*d*e + a*e^2)^4/(5*e^9*(d + e*x)^5) +
((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^4) - (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)))/(3*e^9*(d + e*x)^3)
+ (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)))/(e^9*(d + e*x)^2) - (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))/
(e^9*(d + e*x)) - (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))
*Log[d + e*x])/e^9
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Rubi [A] time = 1.75421, antiderivative size = 414, 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
\[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^9 (d+e x)}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9 (d+e x)^2}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^9 (d+e x)^3}-\frac{4 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{c^2 x \left (-4 c e (6 b d-a e)+6 b^2 e^2+21 c^2 d^2\right )}{e^8}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^4}{5 e^9 (d+e x)^5}-\frac{c^3 x^2 (3 c d-2 b e)}{e^7}+\frac{c^4 x^3}{3 e^6} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^4/(d + e*x)^6,x]
[Out]
(c^2*(21*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(6*b*d - a*e))*x)/e^8 - (c^3*(3*c*d - 2*b*e
)*x^2)/e^7 + (c^4*x^3)/(3*e^6) - (c*d^2 - b*d*e + a*e^2)^4/(5*e^9*(d + e*x)^5) +
((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^4) - (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)))/(3*e^9*(d + e*x)^3)
+ (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)))/(e^9*(d + e*x)^2) - (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))/
(e^9*(d + e*x)) - (4*c*(2*c*d - b*e)*(7*c^2*d^2 + b^2*e^2 - c*e*(7*b*d - 3*a*e))
*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)**6,x)
[Out]
Timed out
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Mathematica [A] time = 0.532114, size = 419, normalized size = 1.01 \[ \frac{\frac{30 (2 c d-b e) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )}{(d+e x)^2}-\frac{15 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{d+e x}+15 c^2 e x \left (4 c e (a e-6 b d)+6 b^2 e^2+21 c^2 d^2\right )-\frac{10 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^3}-60 c (2 c d-b e) \log (d+e x) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )+\frac{15 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^4}-\frac{3 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^5}+15 c^3 e^2 x^2 (2 b e-3 c d)+5 c^4 e^3 x^3}{15 e^9} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^4/(d + e*x)^6,x]
[Out]
(15*c^2*e*(21*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(-6*b*d + a*e))*x + 15*c^3*e^2*(-3*c*d
+ 2*b*e)*x^2 + 5*c^4*e^3*x^3 - (3*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^5 + (
15*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^4 - (10*(14*c^2*d^2 + 3
*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^3 + (30
*(2*c*d - b*e)*(7*c^3*d^4 - 2*c^2*d^2*e*(7*b*d - 5*a*e) + b^2*e^3*(-(b*d) + a*e)
+ c*e^2*(8*b^2*d^2 - 10*a*b*d*e + 3*a^2*e^2)))/(d + e*x)^2 - (15*(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) - 60*c*(2*c*d - b*e)*(7*c^2*d^2
+ b^2*e^2 + c*e*(-7*b*d + 3*a*e))*Log[d + e*x])/(15*e^9)
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Maple [B] time = 0.026, size = 1341, normalized size = 3.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^4/(e*x+d)^6,x)
[Out]
-1/5/e^5/(e*x+d)^5*d^4*b^4-1/5/e^9/(e*x+d)^5*c^4*d^8-6/e^5/(e*x+d)*a^2*c^2-70/e^
9/(e*x+d)*c^4*d^4-1/e^2/(e*x+d)^4*a^3*b+1/e^5/(e*x+d)^4*b^4*d^3+2/e^9/(e*x+d)^4*
c^4*d^7+2*c^3/e^6*x^2*b-3*c^4/e^7*x^2*d+4*c^3/e^6*a*x+6*c^2/e^6*b^2*x+21*c^4/e^8
*d^2*x-2/e^4/(e*x+d)^2*a*b^3+2/e^5/(e*x+d)^2*b^4*d+28/e^9/(e*x+d)^2*c^4*d^5-4/3/
e^3/(e*x+d)^3*a^3*c-2/e^3/(e*x+d)^3*a^2*b^2-2/e^5/(e*x+d)^3*b^4*d^2-28/3/e^9/(e*
x+d)^3*c^4*d^6+4*c/e^6*ln(e*x+d)*b^3-56*c^4/e^9*ln(e*x+d)*d^3-b^4/e^5/(e*x+d)-1/
5/e/(e*x+d)^5*a^4+4/5/e^6/(e*x+d)^5*d^5*b^3*c-6/5/e^7/(e*x+d)^5*b^2*c^2*d^6+4/5/
e^8/(e*x+d)^5*b*c^3*d^7-12/e^5/(e*x+d)*a*b^2*c-60/e^7/(e*x+d)*a*c^3*d^2+20/e^6/(
e*x+d)*b^3*c*d-90/e^7/(e*x+d)*b^2*c^2*d^2+140/e^8/(e*x+d)*b*c^3*d^3-3/e^4/(e*x+d
)^4*a*b^3*d^2+6/e^7/(e*x+d)^4*a*c^3*d^5-5/e^6/(e*x+d)^4*b^3*c*d^4+9/e^7/(e*x+d)^
4*b^2*c^2*d^5-6/e^4/(e*x+d)^2*a^2*b*c+12/e^5/(e*x+d)^2*a^2*d*c^2+40/e^7/(e*x+d)^
2*c^3*d^3*a-20/e^6/(e*x+d)^2*b^3*c*d^2+60/e^7/(e*x+d)^2*b^2*c^2*d^3-70/e^8/(e*x+
d)^2*b*c^3*d^4-7/e^8/(e*x+d)^4*b*c^3*d^6-24*c^3/e^7*b*d*x+6/e^5/(e*x+d)^4*a^2*c^
2*d^3-12/e^5/(e*x+d)^3*a^2*c^2*d^2+4/e^4/(e*x+d)^3*a*b^3*d-20/e^7/(e*x+d)^3*a*c^
3*d^4+40/3/e^6/(e*x+d)^3*b^3*c*d^3-30/e^7/(e*x+d)^3*b^2*c^2*d^4+28/e^8/(e*x+d)^3
*b*c^3*d^5+12*c^2/e^6*ln(e*x+d)*a*b-24*c^3/e^7*ln(e*x+d)*a*d-36*c^2/e^7*ln(e*x+d
)*b^2*d+84*c^3/e^8*ln(e*x+d)*b*d^2+4/5/e^2/(e*x+d)^5*d*a^3*b-4/5/e^3/(e*x+d)^5*a
^3*c*d^2-6/5/e^3/(e*x+d)^5*d^2*a^2*b^2-6/5/e^5/(e*x+d)^5*a^2*c^2*d^4+4/5/e^4/(e*
x+d)^5*d^3*a*b^3-4/5/e^7/(e*x+d)^5*a*c^3*d^6+1/3*c^4*x^3/e^6+2/e^3/(e*x+d)^4*a^3
*c*d+3/e^3/(e*x+d)^4*a^2*b^2*d+12/5/e^6/(e*x+d)^5*d^5*a*b*c^2+24/e^5/(e*x+d)^2*a
*b^2*c*d-60/e^6/(e*x+d)^2*a*b*c^2*d^2+12/e^4/(e*x+d)^3*a^2*b*c*d-24/e^5/(e*x+d)^
3*a*b^2*c*d^2+40/e^6/(e*x+d)^3*a*b*c^2*d^3+12/5/e^4/(e*x+d)^5*d^3*a^2*b*c-12/5/e
^5/(e*x+d)^5*d^4*a*b^2*c-9/e^4/(e*x+d)^4*a^2*b*c*d^2+12/e^5/(e*x+d)^4*a*b^2*c*d^
3-15/e^6/(e*x+d)^4*a*b*c^2*d^4+60/e^6/(e*x+d)*a*b*c^2*d
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Maxima [A] time = 0.85911, size = 1148, 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)^6,x, algorithm="maxima")
[Out]
-1/15*(743*c^4*d^8 - 1377*b*c^3*d^7*e + 3*a^3*b*d*e^7 + 3*a^4*e^8 + 261*(3*b^2*c
^2 + 2*a*c^3)*d^6*e^2 - 137*(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 + 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 + 15*(70*c^4*d^4*e^4 - 140*b*c^3*d^3*e^5 + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 -
20*(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 + 30*(12
6*c^4*d^5*e^3 - 245*b*c^3*d^4*e^4 + 50*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 30*(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 + (a*b^3 + 3*a^2*b*
c)*e^8)*x^3 + 10*(518*c^4*d^6*e^2 - 987*b*c^3*d^5*e^3 + 195*(3*b^2*c^2 + 2*a*c^3
)*d^4*e^4 - 110*(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 + 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 + 5*(638*c
^4*d^7*e - 1197*b*c^3*d^6*e^2 + 3*a^3*b*e^8 + 231*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3
- 125*(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 + 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)/(e^14*x^5 + 5*d*e
^13*x^4 + 10*d^2*e^12*x^3 + 10*d^3*e^11*x^2 + 5*d^4*e^10*x + d^5*e^9) + 1/3*(c^4
*e^2*x^3 - 3*(3*c^4*d*e - 2*b*c^3*e^2)*x^2 + 3*(21*c^4*d^2 - 24*b*c^3*d*e + 2*(3
*b^2*c^2 + 2*a*c^3)*e^2)*x)/e^8 - 4*(14*c^4*d^3 - 21*b*c^3*d^2*e + 3*(3*b^2*c^2
+ 2*a*c^3)*d*e^2 - (b^3*c + 3*a*b*c^2)*e^3)*log(e*x + d)/e^9
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Fricas [A] time = 0.213835, size = 1704, normalized size = 4.12 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4/(e*x + d)^6,x, algorithm="fricas")
[Out]
1/15*(5*c^4*e^8*x^8 - 743*c^4*d^8 + 1377*b*c^3*d^7*e - 3*a^3*b*d*e^7 - 3*a^4*e^8
- 261*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 137*(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 - 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 - 10*(2*c^4*d*e^7 - 3*b*c^3*e^8)*x^7 + 10*(14*c^4*d^2*e^6 - 21
*b*c^3*d*e^7 + 3*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 + 25*(47*c^4*d^3*e^5 - 60*b*c^3*
d^2*e^6 + 6*(3*b^2*c^2 + 2*a*c^3)*d*e^7)*x^5 + 5*(335*c^4*d^4*e^4 - 240*b*c^3*d^
3*e^5 - 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 + 60*(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 - 10*(85*c^4*d^5*e^3 - 390*b*c^3*d^4*e^4 + 1
20*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 90*(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 + 3*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 - 10*(365*c^4*d^6*e^
2 - 810*b*c^3*d^5*e^3 + 180*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 110*(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 + 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 - 5*(575*c^4*d^7*e - 1125*b*c^3*d^6*e^2 +
3*a^3*b*e^8 + 225*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 125*(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 + 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 - 60*(14*c^4*d^8 - 21*b*c^3*d^7*e + 3*(3*b^2*c^2 +
2*a*c^3)*d^6*e^2 - (b^3*c + 3*a*b*c^2)*d^5*e^3 + (14*c^4*d^3*e^5 - 21*b*c^3*d^2
*e^6 + 3*(3*b^2*c^2 + 2*a*c^3)*d*e^7 - (b^3*c + 3*a*b*c^2)*e^8)*x^5 + 5*(14*c^4*
d^4*e^4 - 21*b*c^3*d^3*e^5 + 3*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - (b^3*c + 3*a*b*c^
2)*d*e^7)*x^4 + 10*(14*c^4*d^5*e^3 - 21*b*c^3*d^4*e^4 + 3*(3*b^2*c^2 + 2*a*c^3)*
d^3*e^5 - (b^3*c + 3*a*b*c^2)*d^2*e^6)*x^3 + 10*(14*c^4*d^6*e^2 - 21*b*c^3*d^5*e
^3 + 3*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - (b^3*c + 3*a*b*c^2)*d^3*e^5)*x^2 + 5*(14*
c^4*d^7*e - 21*b*c^3*d^6*e^2 + 3*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - (b^3*c + 3*a*b*
c^2)*d^4*e^4)*x)*log(e*x + d))/(e^14*x^5 + 5*d*e^13*x^4 + 10*d^2*e^12*x^3 + 10*d
^3*e^11*x^2 + 5*d^4*e^10*x + d^5*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)**6,x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.203619, 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)^6,x, algorithm="giac")
[Out]
Done