3.2213 \(\int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^5} \, dx\)

Optimal. Leaf size=330 \[ \frac{5 c (b+2 c x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{5 (b+2 c x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{-x \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )-7 b \left (a e^2+c d^2\right )+12 a c d e+4 b^2 d e}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{10 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

[Out]

-((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^
4) - (4*b^2*d*e + 12*a*c*d*e - 7*b*(c*d^2 + a*e^2) - (14*c^2*d^2 + 3*b^2*e^2 - 2
*c*e*(7*b*d - a*e))*x)/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) - (5*(14*c^2*d^2
+ 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(b + 2*c*x))/(12*(b^2 - 4*a*c)^3*(a + b*x + c
*x^2)^2) + (5*c*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(b + 2*c*x))/(2*(
b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (10*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b
*d - a*e))*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 = 0.83207, antiderivative size = 330, 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 \[ \frac{5 c (b+2 c x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{5 (b+2 c x) \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}-\frac{-x \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )-7 b \left (a e^2+c d^2\right )+12 a c d e+4 b^2 d e}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{10 c^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}-\frac{(d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

-((d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^
4) - (4*b^2*d*e + 12*a*c*d*e - 7*b*(c*d^2 + a*e^2) - (14*c^2*d^2 + 3*b^2*e^2 - 2
*c*e*(7*b*d - a*e))*x)/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) - (5*(14*c^2*d^2
+ 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(b + 2*c*x))/(12*(b^2 - 4*a*c)^3*(a + b*x + c
*x^2)^2) + (5*c*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b*d - a*e))*(b + 2*c*x))/(2*(
b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (10*c^2*(14*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(7*b
*d - a*e))*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 [A]  time = 99.8934, size = 342, normalized size = 1.04 \[ - \frac{10 c^{2} \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{9}{2}}} + \frac{5 c \left (b + 2 c x\right ) \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right )}{2 \left (- 4 a c + b^{2}\right )^{4} \left (a + b x + c x^{2}\right )} - \frac{5 \left (b + 2 c x\right ) \left (2 a c e^{2} + 3 b^{2} e^{2} - 14 b c d e + 14 c^{2} d^{2}\right )}{12 \left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )^{2}} + \frac{\left (d + e x\right ) \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{4}} + \frac{14 a b e^{2} - 24 a c d e - 8 b^{2} d e + 14 b c d^{2} + x \left (4 a c e^{2} + 6 b^{2} e^{2} - 28 b c d e + 28 c^{2} d^{2}\right )}{12 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*c**2*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)*atanh((b + 2*c*x
)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(9/2) + 5*c*(b + 2*c*x)*(2*a*c*e**2 + 3*
b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)/(2*(-4*a*c + b**2)**4*(a + b*x + c*x**2))
- 5*(b + 2*c*x)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)/(12*(-4*
a*c + b**2)**3*(a + b*x + c*x**2)**2) + (d + e*x)*(2*a*e - b*d + x*(b*e - 2*c*d)
)/(4*(-4*a*c + b**2)*(a + b*x + c*x**2)**4) + (14*a*b*e**2 - 24*a*c*d*e - 8*b**2
*d*e + 14*b*c*d**2 + x*(4*a*c*e**2 + 6*b**2*e**2 - 28*b*c*d*e + 28*c**2*d**2))/(
12*(-4*a*c + b**2)**2*(a + b*x + c*x**2)**3)

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Mathematica [A]  time = 1.61565, size = 326, normalized size = 0.99 \[ \frac{1}{12} \left (\frac{30 c (b+2 c x) \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right )}{\left (b^2-4 a c\right )^4 (a+x (b+c x))}-\frac{5 (b+2 c x) \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))^2}+\frac{3 \left (a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )}{c \left (4 a c-b^2\right ) (a+x (b+c x))^4}+\frac{(b+2 c x) \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right )}{c \left (b^2-4 a c\right )^2 (a+x (b+c x))^3}+\frac{120 c^2 \left (2 c e (a e-7 b d)+3 b^2 e^2+14 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{9/2}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(((14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(b + 2*c*x))/(c*(b^2 - 4*a*c)^
2*(a + x*(b + c*x))^3) - (5*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-7*b*d + a*e))*(b +
2*c*x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))^2) + (30*c*(14*c^2*d^2 + 3*b^2*e^2 +
2*c*e*(-7*b*d + a*e))*(b + 2*c*x))/((b^2 - 4*a*c)^4*(a + x*(b + c*x))) + (3*(a*
b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2*d + e*x)))/(c*(
-b^2 + 4*a*c)*(a + x*(b + c*x))^4) + (120*c^2*(14*c^2*d^2 + 3*b^2*e^2 + 2*c*e*(-
7*b*d + a*e))*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 = 0.028, size = 1243, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(5*c^5*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^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)*x^7+35/2*b*c^4*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14
*c^2*d^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)*x^6+5/3*c^
3*(11*a*c+13*b^2)*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^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)*x^5+25/12*b*(22*a*c+5*b^2)*c^2*(2*a*c*
e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*d^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)*x^4+1/3*(73*a^2*c^2+101*a*b^2*c+3*b^4)*c*(2*a*c*e^2+3*b^2*e^2-1
4*b*c*d*e+14*c^2*d^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
)*x^3+1/6*b*(219*a^2*c^2+28*a*b^2*c-b^4)*(2*a*c*e^2+3*b^2*e^2-14*b*c*d*e+14*c^2*
d^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)*x^2-1/3*(30*a^4
*c^3*e^2-279*a^3*b^2*c^2*e^2+558*a^3*b*c^3*d*e-558*a^3*c^4*d^2-28*a^2*b^4*c*e^2+
348*a^2*b^3*c^2*d*e-348*a^2*b^2*c^3*d^2+a*b^6*e^2-38*a*b^5*c*d*e+38*a*b^4*c^2*d^
2+2*b^7*d*e-2*b^6*c*d^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)*x+1/12*(324*a^4*b*c^2*e^2-768*a^4*c^3*d*e+28*a^3*b^3*c*e^2-348*a^3*b^2*c^2*
d*e+1116*a^3*b*c^3*d^2-a^2*b^5*e^2+38*a^2*b^4*c*d*e-326*a^2*b^3*c^2*d^2-2*a*b^6*
d*e+50*a*b^5*c*d^2-3*b^7*d^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))/(c*x^2+b*x+a)^4+20*c^3/(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*e^2+30*c^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)*a
rctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*e^2-140*c^3/(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*d*e+140*c^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)/(
4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^2

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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)^2/(c*x^2 + b*x + a)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244568, 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)^2/(c*x^2 + b*x + a)^5,x, algorithm="fricas")

[Out]

[1/12*(60*(14*a^4*c^4*d^2 - 14*a^4*b*c^3*d*e + (14*c^8*d^2 - 14*b*c^7*d*e + (3*b
^2*c^6 + 2*a*c^7)*e^2)*x^8 + 4*(14*b*c^7*d^2 - 14*b^2*c^6*d*e + (3*b^3*c^5 + 2*a
*b*c^6)*e^2)*x^7 + 2*(14*(3*b^2*c^6 + 2*a*c^7)*d^2 - 14*(3*b^3*c^5 + 2*a*b*c^6)*
d*e + (9*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*e^2)*x^6 + 4*(14*(b^3*c^5 + 3*a*b*c
^6)*d^2 - 14*(b^4*c^4 + 3*a*b^2*c^5)*d*e + (3*b^5*c^3 + 11*a*b^3*c^4 + 6*a^2*b*c
^5)*e^2)*x^5 + (14*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^2 - 14*(b^5*c^3 + 12*a
*b^3*c^4 + 6*a^2*b*c^5)*d*e + (3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^2*b^2*c^4 + 12*a^
3*c^5)*e^2)*x^4 + 4*(14*(a*b^3*c^4 + 3*a^2*b*c^5)*d^2 - 14*(a*b^4*c^3 + 3*a^2*b^
2*c^4)*d*e + (3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*e^2)*x^3 + (3*a^4*b^2*
c^2 + 2*a^5*c^3)*e^2 + 2*(14*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^2 - 14*(3*a^2*b^3*c^3
+ 2*a^3*b*c^4)*d*e + (9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c^4)*e^2)*x^2 + 4*
(14*a^3*b*c^4*d^2 - 14*a^3*b^2*c^3*d*e + (3*a^3*b^3*c^2 + 2*a^4*b*c^3)*e^2)*x)*l
og(-(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)) + (60*(14*c^7*d^2 - 14*b*c^6*d*e + (3*b^2
*c^5 + 2*a*c^6)*e^2)*x^7 + 210*(14*b*c^6*d^2 - 14*b^2*c^5*d*e + (3*b^3*c^4 + 2*a
*b*c^5)*e^2)*x^6 + 20*(14*(13*b^2*c^5 + 11*a*c^6)*d^2 - 14*(13*b^3*c^4 + 11*a*b*
c^5)*d*e + (39*b^4*c^3 + 59*a*b^2*c^4 + 22*a^2*c^5)*e^2)*x^5 + 25*(14*(5*b^3*c^4
+ 22*a*b*c^5)*d^2 - 14*(5*b^4*c^3 + 22*a*b^2*c^4)*d*e + (15*b^5*c^2 + 76*a*b^3*
c^3 + 44*a^2*b*c^4)*e^2)*x^4 + 4*(14*(3*b^4*c^3 + 101*a*b^2*c^4 + 73*a^2*c^5)*d^
2 - 14*(3*b^5*c^2 + 101*a*b^3*c^3 + 73*a^2*b*c^4)*d*e + (9*b^6*c + 309*a*b^4*c^2
+ 421*a^2*b^2*c^3 + 146*a^3*c^4)*e^2)*x^3 - (3*b^7 - 50*a*b^5*c + 326*a^2*b^3*c
^2 - 1116*a^3*b*c^3)*d^2 - 2*(a*b^6 - 19*a^2*b^4*c + 174*a^3*b^2*c^2 + 384*a^4*c
^3)*d*e - (a^2*b^5 - 28*a^3*b^3*c - 324*a^4*b*c^2)*e^2 - 2*(14*(b^5*c^2 - 28*a*b
^3*c^3 - 219*a^2*b*c^4)*d^2 - 14*(b^6*c - 28*a*b^4*c^2 - 219*a^2*b^2*c^3)*d*e +
(3*b^7 - 82*a*b^5*c - 713*a^2*b^3*c^2 - 438*a^3*b*c^3)*e^2)*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^2 - 2*(b^7 - 19*a*b^5*c + 174*a^
2*b^3*c^2 + 279*a^3*b*c^3)*d*e - (a*b^6 - 28*a^2*b^4*c - 279*a^3*b^2*c^2 + 30*a^
4*c^3)*e^2)*x)*sqrt(b^2 - 4*a*c))/((a^4*b^8 - 16*a^5*b^6*c + 96*a^6*b^4*c^2 - 25
6*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)*sqr
t(b^2 - 4*a*c)), 1/12*(120*(14*a^4*c^4*d^2 - 14*a^4*b*c^3*d*e + (14*c^8*d^2 - 14
*b*c^7*d*e + (3*b^2*c^6 + 2*a*c^7)*e^2)*x^8 + 4*(14*b*c^7*d^2 - 14*b^2*c^6*d*e +
(3*b^3*c^5 + 2*a*b*c^6)*e^2)*x^7 + 2*(14*(3*b^2*c^6 + 2*a*c^7)*d^2 - 14*(3*b^3*
c^5 + 2*a*b*c^6)*d*e + (9*b^4*c^4 + 12*a*b^2*c^5 + 4*a^2*c^6)*e^2)*x^6 + 4*(14*(
b^3*c^5 + 3*a*b*c^6)*d^2 - 14*(b^4*c^4 + 3*a*b^2*c^5)*d*e + (3*b^5*c^3 + 11*a*b^
3*c^4 + 6*a^2*b*c^5)*e^2)*x^5 + (14*(b^4*c^4 + 12*a*b^2*c^5 + 6*a^2*c^6)*d^2 - 1
4*(b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*d*e + (3*b^6*c^2 + 38*a*b^4*c^3 + 42*a^
2*b^2*c^4 + 12*a^3*c^5)*e^2)*x^4 + 4*(14*(a*b^3*c^4 + 3*a^2*b*c^5)*d^2 - 14*(a*b
^4*c^3 + 3*a^2*b^2*c^4)*d*e + (3*a*b^5*c^2 + 11*a^2*b^3*c^3 + 6*a^3*b*c^4)*e^2)*
x^3 + (3*a^4*b^2*c^2 + 2*a^5*c^3)*e^2 + 2*(14*(3*a^2*b^2*c^4 + 2*a^3*c^5)*d^2 -
14*(3*a^2*b^3*c^3 + 2*a^3*b*c^4)*d*e + (9*a^2*b^4*c^2 + 12*a^3*b^2*c^3 + 4*a^4*c
^4)*e^2)*x^2 + 4*(14*a^3*b*c^4*d^2 - 14*a^3*b^2*c^3*d*e + (3*a^3*b^3*c^2 + 2*a^4
*b*c^3)*e^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (60*(14*
c^7*d^2 - 14*b*c^6*d*e + (3*b^2*c^5 + 2*a*c^6)*e^2)*x^7 + 210*(14*b*c^6*d^2 - 14
*b^2*c^5*d*e + (3*b^3*c^4 + 2*a*b*c^5)*e^2)*x^6 + 20*(14*(13*b^2*c^5 + 11*a*c^6)
*d^2 - 14*(13*b^3*c^4 + 11*a*b*c^5)*d*e + (39*b^4*c^3 + 59*a*b^2*c^4 + 22*a^2*c^
5)*e^2)*x^5 + 25*(14*(5*b^3*c^4 + 22*a*b*c^5)*d^2 - 14*(5*b^4*c^3 + 22*a*b^2*c^4
)*d*e + (15*b^5*c^2 + 76*a*b^3*c^3 + 44*a^2*b*c^4)*e^2)*x^4 + 4*(14*(3*b^4*c^3 +
101*a*b^2*c^4 + 73*a^2*c^5)*d^2 - 14*(3*b^5*c^2 + 101*a*b^3*c^3 + 73*a^2*b*c^4)
*d*e + (9*b^6*c + 309*a*b^4*c^2 + 421*a^2*b^2*c^3 + 146*a^3*c^4)*e^2)*x^3 - (3*b
^7 - 50*a*b^5*c + 326*a^2*b^3*c^2 - 1116*a^3*b*c^3)*d^2 - 2*(a*b^6 - 19*a^2*b^4*
c + 174*a^3*b^2*c^2 + 384*a^4*c^3)*d*e - (a^2*b^5 - 28*a^3*b^3*c - 324*a^4*b*c^2
)*e^2 - 2*(14*(b^5*c^2 - 28*a*b^3*c^3 - 219*a^2*b*c^4)*d^2 - 14*(b^6*c - 28*a*b^
4*c^2 - 219*a^2*b^2*c^3)*d*e + (3*b^7 - 82*a*b^5*c - 713*a^2*b^3*c^2 - 438*a^3*b
*c^3)*e^2)*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^2
- 2*(b^7 - 19*a*b^5*c + 174*a^2*b^3*c^2 + 279*a^3*b*c^3)*d*e - (a*b^6 - 28*a^2*
b^4*c - 279*a^3*b^2*c^2 + 30*a^4*c^3)*e^2)*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))]

_______________________________________________________________________________________

Sympy [A]  time = 69.9162, size = 2407, normalized size = 7.29 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*c**2*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c
**2*d**2)*log(x + (-5120*a**5*c**7*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b*
*2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 6400*a**4*b**2*c**6*sqrt(-1/(4*a*c - b**2
)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) - 3200*a**3*b**4*c*
*5*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d
**2) + 800*a**2*b**6*c**4*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 -
14*b*c*d*e + 14*c**2*d**2) - 100*a*b**8*c**3*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*
e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 10*a*b*c**3*e**2 + 5*b**10*c**
2*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d*
*2) + 15*b**3*c**2*e**2 - 70*b**2*c**3*d*e + 70*b*c**4*d**2)/(20*a*c**4*e**2 + 3
0*b**2*c**3*e**2 - 140*b*c**4*d*e + 140*c**5*d**2)) + 5*c**2*sqrt(-1/(4*a*c - b*
*2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)*log(x + (5120*a**
5*c**7*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c*
*2*d**2) - 6400*a**4*b**2*c**6*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e
**2 - 14*b*c*d*e + 14*c**2*d**2) + 3200*a**3*b**4*c**5*sqrt(-1/(4*a*c - b**2)**9
)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) - 800*a**2*b**6*c**4*sq
rt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2)
+ 100*a*b**8*c**3*sqrt(-1/(4*a*c - b**2)**9)*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*
d*e + 14*c**2*d**2) + 10*a*b*c**3*e**2 - 5*b**10*c**2*sqrt(-1/(4*a*c - b**2)**9)
*(2*a*c*e**2 + 3*b**2*e**2 - 14*b*c*d*e + 14*c**2*d**2) + 15*b**3*c**2*e**2 - 70
*b**2*c**3*d*e + 70*b*c**4*d**2)/(20*a*c**4*e**2 + 30*b**2*c**3*e**2 - 140*b*c**
4*d*e + 140*c**5*d**2)) + (324*a**4*b*c**2*e**2 - 768*a**4*c**3*d*e + 28*a**3*b*
*3*c*e**2 - 348*a**3*b**2*c**2*d*e + 1116*a**3*b*c**3*d**2 - a**2*b**5*e**2 + 38
*a**2*b**4*c*d*e - 326*a**2*b**3*c**2*d**2 - 2*a*b**6*d*e + 50*a*b**5*c*d**2 - 3
*b**7*d**2 + x**7*(120*a*c**6*e**2 + 180*b**2*c**5*e**2 - 840*b*c**6*d*e + 840*c
**7*d**2) + x**6*(420*a*b*c**5*e**2 + 630*b**3*c**4*e**2 - 2940*b**2*c**5*d*e +
2940*b*c**6*d**2) + x**5*(440*a**2*c**5*e**2 + 1180*a*b**2*c**4*e**2 - 3080*a*b*
c**5*d*e + 3080*a*c**6*d**2 + 780*b**4*c**3*e**2 - 3640*b**3*c**4*d*e + 3640*b**
2*c**5*d**2) + x**4*(1100*a**2*b*c**4*e**2 + 1900*a*b**3*c**3*e**2 - 7700*a*b**2
*c**4*d*e + 7700*a*b*c**5*d**2 + 375*b**5*c**2*e**2 - 1750*b**4*c**3*d*e + 1750*
b**3*c**4*d**2) + x**3*(584*a**3*c**4*e**2 + 1684*a**2*b**2*c**3*e**2 - 4088*a**
2*b*c**4*d*e + 4088*a**2*c**5*d**2 + 1236*a*b**4*c**2*e**2 - 5656*a*b**3*c**3*d*
e + 5656*a*b**2*c**4*d**2 + 36*b**6*c*e**2 - 168*b**5*c**2*d*e + 168*b**4*c**3*d
**2) + x**2*(876*a**3*b*c**3*e**2 + 1426*a**2*b**3*c**2*e**2 - 6132*a**2*b**2*c*
*3*d*e + 6132*a**2*b*c**4*d**2 + 164*a*b**5*c*e**2 - 784*a*b**4*c**2*d*e + 784*a
*b**3*c**3*d**2 - 6*b**7*e**2 + 28*b**6*c*d*e - 28*b**5*c**2*d**2) + x*(-120*a**
4*c**3*e**2 + 1116*a**3*b**2*c**2*e**2 - 2232*a**3*b*c**3*d*e + 2232*a**3*c**4*d
**2 + 112*a**2*b**4*c*e**2 - 1392*a**2*b**3*c**2*d*e + 1392*a**2*b**2*c**3*d**2
- 4*a*b**6*e**2 + 152*a*b**5*c*d*e - 152*a*b**4*c**2*d**2 - 8*b**7*d*e + 8*b**6*
c*d**2))/(3072*a**8*c**4 - 3072*a**7*b**2*c**3 + 1152*a**6*b**4*c**2 - 192*a**5*
b**6*c + 12*a**4*b**8 + x**8*(3072*a**4*c**8 - 3072*a**3*b**2*c**7 + 1152*a**2*b
**4*c**6 - 192*a*b**6*c**5 + 12*b**8*c**4) + x**7*(12288*a**4*b*c**7 - 12288*a**
3*b**3*c**6 + 4608*a**2*b**5*c**5 - 768*a*b**7*c**4 + 48*b**9*c**3) + x**6*(1228
8*a**5*c**7 + 6144*a**4*b**2*c**6 - 13824*a**3*b**4*c**5 + 6144*a**2*b**6*c**4 -
1104*a*b**8*c**3 + 72*b**10*c**2) + x**5*(36864*a**5*b*c**6 - 24576*a**4*b**3*c
**5 + 1536*a**3*b**5*c**4 + 2304*a**2*b**7*c**3 - 624*a*b**9*c**2 + 48*b**11*c)
+ x**4*(18432*a**6*c**6 + 18432*a**5*b**2*c**5 - 26880*a**4*b**4*c**4 + 9600*a**
3*b**6*c**3 - 1080*a**2*b**8*c**2 - 48*a*b**10*c + 12*b**12) + x**3*(36864*a**6*
b*c**5 - 24576*a**5*b**3*c**4 + 1536*a**4*b**5*c**3 + 2304*a**3*b**7*c**2 - 624*
a**2*b**9*c + 48*a*b**11) + x**2*(12288*a**7*c**5 + 6144*a**6*b**2*c**4 - 13824*
a**5*b**4*c**3 + 6144*a**4*b**6*c**2 - 1104*a**3*b**8*c + 72*a**2*b**10) + x*(12
288*a**7*b*c**4 - 12288*a**6*b**3*c**3 + 4608*a**5*b**5*c**2 - 768*a**4*b**7*c +
48*a**3*b**9))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.208892, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done