3.2143 \(\int \frac{\left (a+b x+c x^2\right )^4}{(d+e x)^4} \, dx\)
Optimal. Leaf size=417 \[ \frac{x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}-\frac{2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac{c^3 x^4 (c d-b e)}{e^5}+\frac{c^4 x^5}{5 e^4} \]
[Out]
((35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) - 40*c^3*d^2*e*(2*b*d - a*e
) + 6*c^2*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*x)/e^8 - (2*c*(5*c^3*d^3 - b^3
*e^3 - 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(2*b*d - a*e))*x^2)/e^7 + (2*c^2*(5
*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(4*b*d - a*e))*x^3)/(3*e^6) - (c^3*(c*d - b*e)*x^4)
/e^5 + (c^4*x^5)/(5*e^4) - (c*d^2 - b*d*e + a*e^2)^4/(3*e^9*(d + e*x)^3) + (2*(2
*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^2) - (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)))/(e^9*(d + e*x)) - (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))*L
og[d + e*x])/e^9
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Rubi [A] time = 1.88438, antiderivative size = 417, 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
\[ \frac{x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)-40 c^3 d^2 e (2 b d-a e)+b^4 e^4+35 c^4 d^4\right )}{e^8}-\frac{2 c x^2 \left (-2 c^2 d e (5 b d-2 a e)+3 b c e^2 (2 b d-a e)-b^3 e^3+5 c^3 d^3\right )}{e^7}-\frac{2 \left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{e^9 (d+e x)}-\frac{4 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^9}+\frac{2 c^2 x^3 \left (-2 c e (4 b d-a e)+3 b^2 e^2+5 c^2 d^2\right )}{3 e^6}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{e^9 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^4}{3 e^9 (d+e x)^3}-\frac{c^3 x^4 (c d-b e)}{e^5}+\frac{c^4 x^5}{5 e^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^4/(d + e*x)^4,x]
[Out]
((35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) - 40*c^3*d^2*e*(2*b*d - a*e
) + 6*c^2*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*x)/e^8 - (2*c*(5*c^3*d^3 - b^3
*e^3 - 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(2*b*d - a*e))*x^2)/e^7 + (2*c^2*(5
*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(4*b*d - a*e))*x^3)/(3*e^6) - (c^3*(c*d - b*e)*x^4)
/e^5 + (c^4*x^5)/(5*e^4) - (c*d^2 - b*d*e + a*e^2)^4/(3*e^9*(d + e*x)^3) + (2*(2
*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(e^9*(d + e*x)^2) - (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)))/(e^9*(d + e*x)) - (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))*L
og[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)**4,x)
[Out]
Timed out
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Mathematica [A] time = 0.458252, size = 425, normalized size = 1.02 \[ \frac{-60 (2 c d-b e) \log (d+e x) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )+15 e x \left (6 c^2 e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-4 b^2 c e^3 (4 b d-3 a e)+40 c^3 d^2 e (a e-2 b d)+b^4 e^4+35 c^4 d^4\right )+30 c e^2 x^2 \left (2 c^2 d e (5 b d-2 a e)+3 b c e^2 (a e-2 b d)+b^3 e^3-5 c^3 d^3\right )-\frac{30 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{d+e x}+10 c^2 e^3 x^3 \left (2 c e (a e-4 b d)+3 b^2 e^2+5 c^2 d^2\right )+\frac{30 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}-\frac{5 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^3}+15 c^3 e^4 x^4 (b e-c d)+3 c^4 e^5 x^5}{15 e^9} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^4/(d + e*x)^4,x]
[Out]
(15*e*(35*c^4*d^4 + b^4*e^4 - 4*b^2*c*e^3*(4*b*d - 3*a*e) + 40*c^3*d^2*e*(-2*b*d
+ a*e) + 6*c^2*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2))*x + 30*c*e^2*(-5*c^3*d^3
+ b^3*e^3 + 2*c^2*d*e*(5*b*d - 2*a*e) + 3*b*c*e^2*(-2*b*d + a*e))*x^2 + 10*c^2*
e^3*(5*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-4*b*d + a*e))*x^3 + 15*c^3*e^4*(-(c*d) + b*
e)*x^4 + 3*c^4*e^5*x^5 - (5*(c*d^2 + e*(-(b*d) + a*e))^4)/(d + e*x)^3 + (30*(2*c
*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^2 - (30*(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) - 60*(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))*Log[d + e*x])/(15*e^9)
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Maple [B] time = 0.027, size = 1265, normalized size = 3. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^4/(e*x+d)^4,x)
[Out]
20/e^6*x^2*b*c^3*d^2+12/e^4*a*b^2*c*x-8/e^5*x^2*a*c^3*d-12/e^5*x^2*b^2*c^2*d+4/3
/e^4/(e*x+d)^3*d^3*a*b^3+6*b^2/e^3/(e*x+d)^2*a^2*d+b^4*x/e^4+4/3/e^2/(e*x+d)^3*d
*a^3*b-2/e^3/(e*x+d)^3*d^2*a^2*b^2+24/e^5/(e*x+d)^2*a*b^2*c*d^3-30/e^6/(e*x+d)^2
*a*b*c^2*d^4+4/e^4/(e*x+d)^3*d^3*a^2*b*c-4/e^5/(e*x+d)^3*a*b^2*c*d^4+4/e^6/(e*x+
d)^3*a*b*c^2*d^5-48/e^5*ln(e*x+d)*a*b^2*c*d+120/e^6*ln(e*x+d)*a*b*c^2*d^2+36/e^4
/(e*x+d)*a^2*b*c*d-72/e^5/(e*x+d)*a*b^2*c*d^2+120/e^6/(e*x+d)*a*b*c^2*d^3-48/e^5
*a*b*c^2*d*x-18/e^4/(e*x+d)^2*a^2*b*c*d^2+40/e^6*a*c^3*d^2*x-16/e^5*b^3*c*d*x+60
/e^6*b^2*c^2*d^2*x+12/e^4*ln(e*x+d)*a^2*b*c-24/e^5*ln(e*x+d)*a^2*d*c^2-80/e^7*ln
(e*x+d)*c^3*d^3*a+40/e^6*ln(e*x+d)*b^3*c*d^2-120/e^7*ln(e*x+d)*b^2*c^2*d^3+140/e
^8*ln(e*x+d)*b*c^3*d^4-36/e^5/(e*x+d)*a^2*c^2*d^2-60/e^7/(e*x+d)*a*c^3*d^4+40/e^
6/(e*x+d)*b^3*c*d^3-90/e^7/(e*x+d)*b^2*c^2*d^4+84/e^8/(e*x+d)*b*c^3*d^5-16/3/e^5
*x^3*b*c^3*d+6/e^4*x^2*a*b*c^2-10/e^6/(e*x+d)^2*b^3*c*d^4+18/e^7/(e*x+d)^2*b^2*c
^2*d^5-14/e^8/(e*x+d)^2*b*c^3*d^6-4/3/e^3/(e*x+d)^3*a^3*c*d^2-2/e^5/(e*x+d)^3*a^
2*c^2*d^4-4/3/e^7/(e*x+d)^3*a*c^3*d^6+4/3/e^6/(e*x+d)^3*b^3*c*d^5-2/e^7/(e*x+d)^
3*b^2*c^2*d^6+4/3/e^8/(e*x+d)^3*b*c^3*d^7-1/3/e^5/(e*x+d)^3*b^4*d^4+4*b^3/e^4*ln
(e*x+d)*a-4*b^4/e^5*ln(e*x+d)*d-2*b/e^2/(e*x+d)^2*a^3+2*b^4/e^5/(e*x+d)^2*d^3-6*
b^2/e^3/(e*x+d)*a^2-6*b^4/e^5/(e*x+d)*d^2-6*b^3/e^4/(e*x+d)^2*a*d^2+12*b^3/e^4/(
e*x+d)*d*a-80/e^7*b*c^3*d^3*x+4/e^3/(e*x+d)^2*a^3*c*d+12/e^5/(e*x+d)^2*a^2*c^2*d
^3+12/e^7/(e*x+d)^2*a*c^3*d^5+1/5*c^4*x^5/e^4+1/e^4*x^4*b*c^3-1/e^5*x^4*c^4*d+4/
3/e^4*x^3*a*c^3+2/e^4*x^3*b^2*c^2+10/3/e^6*x^3*c^4*d^2+2/e^4*x^2*b^3*c-10/e^7*x^
2*c^4*d^3+6/e^4*a^2*c^2*x+35/e^8*c^4*d^4*x+4/e^9/(e*x+d)^2*c^4*d^7-1/3/e^9/(e*x+
d)^3*c^4*d^8-56/e^9*ln(e*x+d)*c^4*d^5-4/e^3/(e*x+d)*a^3*c-28/e^9/(e*x+d)*c^4*d^6
-1/3/e/(e*x+d)^3*a^4
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Maxima [A] time = 0.854754, size = 1116, normalized size = 2.68 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4/(e*x + d)^4,x, algorithm="maxima")
[Out]
-1/3*(73*c^4*d^8 - 214*b*c^3*d^7*e + 2*a^3*b*d*e^7 + a^4*e^8 + 74*(3*b^2*c^2 + 2
*a*c^3)*d^6*e^2 - 94*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 13*(b^4 + 12*a*b^2*c + 6*a^2*
c^2)*d^4*e^4 - 22*(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
+ 6*(14*c^4*d^6*e^2 - 42*b*c^3*d^5*e^3 + 15*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 20*(
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 - 6*(a*b^3
+ 3*a^2*b*c)*d*e^7 + (3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 + 6*(26*c^4*d^7*e - 77*b*c^
3*d^6*e^2 + a^3*b*e^8 + 27*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 35*(b^3*c + 3*a*b*c^2
)*d^4*e^4 + 5*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 9*(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^12*x^3 + 3*d*e^11*x^2 + 3*d^2*e^10*x +
d^3*e^9) + 1/15*(3*c^4*e^4*x^5 - 15*(c^4*d*e^3 - b*c^3*e^4)*x^4 + 10*(5*c^4*d^2
*e^2 - 8*b*c^3*d*e^3 + (3*b^2*c^2 + 2*a*c^3)*e^4)*x^3 - 30*(5*c^4*d^3*e - 10*b*c
^3*d^2*e^2 + 2*(3*b^2*c^2 + 2*a*c^3)*d*e^3 - (b^3*c + 3*a*b*c^2)*e^4)*x^2 + 15*(
35*c^4*d^4 - 80*b*c^3*d^3*e + 20*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 16*(b^3*c + 3*a
*b*c^2)*d*e^3 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^4)*x)/e^8 - 4*(14*c^4*d^5 - 35*
b*c^3*d^4*e + 10*(3*b^2*c^2 + 2*a*c^3)*d^3*e^2 - 10*(b^3*c + 3*a*b*c^2)*d^2*e^3
+ (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^4 - (a*b^3 + 3*a^2*b*c)*e^5)*log(e*x + d)/e
^9
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Fricas [A] time = 0.223648, size = 1731, normalized size = 4.15 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^4/(e*x + d)^4,x, algorithm="fricas")
[Out]
1/15*(3*c^4*e^8*x^8 - 365*c^4*d^8 + 1070*b*c^3*d^7*e - 10*a^3*b*d*e^7 - 5*a^4*e^
8 - 370*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 + 470*(b^3*c + 3*a*b*c^2)*d^5*e^3 - 65*(b^
4 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 + 110*(a*b^3 + 3*a^2*b*c)*d^3*e^5 - 10*(3*a^
2*b^2 + 2*a^3*c)*d^2*e^6 - 3*(2*c^4*d*e^7 - 5*b*c^3*e^8)*x^7 + (14*c^4*d^2*e^6 -
35*b*c^3*d*e^7 + 10*(3*b^2*c^2 + 2*a*c^3)*e^8)*x^6 - 3*(14*c^4*d^3*e^5 - 35*b*c
^3*d^2*e^6 + 10*(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 +
15*(14*c^4*d^4*e^4 - 35*b*c^3*d^3*e^5 + 10*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*e^8)*x^4 + 5*(235*c^4*d
^5*e^3 - 556*b*c^3*d^4*e^4 + 146*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 126*(b^3*c + 3*
a*b*c^2)*d^2*e^6 + 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d*e^7)*x^3 + 15*(67*c^4*d^6*
e^2 - 136*b*c^3*d^5*e^3 + 26*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 6*(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 + 12*(a*b^3 + 3*a^2*b*c)*d
*e^7 - 2*(3*a^2*b^2 + 2*a^3*c)*e^8)*x^2 - 15*(17*c^4*d^7*e - 74*b*c^3*d^6*e^2 +
2*a^3*b*e^8 + 34*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 54*(b^3*c + 3*a*b*c^2)*d^4*e^4
+ 9*(b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 18*(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 - 60*(14*c^4*d^8 - 35*b*c^3*d^7*e + 10*(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 + (b^4 + 12*a*b^2*c + 6*a^2
*c^2)*d^4*e^4 - (a*b^3 + 3*a^2*b*c)*d^3*e^5 + (14*c^4*d^5*e^3 - 35*b*c^3*d^4*e^4
+ 10*(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 + (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 + 3*(14*c^4*d^6*e^2 -
35*b*c^3*d^5*e^3 + 10*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6 - (a*b^3 + 3*a^2*b*c)*d*e^7)*x^2
+ 3*(14*c^4*d^7*e - 35*b*c^3*d^6*e^2 + 10*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - (a*b^3 + 3*a
^2*b*c)*d^2*e^6)*x)*log(e*x + d))/(e^12*x^3 + 3*d*e^11*x^2 + 3*d^2*e^10*x + d^3*
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)**4,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.204237, 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)^4,x, algorithm="giac")
[Out]
Done