3.2211 \(\int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^5} \, dx\)

Optimal. Leaf size=545 \[ -\frac{2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{-x \left (2 c^2 e^2 \left (-18 a^2 e^2-40 a b d e+305 b^2 d^2\right )-38 b^2 c e^3 (5 b d-a e)-40 c^3 d^2 e (21 b d-2 a e)+19 b^4 e^4+420 c^4 d^4\right )-10 b c \left (11 a^2 e^4+88 a c d^2 e^2+21 c^2 d^4\right )-5 b^3 \left (5 a e^4+19 c d^2 e^2\right )+4 b^2 c d e \left (83 a e^2+70 c d^2\right )+16 a c^2 d e \left (16 a e^2+35 c d^2\right )+6 b^4 d e^3}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (-6 a c e-2 b^2 e+7 c x (2 c d-b e)+7 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (-c x \left (-2 c e (35 b d-9 a e)+13 b^2 e^2+70 c^2 d^2\right )-b c \left (23 a e^2+35 c d^2\right )+28 a c^2 d e-3 b^3 e^2+28 b^2 c d e\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \]

[Out]

-((b + 2*c*x)*(d + e*x)^4)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^3*
(7*b*c*d - 2*b^2*e - 6*a*c*e + 7*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b*x
+ c*x^2)^3) + ((d + e*x)^2*(28*b^2*c*d*e + 28*a*c^2*d*e - 3*b^3*e^2 - b*c*(35*c
*d^2 + 23*a*e^2) - c*(70*c^2*d^2 + 13*b^2*e^2 - 2*c*e*(35*b*d - 9*a*e))*x))/(6*(
b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) - (6*b^4*d*e^3 + 16*a*c^2*d*e*(35*c*d^2 + 16
*a*e^2) + 4*b^2*c*d*e*(70*c*d^2 + 83*a*e^2) - 5*b^3*(19*c*d^2*e^2 + 5*a*e^4) - 1
0*b*c*(21*c^2*d^4 + 88*a*c*d^2*e^2 + 11*a^2*e^4) - (420*c^4*d^4 + 19*b^4*e^4 - 4
0*c^3*d^2*e*(21*b*d - 2*a*e) - 38*b^2*c*e^3*(5*b*d - a*e) + 2*c^2*e^2*(305*b^2*d
^2 - 40*a*b*d*e - 18*a^2*e^2))*x)/(6*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (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))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 -
4*a*c]])/(b^2 - 4*a*c)^(9/2)

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Rubi [A]  time = 1.94173, antiderivative size = 545, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{-x \left (2 c^2 e^2 \left (-18 a^2 e^2-40 a b d e+305 b^2 d^2\right )-38 b^2 c e^3 (5 b d-a e)-40 c^3 d^2 e (21 b d-2 a e)+19 b^4 e^4+420 c^4 d^4\right )-10 b c \left (11 a^2 e^4+88 a c d^2 e^2+21 c^2 d^4\right )-5 b^3 \left (5 a e^4+19 c d^2 e^2\right )+4 b^2 c d e \left (83 a e^2+70 c d^2\right )+16 a c^2 d e \left (16 a e^2+35 c d^2\right )+6 b^4 d e^3}{6 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{(b+2 c x) (d+e x)^4}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4}+\frac{(d+e x)^3 \left (-6 a c e-2 b^2 e+7 c x (2 c d-b e)+7 b c d\right )}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}+\frac{(d+e x)^2 \left (-c x \left (-2 c e (35 b d-9 a e)+13 b^2 e^2+70 c^2 d^2\right )-b c \left (23 a e^2+35 c d^2\right )+28 a c^2 d e-3 b^3 e^2+28 b^2 c d e\right )}{6 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b + 2*c*x)*(d + e*x)^4)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + ((d + e*x)^3*
(7*b*c*d - 2*b^2*e - 6*a*c*e + 7*c*(2*c*d - b*e)*x))/(6*(b^2 - 4*a*c)^2*(a + b*x
+ c*x^2)^3) + ((d + e*x)^2*(28*b^2*c*d*e + 28*a*c^2*d*e - 3*b^3*e^2 - b*c*(35*c
*d^2 + 23*a*e^2) - c*(70*c^2*d^2 + 13*b^2*e^2 - 2*c*e*(35*b*d - 9*a*e))*x))/(6*(
b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) - (6*b^4*d*e^3 + 16*a*c^2*d*e*(35*c*d^2 + 16
*a*e^2) + 4*b^2*c*d*e*(70*c*d^2 + 83*a*e^2) - 5*b^3*(19*c*d^2*e^2 + 5*a*e^4) - 1
0*b*c*(21*c^2*d^4 + 88*a*c*d^2*e^2 + 11*a^2*e^4) - (420*c^4*d^4 + 19*b^4*e^4 - 4
0*c^3*d^2*e*(21*b*d - 2*a*e) - 38*b^2*c*e^3*(5*b*d - a*e) + 2*c^2*e^2*(305*b^2*d
^2 - 40*a*b*d*e - 18*a^2*e^2))*x)/(6*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (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))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 -
4*a*c]])/(b^2 - 4*a*c)^(9/2)

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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((e*x+d)**4/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

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Mathematica [A]  time = 3.51359, size = 713, normalized size = 1.31 \[ \frac{1}{12} \left (\frac{6 (b+2 c x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{c \left (b^2-4 a c\right )^4 (a+x (b+c x))}+\frac{(b+2 c x) \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right )}{c^2 \left (4 a c-b^2\right )^3 (a+x (b+c x))^2}+\frac{24 \left (6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{9/2}}+\frac{3 \left (b c \left (-3 a^2 e^4+6 a c d e^2 (d+2 e x)+c^2 d^3 (d-4 e x)\right )+2 c^2 \left (a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x\right )+b^3 e^3 (a e-4 c d x)+2 b^2 c e^2 \left (3 c d^2 x-2 a e (d+e x)\right )+b^4 e^4 x\right )}{c^3 \left (4 a c-b^2\right ) (a+x (b+c x))^4}+\frac{2 b c^2 \left (23 a^2 e^4+6 a c d e^2 (d-2 e x)+7 c^2 d^3 (d-4 e x)\right )-4 c^3 \left (a^2 e^3 (32 d+9 e x)-6 a c d^2 e^2 x-7 c^2 d^4 x\right )+2 b^3 c e^2 \left (c d (9 d-4 e x)-10 a e^2\right )+4 b^2 c^2 e \left (a e^2 (13 d+6 e x)+c d^2 (9 e x-7 d)\right )+3 b^5 e^4-2 b^4 c e^3 (6 d+e x)}{c^3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(((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))*(b + 2*c*x))/(c^2*(-b^2 +
4*a*c)^3*(a + x*(b + c*x))^2) + (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))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (3*(b^4*e^4*x + b^3*
e^3*(a*e - 4*c*d*x) + 2*b^2*c*e^2*(3*c*d^2*x - 2*a*e*(d + e*x)) + b*c*(-3*a^2*e^
4 + c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d + 2*e*x)) + 2*c^2*(c^2*d^4*x + a^2*e^3*
(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x))))/(c^3*(-b^2 + 4*a*c)*(a + x*(b + c*x))
^4) + (3*b^5*e^4 - 2*b^4*c*e^3*(6*d + e*x) + 2*b^3*c*e^2*(-10*a*e^2 + c*d*(9*d -
4*e*x)) + 2*b*c^2*(23*a^2*e^4 + 7*c^2*d^3*(d - 4*e*x) + 6*a*c*d*e^2*(d - 2*e*x)
) + 4*b^2*c^2*e*(a*e^2*(13*d + 6*e*x) + c*d^2*(-7*d + 9*e*x)) - 4*c^3*(-7*c^2*d^
4*x - 6*a*c*d^2*e^2*x + a^2*e^3*(32*d + 9*e*x)))/(c^3*(b^2 - 4*a*c)^2*(a + x*(b
+ c*x))^3) + (24*(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))*ArcTan[(b +
2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(9/2))/12

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Maple [B]  time = 0.029, size = 2430, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((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)/(256*a^4*c^4-256*a^3*b^2*
c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*c^3*x^7+7/2*b*c^2*(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^2-1
40*b*c^3*d^3*e+70*c^4*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*
c+b^8)*x^6+1/3*c*(11*a*c+13*b^2)*(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)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^5+5/12*b*(2
2*a*c+5*b^2)*(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)/(256*a^4*c^4-
256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^4-1/3*(66*a^4*c^3*e^4-450*a^3*b
^2*c^2*e^4+876*a^3*b*c^3*d*e^3-876*a^3*c^4*d^2*e^2-203*a^2*b^4*c*e^4+1504*a^2*b^
3*c^2*d*e^3-2526*a^2*b^2*c^3*d^2*e^2+2044*a^2*b*c^4*d^3*e-1022*a^2*c^5*d^4-37*a*
b^6*e^4+440*a*b^5*c*d*e^3-1854*a*b^4*c^2*d^2*e^2+2828*a*b^3*c^3*d^3*e-1414*a*b^2
*c^4*d^4+12*b^7*d*e^3-54*b^6*c*d^2*e^2+84*b^5*c^2*d^3*e-42*b^4*c^3*d^4)/(256*a^4
*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^3+1/6*(314*a^4*b*c^2*e^4-1
024*a^4*c^3*d*e^3+508*a^3*b^3*c*e^4-1604*a^3*b^2*c^2*d*e^3+2628*a^3*b*c^3*d^2*e^
2+129*a^2*b^5*e^4-1596*a^2*b^4*c*d*e^3+4278*a^2*b^3*c^2*d^2*e^2-6132*a^2*b^2*c^3
*d^3*e+3066*a^2*b*c^4*d^4-36*a*b^6*d*e^3+492*a*b^5*c*d^2*e^2-784*a*b^4*c^2*d^3*e
+392*a*b^3*c^3*d^4-18*b^7*d^2*e^2+28*b^6*c*d^3*e-14*b^5*c^2*d^4)/(256*a^4*c^4-25
6*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x^2-1/3*(18*a^5*c^2*e^4-184*a^4*b^2
*c*e^4+332*a^4*b*c^2*d*e^3+180*a^4*c^3*d^2*e^2-47*a^3*b^4*e^4+604*a^3*b^3*c*d*e^
3-1674*a^3*b^2*c^2*d^2*e^2+1116*a^3*b*c^3*d^3*e-558*a^3*c^4*d^4+12*a^2*b^5*d*e^3
-168*a^2*b^4*c*d^2*e^2+696*a^2*b^3*c^2*d^3*e-348*a^2*b^2*c^3*d^4+6*a*b^6*d^2*e^2
-76*a*b^5*c*d^3*e+38*a*b^4*c^2*d^4+4*b^7*d^3*e-2*b^6*c*d^4)/(256*a^4*c^4-256*a^3
*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)*x+1/12*(220*a^5*b*c*e^4-512*a^5*c^2*d*e^
3+50*a^4*b^3*e^4-664*a^4*b^2*c*d*e^3+1944*a^4*b*c^2*d^2*e^2-1536*a^4*c^3*d^3*e-1
2*a^3*b^4*d*e^3+168*a^3*b^3*c*d^2*e^2-696*a^3*b^2*c^2*d^3*e+1116*a^3*b*c^3*d^4-6
*a^2*b^5*d^2*e^2+76*a^2*b^4*c*d^3*e-326*a^2*b^3*c^2*d^4-4*a*b^6*d^3*e+50*a*b^5*c
*d^4-3*b^7*d^4)/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8))/(c*
x^2+b*x+a)^4+12/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a
*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*e^4*a^2*c^2+24/(256*a^4*c^4-25
6*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/
(4*a*c-b^2)^(1/2))*a*b^2*c*e^4-120/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-1
6*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*c^2*d*e
^3+120/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(
1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c^3*e^2*d^2+2/(256*a^4*c^4-256*a^3*b^
2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b
^2)^(1/2))*b^4*e^4-40/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8
)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*c*d*e^3+180/(256*a^4
*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*
c*x+b)/(4*a*c-b^2)^(1/2))*b^2*c^2*d^2*e^2-280/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^
2*b^4*c^2-16*a*b^6*c+b^8)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*
b*c^3*d^3*e+140/(256*a^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/(4*a
*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*c^4*d^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266719, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(12*(70*a^4*c^4*d^4 - 140*a^4*b*c^3*d^3*e + (70*c^8*d^4 - 140*b*c^7*d^3*e
+ 30*(3*b^2*c^6 + 2*a*c^7)*d^2*e^2 - 20*(b^3*c^5 + 3*a*b*c^6)*d*e^3 + (b^4*c^4 +
12*a*b^2*c^5 + 6*a^2*c^6)*e^4)*x^8 + 4*(70*b*c^7*d^4 - 140*b^2*c^6*d^3*e + 30*(
3*b^3*c^5 + 2*a*b*c^6)*d^2*e^2 - 20*(b^4*c^4 + 3*a*b^2*c^5)*d*e^3 + (b^5*c^3 + 1
2*a*b^3*c^4 + 6*a^2*b*c^5)*e^4)*x^7 + 2*(70*(3*b^2*c^6 + 2*a*c^7)*d^4 - 140*(3*b
^3*c^5 + 2*a*b*c^6)*d^3*e + 30*(9*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*d^2*e^2 -
20*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d*e^3 + (3*b^6*c^2 + 38*a*b^4*c^3 +
42*a^2*b^2*c^4 + 12*a^3*c^5)*e^4)*x^6 + 4*(70*(b^3*c^5 + 3*a*b*c^6)*d^4 - 140*(b
^4*c^4 + 3*a*b^2*c^5)*d^3*e + 30*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*e^
2 - 20*(b^6*c^2 + 6*a*b^4*c^3 + 9*a^2*b^2*c^4)*d*e^3 + (b^7*c + 15*a*b^5*c^2 + 4
2*a^2*b^3*c^3 + 18*a^3*b*c^4)*e^4)*x^5 + 30*(3*a^4*b^2*c^2 + 2*a^5*c^3)*d^2*e^2
- 20*(a^4*b^3*c + 3*a^5*b*c^2)*d*e^3 + (a^4*b^4 + 12*a^5*b^2*c + 6*a^6*c^2)*e^4
+ (70*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^4 - 140*(b^5*c^3 + 12*a*b^3*c^4 + 6
*a^2*b*c^5)*d^3*e + 30*(3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*
d^2*e^2 - 20*(b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*d*e^3 + (b^8
+ 24*a*b^6*c + 156*a^2*b^4*c^2 + 144*a^3*b^2*c^3 + 36*a^4*c^4)*e^4)*x^4 + 4*(70
*(a*b^3*c^4 + 3*a^2*b*c^5)*d^4 - 140*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d^3*e + 30*(3*a
*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*d^2*e^2 - 20*(a*b^6*c + 6*a^2*b^4*c^2 +
9*a^3*b^2*c^3)*d*e^3 + (a*b^7 + 15*a^2*b^5*c + 42*a^3*b^3*c^2 + 18*a^4*b*c^3)*e
^4)*x^3 + 2*(70*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^4 - 140*(3*a^2*b^3*c^3 + 2*a^3*b*c
^4)*d^3*e + 30*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*d^2*e^2 - 20*(3*a^2*
b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*d*e^3 + (3*a^2*b^6 + 38*a^3*b^4*c + 42*a^4
*b^2*c^2 + 12*a^5*c^3)*e^4)*x^2 + 4*(70*a^3*b*c^4*d^4 - 140*a^3*b^2*c^3*d^3*e +
30*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d^2*e^2 - 20*(a^3*b^4*c + 3*a^4*b^2*c^2)*d*e^3
+ (a^3*b^5 + 12*a^4*b^3*c + 6*a^5*b*c^2)*e^4)*x)*log(-(b^3 - 4*a*b*c + 2*(b^2*c
- 4*a*c^2)*x - (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^2 + b
*x + a)) + (12*(70*c^7*d^4 - 140*b*c^6*d^3*e + 30*(3*b^2*c^5 + 2*a*c^6)*d^2*e^2
- 20*(b^3*c^4 + 3*a*b*c^5)*d*e^3 + (b^4*c^3 + 12*a*b^2*c^4 + 6*a^2*c^5)*e^4)*x^7
+ 42*(70*b*c^6*d^4 - 140*b^2*c^5*d^3*e + 30*(3*b^3*c^4 + 2*a*b*c^5)*d^2*e^2 - 2
0*(b^4*c^3 + 3*a*b^2*c^4)*d*e^3 + (b^5*c^2 + 12*a*b^3*c^3 + 6*a^2*b*c^4)*e^4)*x^
6 + 4*(70*(13*b^2*c^5 + 11*a*c^6)*d^4 - 140*(13*b^3*c^4 + 11*a*b*c^5)*d^3*e + 30
*(39*b^4*c^3 + 59*a*b^2*c^4 + 22*a^2*c^5)*d^2*e^2 - 20*(13*b^5*c^2 + 50*a*b^3*c^
3 + 33*a^2*b*c^4)*d*e^3 + (13*b^6*c + 167*a*b^4*c^2 + 210*a^2*b^2*c^3 + 66*a^3*c
^4)*e^4)*x^5 - (3*b^7 - 50*a*b^5*c + 326*a^2*b^3*c^2 - 1116*a^3*b*c^3)*d^4 - 4*(
a*b^6 - 19*a^2*b^4*c + 174*a^3*b^2*c^2 + 384*a^4*c^3)*d^3*e - 6*(a^2*b^5 - 28*a^
3*b^3*c - 324*a^4*b*c^2)*d^2*e^2 - 4*(3*a^3*b^4 + 166*a^4*b^2*c + 128*a^5*c^2)*d
*e^3 + 10*(5*a^4*b^3 + 22*a^5*b*c)*e^4 + 5*(70*(5*b^3*c^4 + 22*a*b*c^5)*d^4 - 14
0*(5*b^4*c^3 + 22*a*b^2*c^4)*d^3*e + 30*(15*b^5*c^2 + 76*a*b^3*c^3 + 44*a^2*b*c^
4)*d^2*e^2 - 20*(5*b^6*c + 37*a*b^4*c^2 + 66*a^2*b^2*c^3)*d*e^3 + (5*b^7 + 82*a*
b^5*c + 294*a^2*b^3*c^2 + 132*a^3*b*c^3)*e^4)*x^4 + 4*(14*(3*b^4*c^3 + 101*a*b^2
*c^4 + 73*a^2*c^5)*d^4 - 28*(3*b^5*c^2 + 101*a*b^3*c^3 + 73*a^2*b*c^4)*d^3*e + 6
*(9*b^6*c + 309*a*b^4*c^2 + 421*a^2*b^2*c^3 + 146*a^3*c^4)*d^2*e^2 - 4*(3*b^7 +
110*a*b^5*c + 376*a^2*b^3*c^2 + 219*a^3*b*c^3)*d*e^3 + (37*a*b^6 + 203*a^2*b^4*c
+ 450*a^3*b^2*c^2 - 66*a^4*c^3)*e^4)*x^3 - 2*(14*(b^5*c^2 - 28*a*b^3*c^3 - 219*
a^2*b*c^4)*d^4 - 28*(b^6*c - 28*a*b^4*c^2 - 219*a^2*b^2*c^3)*d^3*e + 6*(3*b^7 -
82*a*b^5*c - 713*a^2*b^3*c^2 - 438*a^3*b*c^3)*d^2*e^2 + 4*(9*a*b^6 + 399*a^2*b^4
*c + 401*a^3*b^2*c^2 + 256*a^4*c^3)*d*e^3 - (129*a^2*b^5 + 508*a^3*b^3*c + 314*a
^4*b*c^2)*e^4)*x^2 + 4*(2*(b^6*c - 19*a*b^4*c^2 + 174*a^2*b^2*c^3 + 279*a^3*c^4)
*d^4 - 4*(b^7 - 19*a*b^5*c + 174*a^2*b^3*c^2 + 279*a^3*b*c^3)*d^3*e - 6*(a*b^6 -
28*a^2*b^4*c - 279*a^3*b^2*c^2 + 30*a^4*c^3)*d^2*e^2 - 4*(3*a^2*b^5 + 151*a^3*b
^3*c + 83*a^4*b*c^2)*d*e^3 + (47*a^3*b^4 + 184*a^4*b^2*c - 18*a^5*c^2)*e^4)*x)*s
qrt(b^2 - 4*a*c))/((a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256*a^7*b^2*c^3 +
256*a^8*c^4 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a
^4*c^8)*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256
*a^4*b*c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3 + 256*a^2*b^6*c^4 - 576*a^3*b^4*c
^5 + 256*a^4*b^2*c^6 + 512*a^5*c^7)*x^6 + 4*(b^11*c - 13*a*b^9*c^2 + 48*a^2*b^7*
c^3 + 32*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 768*a^5*b*c^6)*x^5 + (b^12 - 4*a*b^10*c
- 90*a^2*b^8*c^2 + 800*a^3*b^6*c^3 - 2240*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 1536
*a^6*c^6)*x^4 + 4*(a*b^11 - 13*a^2*b^9*c + 48*a^3*b^7*c^2 + 32*a^4*b^5*c^3 - 512
*a^5*b^3*c^4 + 768*a^6*b*c^5)*x^3 + 2*(3*a^2*b^10 - 46*a^3*b^8*c + 256*a^4*b^6*c
^2 - 576*a^5*b^4*c^3 + 256*a^6*b^2*c^4 + 512*a^7*c^5)*x^2 + 4*(a^3*b^9 - 16*a^4*
b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)*sqrt(b^2 - 4*a*c)),
1/12*(24*(70*a^4*c^4*d^4 - 140*a^4*b*c^3*d^3*e + (70*c^8*d^4 - 140*b*c^7*d^3*e
+ 30*(3*b^2*c^6 + 2*a*c^7)*d^2*e^2 - 20*(b^3*c^5 + 3*a*b*c^6)*d*e^3 + (b^4*c^4 +
12*a*b^2*c^5 + 6*a^2*c^6)*e^4)*x^8 + 4*(70*b*c^7*d^4 - 140*b^2*c^6*d^3*e + 30*(
3*b^3*c^5 + 2*a*b*c^6)*d^2*e^2 - 20*(b^4*c^4 + 3*a*b^2*c^5)*d*e^3 + (b^5*c^3 + 1
2*a*b^3*c^4 + 6*a^2*b*c^5)*e^4)*x^7 + 2*(70*(3*b^2*c^6 + 2*a*c^7)*d^4 - 140*(3*b
^3*c^5 + 2*a*b*c^6)*d^3*e + 30*(9*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*d^2*e^2 -
20*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d*e^3 + (3*b^6*c^2 + 38*a*b^4*c^3 +
42*a^2*b^2*c^4 + 12*a^3*c^5)*e^4)*x^6 + 4*(70*(b^3*c^5 + 3*a*b*c^6)*d^4 - 140*(b
^4*c^4 + 3*a*b^2*c^5)*d^3*e + 30*(3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c^5)*d^2*e^
2 - 20*(b^6*c^2 + 6*a*b^4*c^3 + 9*a^2*b^2*c^4)*d*e^3 + (b^7*c + 15*a*b^5*c^2 + 4
2*a^2*b^3*c^3 + 18*a^3*b*c^4)*e^4)*x^5 + 30*(3*a^4*b^2*c^2 + 2*a^5*c^3)*d^2*e^2
- 20*(a^4*b^3*c + 3*a^5*b*c^2)*d*e^3 + (a^4*b^4 + 12*a^5*b^2*c + 6*a^6*c^2)*e^4
+ (70*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^4 - 140*(b^5*c^3 + 12*a*b^3*c^4 + 6
*a^2*b*c^5)*d^3*e + 30*(3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^3*c^5)*
d^2*e^2 - 20*(b^7*c + 15*a*b^5*c^2 + 42*a^2*b^3*c^3 + 18*a^3*b*c^4)*d*e^3 + (b^8
+ 24*a*b^6*c + 156*a^2*b^4*c^2 + 144*a^3*b^2*c^3 + 36*a^4*c^4)*e^4)*x^4 + 4*(70
*(a*b^3*c^4 + 3*a^2*b*c^5)*d^4 - 140*(a*b^4*c^3 + 3*a^2*b^2*c^4)*d^3*e + 30*(3*a
*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*d^2*e^2 - 20*(a*b^6*c + 6*a^2*b^4*c^2 +
9*a^3*b^2*c^3)*d*e^3 + (a*b^7 + 15*a^2*b^5*c + 42*a^3*b^3*c^2 + 18*a^4*b*c^3)*e
^4)*x^3 + 2*(70*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^4 - 140*(3*a^2*b^3*c^3 + 2*a^3*b*c
^4)*d^3*e + 30*(9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*d^2*e^2 - 20*(3*a^2*
b^5*c + 11*a^3*b^3*c^2 + 6*a^4*b*c^3)*d*e^3 + (3*a^2*b^6 + 38*a^3*b^4*c + 42*a^4
*b^2*c^2 + 12*a^5*c^3)*e^4)*x^2 + 4*(70*a^3*b*c^4*d^4 - 140*a^3*b^2*c^3*d^3*e +
30*(3*a^3*b^3*c^2 + 2*a^4*b*c^3)*d^2*e^2 - 20*(a^3*b^4*c + 3*a^4*b^2*c^2)*d*e^3
+ (a^3*b^5 + 12*a^4*b^3*c + 6*a^5*b*c^2)*e^4)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c
*x + b)/(b^2 - 4*a*c)) + (12*(70*c^7*d^4 - 140*b*c^6*d^3*e + 30*(3*b^2*c^5 + 2*a
*c^6)*d^2*e^2 - 20*(b^3*c^4 + 3*a*b*c^5)*d*e^3 + (b^4*c^3 + 12*a*b^2*c^4 + 6*a^2
*c^5)*e^4)*x^7 + 42*(70*b*c^6*d^4 - 140*b^2*c^5*d^3*e + 30*(3*b^3*c^4 + 2*a*b*c^
5)*d^2*e^2 - 20*(b^4*c^3 + 3*a*b^2*c^4)*d*e^3 + (b^5*c^2 + 12*a*b^3*c^3 + 6*a^2*
b*c^4)*e^4)*x^6 + 4*(70*(13*b^2*c^5 + 11*a*c^6)*d^4 - 140*(13*b^3*c^4 + 11*a*b*c
^5)*d^3*e + 30*(39*b^4*c^3 + 59*a*b^2*c^4 + 22*a^2*c^5)*d^2*e^2 - 20*(13*b^5*c^2
+ 50*a*b^3*c^3 + 33*a^2*b*c^4)*d*e^3 + (13*b^6*c + 167*a*b^4*c^2 + 210*a^2*b^2*
c^3 + 66*a^3*c^4)*e^4)*x^5 - (3*b^7 - 50*a*b^5*c + 326*a^2*b^3*c^2 - 1116*a^3*b*
c^3)*d^4 - 4*(a*b^6 - 19*a^2*b^4*c + 174*a^3*b^2*c^2 + 384*a^4*c^3)*d^3*e - 6*(a
^2*b^5 - 28*a^3*b^3*c - 324*a^4*b*c^2)*d^2*e^2 - 4*(3*a^3*b^4 + 166*a^4*b^2*c +
128*a^5*c^2)*d*e^3 + 10*(5*a^4*b^3 + 22*a^5*b*c)*e^4 + 5*(70*(5*b^3*c^4 + 22*a*b
*c^5)*d^4 - 140*(5*b^4*c^3 + 22*a*b^2*c^4)*d^3*e + 30*(15*b^5*c^2 + 76*a*b^3*c^3
+ 44*a^2*b*c^4)*d^2*e^2 - 20*(5*b^6*c + 37*a*b^4*c^2 + 66*a^2*b^2*c^3)*d*e^3 +
(5*b^7 + 82*a*b^5*c + 294*a^2*b^3*c^2 + 132*a^3*b*c^3)*e^4)*x^4 + 4*(14*(3*b^4*c
^3 + 101*a*b^2*c^4 + 73*a^2*c^5)*d^4 - 28*(3*b^5*c^2 + 101*a*b^3*c^3 + 73*a^2*b*
c^4)*d^3*e + 6*(9*b^6*c + 309*a*b^4*c^2 + 421*a^2*b^2*c^3 + 146*a^3*c^4)*d^2*e^2
- 4*(3*b^7 + 110*a*b^5*c + 376*a^2*b^3*c^2 + 219*a^3*b*c^3)*d*e^3 + (37*a*b^6 +
203*a^2*b^4*c + 450*a^3*b^2*c^2 - 66*a^4*c^3)*e^4)*x^3 - 2*(14*(b^5*c^2 - 28*a*
b^3*c^3 - 219*a^2*b*c^4)*d^4 - 28*(b^6*c - 28*a*b^4*c^2 - 219*a^2*b^2*c^3)*d^3*e
+ 6*(3*b^7 - 82*a*b^5*c - 713*a^2*b^3*c^2 - 438*a^3*b*c^3)*d^2*e^2 + 4*(9*a*b^6
+ 399*a^2*b^4*c + 401*a^3*b^2*c^2 + 256*a^4*c^3)*d*e^3 - (129*a^2*b^5 + 508*a^3
*b^3*c + 314*a^4*b*c^2)*e^4)*x^2 + 4*(2*(b^6*c - 19*a*b^4*c^2 + 174*a^2*b^2*c^3
+ 279*a^3*c^4)*d^4 - 4*(b^7 - 19*a*b^5*c + 174*a^2*b^3*c^2 + 279*a^3*b*c^3)*d^3*
e - 6*(a*b^6 - 28*a^2*b^4*c - 279*a^3*b^2*c^2 + 30*a^4*c^3)*d^2*e^2 - 4*(3*a^2*b
^5 + 151*a^3*b^3*c + 83*a^4*b*c^2)*d*e^3 + (47*a^3*b^4 + 184*a^4*b^2*c - 18*a^5*
c^2)*e^4)*x)*sqrt(-b^2 + 4*a*c))/((a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 256
*a^7*b^2*c^3 + 256*a^8*c^4 + (b^8*c^4 - 16*a*b^6*c^5 + 96*a^2*b^4*c^6 - 256*a^3*
b^2*c^7 + 256*a^4*c^8)*x^8 + 4*(b^9*c^3 - 16*a*b^7*c^4 + 96*a^2*b^5*c^5 - 256*a^
3*b^3*c^6 + 256*a^4*b*c^7)*x^7 + 2*(3*b^10*c^2 - 46*a*b^8*c^3 + 256*a^2*b^6*c^4
- 576*a^3*b^4*c^5 + 256*a^4*b^2*c^6 + 512*a^5*c^7)*x^6 + 4*(b^11*c - 13*a*b^9*c^
2 + 48*a^2*b^7*c^3 + 32*a^3*b^5*c^4 - 512*a^4*b^3*c^5 + 768*a^5*b*c^6)*x^5 + (b^
12 - 4*a*b^10*c - 90*a^2*b^8*c^2 + 800*a^3*b^6*c^3 - 2240*a^4*b^4*c^4 + 1536*a^5
*b^2*c^5 + 1536*a^6*c^6)*x^4 + 4*(a*b^11 - 13*a^2*b^9*c + 48*a^3*b^7*c^2 + 32*a^
4*b^5*c^3 - 512*a^5*b^3*c^4 + 768*a^6*b*c^5)*x^3 + 2*(3*a^2*b^10 - 46*a^3*b^8*c
+ 256*a^4*b^6*c^2 - 576*a^5*b^4*c^3 + 256*a^6*b^2*c^4 + 512*a^7*c^5)*x^2 + 4*(a^
3*b^9 - 16*a^4*b^7*c + 96*a^5*b^5*c^2 - 256*a^6*b^3*c^3 + 256*a^7*b*c^4)*x)*sqrt
(-b^2 + 4*a*c))]

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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((e*x+d)**4/(c*x**2+b*x+a)**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214037, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done