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3.2144 \(\int \frac{\left (a+b x+c x^2\right )^4}{(d+e x)^5} \, dx\)

Optimal. Leaf size=426 \[ \frac{\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9}-\frac{c x \left (-20 c^2 d e (3 b d-a e)+6 b c e^2 (5 b d-2 a e)-4 b^3 e^3+35 c^3 d^3\right )}{e^8}-\frac{\left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )^2}{e^9 (d+e x)^2}+\frac{4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^9 (d+e x)}+\frac{c^2 x^2 \left (-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right )}{2 e^7}-\frac{\left (a e^2-b d e+c d^2\right )^4}{4 e^9 (d+e x)^4}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^3}-\frac{c^3 x^3 (5 c d-4 b e)}{3 e^6}+\frac{c^4 x^4}{4 e^5} \]

[Out]

-((c*(35*c^3*d^3 - 4*b^3*e^3 + 6*b*c*e^2*(5*b*d - 2*a*e) - 20*c^2*d*e*(3*b*d - a
*e))*x)/e^8) + (c^2*(15*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(5*b*d - a*e))*x^2)/(2*e^7)
- (c^3*(5*c*d - 4*b*e)*x^3)/(3*e^6) + (c^4*x^4)/(4*e^5) - (c*d^2 - b*d*e + a*e^2
)^4/(4*e^9*(d + e*x)^4) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^9*(d
+ e*x)^3) - ((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)^2) + (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)))/(e^9*(d + e*x)) + ((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))*Log[d + e*x])/e^9

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Rubi [A]  time = 1.84338, antiderivative size = 426, 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 \[ \frac{\log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{e^9}-\frac{c x \left (-20 c^2 d e (3 b d-a e)+6 b c e^2 (5 b d-2 a e)-4 b^3 e^3+35 c^3 d^3\right )}{e^8}-\frac{\left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )^2}{e^9 (d+e x)^2}+\frac{4 (2 c d-b e) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^9 (d+e x)}+\frac{c^2 x^2 \left (-4 c e (5 b d-a e)+6 b^2 e^2+15 c^2 d^2\right )}{2 e^7}-\frac{\left (a e^2-b d e+c d^2\right )^4}{4 e^9 (d+e x)^4}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{3 e^9 (d+e x)^3}-\frac{c^3 x^3 (5 c d-4 b e)}{3 e^6}+\frac{c^4 x^4}{4 e^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^4/(d + e*x)^5,x]

[Out]

-((c*(35*c^3*d^3 - 4*b^3*e^3 + 6*b*c*e^2*(5*b*d - 2*a*e) - 20*c^2*d*e*(3*b*d - a
*e))*x)/e^8) + (c^2*(15*c^2*d^2 + 6*b^2*e^2 - 4*c*e*(5*b*d - a*e))*x^2)/(2*e^7)
- (c^3*(5*c*d - 4*b*e)*x^3)/(3*e^6) + (c^4*x^4)/(4*e^5) - (c*d^2 - b*d*e + a*e^2
)^4/(4*e^9*(d + e*x)^4) + (4*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^3)/(3*e^9*(d
+ e*x)^3) - ((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)^2) + (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)))/(e^9*(d + e*x)) + ((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))*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)**5,x)

[Out]

Timed out

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Mathematica [A]  time = 0.468212, size = 430, normalized size = 1.01 \[ \frac{\frac{48 (2 c d-b e) \left (c e^2 \left (3 a^2 e^2-10 a b d e+8 b^2 d^2\right )+b^2 e^3 (a e-b d)-2 c^2 d^2 e (7 b d-5 a e)+7 c^3 d^4\right )}{d+e x}+12 \log (d+e x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )+12 c e x \left (20 c^2 d e (3 b d-a e)-6 b c e^2 (5 b d-2 a e)+4 b^3 e^3-35 c^3 d^3\right )+6 c^2 e^2 x^2 \left (4 c e (a e-5 b d)+6 b^2 e^2+15 c^2 d^2\right )-\frac{12 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \left (e (a e-b d)+c d^2\right )^2}{(d+e x)^2}-\frac{3 \left (e (a e-b d)+c d^2\right )^4}{(d+e x)^4}+\frac{16 (2 c d-b e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^3}+4 c^3 e^3 x^3 (4 b e-5 c d)+3 c^4 e^4 x^4}{12 e^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^4/(d + e*x)^5,x]

[Out]

(12*c*e*(-35*c^3*d^3 + 4*b^3*e^3 - 6*b*c*e^2*(5*b*d - 2*a*e) + 20*c^2*d*e*(3*b*d
 - a*e))*x + 6*c^2*e^2*(15*c^2*d^2 + 6*b^2*e^2 + 4*c*e*(-5*b*d + a*e))*x^2 + 4*c
^3*e^3*(-5*c*d + 4*b*e)*x^3 + 3*c^4*e^4*x^4 - (3*(c*d^2 + e*(-(b*d) + a*e))^4)/(
d + e*x)^4 + (16*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^3 - (12*(
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)^2 + (48*(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) + 12*(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))*Log[d + e*x])/(12*e^9)

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Maple [B]  time = 0.026, size = 1306, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^4/(e*x+d)^5,x)

[Out]

b^4*ln(e*x+d)/e^5-10*c^3/e^6*x^2*b*d+12*c^2/e^5*a*b*x+1/e^4/(e*x+d)^4*d^3*a*b^3-
28/3/e^8/(e*x+d)^3*b*c^3*d^6+12/e^5*ln(e*x+d)*a*b^2*c+60/e^7*ln(e*x+d)*a*c^3*d^2
-20/e^6*ln(e*x+d)*b^3*c*d+90/e^7*ln(e*x+d)*b^2*c^2*d^2-140/e^8*ln(e*x+d)*b*c^3*d
^3-12/e^4/(e*x+d)*a^2*b*c+24/e^5/(e*x+d)*a^2*d*c^2+80/e^7/(e*x+d)*c^3*d^3*a-40/e
^6/(e*x+d)*b^3*c*d^2+120/e^7/(e*x+d)*b^2*c^2*d^3-140/e^8/(e*x+d)*b*c^3*d^4-1/e^3
/(e*x+d)^4*a^3*c*d^2+1/e^2/(e*x+d)^4*d*a^3*b-3/2/e^3/(e*x+d)^4*d^2*a^2*b^2-36/e^
5/(e*x+d)^2*a*b^2*c*d^2+60/e^6/(e*x+d)^2*a*b*c^2*d^3-12/e^4/(e*x+d)^3*a^2*b*c*d^
2+16/e^5/(e*x+d)^3*a*b^2*c*d^3-20/e^6/(e*x+d)^3*a*b*c^2*d^4-60/e^6*ln(e*x+d)*a*b
*c^2*d+48/e^5/(e*x+d)*a*b^2*c*d-120/e^6/(e*x+d)*a*b*c^2*d^2+3/e^4/(e*x+d)^4*d^3*
a^2*b*c-3/e^5/(e*x+d)^4*d^4*a*b^2*c+3/e^6/(e*x+d)^4*a*b*c^2*d^5-1/4/e/(e*x+d)^4*
a^4+56/e^9/(e*x+d)*c^4*d^5-1/4/e^9/(e*x+d)^4*c^4*d^8+4/3*c^3/e^5*x^3*b-5/3*c^4/e
^6*x^3*d+2*c^3/e^5*x^2*a+3*c^2/e^5*x^2*b^2+15/2*c^4/e^7*x^2*d^2+4*c/e^5*b^3*x-35
*c^4/e^8*d^3*x-2/e^3/(e*x+d)^2*a^3*c-14/e^9/(e*x+d)^2*c^4*d^6+8/3/e^9/(e*x+d)^3*
c^4*d^7+6/e^5*ln(e*x+d)*a^2*c^2+70/e^9*ln(e*x+d)*c^4*d^4-3/2/e^5/(e*x+d)^4*a^2*c
^2*d^4-1/e^7/(e*x+d)^4*a*c^3*d^6+1/e^6/(e*x+d)^4*b^3*c*d^5-3/2/e^7/(e*x+d)^4*b^2
*c^2*d^6+1/e^8/(e*x+d)^4*b*c^3*d^7-20*c^3/e^6*a*d*x-30*c^2/e^6*b^2*d*x+60*c^3/e^
7*b*d^2*x-18/e^5/(e*x+d)^2*a^2*c^2*d^2-30/e^7/(e*x+d)^2*a*c^3*d^4+20/e^6/(e*x+d)
^2*b^3*c*d^3-45/e^7/(e*x+d)^2*b^2*c^2*d^4+42/e^8/(e*x+d)^2*b*c^3*d^5+8/3/e^3/(e*
x+d)^3*a^3*c*d+8/e^5/(e*x+d)^3*a^2*c^2*d^3+8/e^7/(e*x+d)^3*a*c^3*d^5-20/3/e^6/(e
*x+d)^3*b^3*c*d^4+12/e^7/(e*x+d)^3*b^2*c^2*d^5-1/4/e^5/(e*x+d)^4*b^4*d^4-3*b^2/e
^3/(e*x+d)^2*a^2-3*b^4/e^5/(e*x+d)^2*d^2-4*b^3/e^4/(e*x+d)*a+4*b^4/e^5/(e*x+d)*d
+18/e^4/(e*x+d)^2*a^2*b*c*d+1/4*c^4*x^4/e^5+6*b^3/e^4/(e*x+d)^2*d*a+4/3*b^4/e^5/
(e*x+d)^3*d^3+4*b^2/e^3/(e*x+d)^3*a^2*d-4*b^3/e^4/(e*x+d)^3*a*d^2-4/3*b/e^2/(e*x
+d)^3*a^3

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Maxima [A]  time = 0.835296, size = 1138, normalized size = 2.67 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^4/(e*x + d)^5,x, algorithm="maxima")

[Out]

1/12*(533*c^4*d^8 - 1276*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 3*a^4*e^8 + 342*(3*b^2*c^
2 + 2*a*c^3)*d^6*e^2 - 308*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 25*(b^4 + 12*a*b^2*c +
6*a^2*c^2)*d^4*e^4 - 12*(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 + 48*(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 + 12*(154*c^4*d^6*e^2 - 378*b*c^3*d^5*e^3 + 105*(3*b^2*c
^2 + 2*a*c^3)*d^4*e^4 - 100*(b^3*c + 3*a*b*c^2)*d^3*e^5 + 9*(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 + 8*(214*c^4*d^7*e - 518*b*c^3*d^6*e^2 - 2*a^3*b*e^8 + 141*(3*b^2*c^2 + 2*a*c^
3)*d^5*e^3 - 130*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 11*(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 - (3*a^2*b^2 + 2*a^3*c)*d*e^7)*x)/(e^13
*x^4 + 4*d*e^12*x^3 + 6*d^2*e^11*x^2 + 4*d^3*e^10*x + d^4*e^9) + 1/12*(3*c^4*e^3
*x^4 - 4*(5*c^4*d*e^2 - 4*b*c^3*e^3)*x^3 + 6*(15*c^4*d^2*e - 20*b*c^3*d*e^2 + 2*
(3*b^2*c^2 + 2*a*c^3)*e^3)*x^2 - 12*(35*c^4*d^3 - 60*b*c^3*d^2*e + 10*(3*b^2*c^2
 + 2*a*c^3)*d*e^2 - 4*(b^3*c + 3*a*b*c^2)*e^3)*x)/e^8 + (70*c^4*d^4 - 140*b*c^3*
d^3*e + 30*(3*b^2*c^2 + 2*a*c^3)*d^2*e^2 - 20*(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)*log(e*x + d)/e^9

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Fricas [A]  time = 0.219903, size = 1759, normalized size = 4.13 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^4/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/12*(3*c^4*e^8*x^8 + 533*c^4*d^8 - 1276*b*c^3*d^7*e - 4*a^3*b*d*e^7 - 3*a^4*e^8
 + 342*(3*b^2*c^2 + 2*a*c^3)*d^6*e^2 - 308*(b^3*c + 3*a*b*c^2)*d^5*e^3 + 25*(b^4
 + 12*a*b^2*c + 6*a^2*c^2)*d^4*e^4 - 12*(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 - 8*(c^4*d*e^7 - 2*b*c^3*e^8)*x^7 + 4*(7*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 - 24*(7*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 - 2*(b^3*c + 3*a*b*c^2)*e^8)*x^5 - (1217*c
^4*d^4*e^4 - 2224*b*c^3*d^3*e^5 + 408*(3*b^2*c^2 + 2*a*c^3)*d^2*e^6 - 192*(b^3*c
 + 3*a*b*c^2)*d*e^7)*x^4 - 4*(377*c^4*d^5*e^3 - 544*b*c^3*d^4*e^4 + 48*(3*b^2*c^
2 + 2*a*c^3)*d^3*e^5 + 48*(b^3*c + 3*a*b*c^2)*d^2*e^6 - 12*(b^4 + 12*a*b^2*c + 6
*a^2*c^2)*d*e^7 + 12*(a*b^3 + 3*a^2*b*c)*e^8)*x^3 + 6*(43*c^4*d^6*e^2 - 296*b*c^
3*d^5*e^3 + 132*(3*b^2*c^2 + 2*a*c^3)*d^4*e^4 - 168*(b^3*c + 3*a*b*c^2)*d^3*e^5
+ 18*(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 + 4*(323*c^4*d^7*e - 856*b*c^3*d^6*e^2 - 4*a^3*b*e
^8 + 252*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 248*(b^3*c + 3*a*b*c^2)*d^4*e^4 + 22*(b
^4 + 12*a*b^2*c + 6*a^2*c^2)*d^3*e^5 - 12*(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 + 12*(70*c^4*d^8 - 140*b*c^3*d^7*e + 30*(3*b^2*c^2 + 2*
a*c^3)*d^6*e^2 - 20*(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 + (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 + 4*(
70*c^4*d^5*e^3 - 140*b*c^3*d^4*e^4 + 30*(3*b^2*c^2 + 2*a*c^3)*d^3*e^5 - 20*(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)*x^3 + 6*(70*c^4*d
^6*e^2 - 140*b*c^3*d^5*e^3 + 30*(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 + (b^4 + 12*a*b^2*c + 6*a^2*c^2)*d^2*e^6)*x^2 + 4*(70*c^4*d^7*e -
 140*b*c^3*d^6*e^2 + 30*(3*b^2*c^2 + 2*a*c^3)*d^5*e^3 - 20*(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)*x)*log(e*x + d))/(e^13*x^4 + 4*
d*e^12*x^3 + 6*d^2*e^11*x^2 + 4*d^3*e^10*x + d^4*e^9)

_______________________________________________________________________________________

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)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211129, 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)^5,x, algorithm="giac")

[Out]

Done

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