3.2214 \(\int \frac{d+e x}{\left (a+b x+c x^2\right )^5} \, dx\)

Optimal. Leaf size=219 \[ -\frac{70 c^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac{35 c^2 (b+2 c x) (2 c d-b e)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{35 c (b+2 c x) (2 c d-b e)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{7 (b+2 c x) (2 c d-b e)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{-2 a e+x (2 c d-b e)+b d}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

[Out]

-(b*d - 2*a*e + (2*c*d - b*e)*x)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + (7*(2*c
*d - b*e)*(b + 2*c*x))/(12*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) - (35*c*(2*c*d -
b*e)*(b + 2*c*x))/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (35*c^2*(2*c*d - b
*e)*(b + 2*c*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (70*c^3*(2*c*d - b*e)*A
rcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(9/2)

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Rubi [A]  time = 0.253705, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{70 c^3 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2}}+\frac{35 c^2 (b+2 c x) (2 c d-b e)}{2 \left (b^2-4 a c\right )^4 \left (a+b x+c x^2\right )}-\frac{35 c (b+2 c x) (2 c d-b e)}{12 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )^2}+\frac{7 (b+2 c x) (2 c d-b e)}{12 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^3}-\frac{-2 a e+x (2 c d-b e)+b d}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

-(b*d - 2*a*e + (2*c*d - b*e)*x)/(4*(b^2 - 4*a*c)*(a + b*x + c*x^2)^4) + (7*(2*c
*d - b*e)*(b + 2*c*x))/(12*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^3) - (35*c*(2*c*d -
b*e)*(b + 2*c*x))/(12*(b^2 - 4*a*c)^3*(a + b*x + c*x^2)^2) + (35*c^2*(2*c*d - b
*e)*(b + 2*c*x))/(2*(b^2 - 4*a*c)^4*(a + b*x + c*x^2)) - (70*c^3*(2*c*d - b*e)*A
rcTanh[(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 = 37.6138, size = 212, normalized size = 0.97 \[ \frac{280 c^{3} \left (\frac{b e}{4} - \frac{c d}{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{9}{2}}} - \frac{35 c^{2} \left (b + 2 c x\right ) \left (b e - 2 c d\right )}{2 \left (- 4 a c + b^{2}\right )^{4} \left (a + b x + c x^{2}\right )} + \frac{35 c \left (b + 2 c x\right ) \left (b e - 2 c d\right )}{12 \left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )^{2}} - \frac{7 \left (b + 2 c x\right ) \left (b e - 2 c d\right )}{12 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{3}} + \frac{2 a e - b d + x \left (b e - 2 c d\right )}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

280*c**3*(b*e/4 - c*d/2)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)*
*(9/2) - 35*c**2*(b + 2*c*x)*(b*e - 2*c*d)/(2*(-4*a*c + b**2)**4*(a + b*x + c*x*
*2)) + 35*c*(b + 2*c*x)*(b*e - 2*c*d)/(12*(-4*a*c + b**2)**3*(a + b*x + c*x**2)*
*2) - 7*(b + 2*c*x)*(b*e - 2*c*d)/(12*(-4*a*c + b**2)**2*(a + b*x + c*x**2)**3)
+ (2*a*e - b*d + x*(b*e - 2*c*d))/(4*(-4*a*c + b**2)*(a + b*x + c*x**2)**4)

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Mathematica [A]  time = 0.56988, size = 209, normalized size = 0.95 \[ \frac{-\frac{840 c^3 (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{35 c \left (b^2-4 a c\right ) (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^2}-\frac{7 \left (b^2-4 a c\right )^2 (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^3}+\frac{3 \left (b^2-4 a c\right )^3 (2 a e-b d+b e x-2 c d x)}{(a+x (b+c x))^4}+\frac{210 c^2 (b+2 c x) (2 c d-b e)}{a+x (b+c x)}}{12 \left (b^2-4 a c\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

((3*(b^2 - 4*a*c)^3*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x))^4 - (7
*(b^2 - 4*a*c)^2*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x))^3 + (35*c*(b^2 -
4*a*c)*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (210*c^2*(2*c*d - b*e)*
(b + 2*c*x))/(a + x*(b + c*x)) - (840*c^3*(-2*c*d + b*e)*ArcTan[(b + 2*c*x)/Sqrt
[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(12*(b^2 - 4*a*c)^4)

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Maple [B]  time = 0.013, size = 496, normalized size = 2.3 \[{\frac{bd-2\,ae+ \left ( -be+2\,cd \right ) x}{ \left ( 16\,ac-4\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{4}}}-{\frac{7\,xbce}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{7\,x{c}^{2}d}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}-{\frac{7\,{b}^{2}e}{12\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}+{\frac{7\,bcd}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{3}}}-{\frac{35\,{c}^{2}xbe}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{35\,{c}^{3}dx}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{35\,{b}^{2}ce}{12\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{35\,{c}^{2}bd}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-35\,{\frac{{c}^{3}xbe}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}+70\,{\frac{x{c}^{4}d}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{35\,{b}^{2}{c}^{2}e}{2\, \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}+35\,{\frac{{c}^{3}db}{ \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) }}-70\,{\frac{b{c}^{3}e}{ \left ( 4\,ac-{b}^{2} \right ) ^{9/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+140\,{\frac{{c}^{4}d}{ \left ( 4\,ac-{b}^{2} \right ) ^{9/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+b*x+a)^5,x)

[Out]

1/4*(b*d-2*a*e+(-b*e+2*c*d)*x)/(4*a*c-b^2)/(c*x^2+b*x+a)^4-7/6/(4*a*c-b^2)^2/(c*
x^2+b*x+a)^3*c*x*b*e+7/3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^3*c^2*x*d-7/12/(4*a*c-b^2)^
2/(c*x^2+b*x+a)^3*b^2*e+7/6/(4*a*c-b^2)^2/(c*x^2+b*x+a)^3*b*c*d-35/6/(4*a*c-b^2)
^3*c^2/(c*x^2+b*x+a)^2*x*b*e+35/3/(4*a*c-b^2)^3*c^3/(c*x^2+b*x+a)^2*x*d-35/12/(4
*a*c-b^2)^3*c/(c*x^2+b*x+a)^2*b^2*e+35/6/(4*a*c-b^2)^3*c^2/(c*x^2+b*x+a)^2*b*d-3
5/(4*a*c-b^2)^4*c^3/(c*x^2+b*x+a)*x*b*e+70/(4*a*c-b^2)^4*c^4/(c*x^2+b*x+a)*x*d-3
5/2/(4*a*c-b^2)^4*c^2/(c*x^2+b*x+a)*b^2*e+35/(4*a*c-b^2)^4*c^3/(c*x^2+b*x+a)*b*d
-70/(4*a*c-b^2)^(9/2)*c^3*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*e+140/(4*a*c-b^2
)^(9/2)*c^4*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238024, 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)/(c*x^2 + b*x + a)^5,x, algorithm="fricas")

[Out]

[-1/12*(420*(2*a^4*c^4*d - a^4*b*c^3*e + (2*c^8*d - b*c^7*e)*x^8 + 4*(2*b*c^7*d
- b^2*c^6*e)*x^7 + 2*(2*(3*b^2*c^6 + 2*a*c^7)*d - (3*b^3*c^5 + 2*a*b*c^6)*e)*x^6
+ 4*(2*(b^3*c^5 + 3*a*b*c^6)*d - (b^4*c^4 + 3*a*b^2*c^5)*e)*x^5 + (2*(b^4*c^4 +
12*a*b^2*c^5 + 6*a^2*c^6)*d - (b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*e)*x^4 + 4
*(2*(a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3 + 3*a^2*b^2*c^4)*e)*x^3 + 2*(2*(3*a
^2*b^2*c^4 + 2*a^3*c^5)*d - (3*a^2*b^3*c^3 + 2*a^3*b*c^4)*e)*x^2 + 4*(2*a^3*b*c^
4*d - a^3*b^2*c^3*e)*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)) - (420*(2*c^7*d -
b*c^6*e)*x^7 + 1470*(2*b*c^6*d - b^2*c^5*e)*x^6 + 140*(2*(13*b^2*c^5 + 11*a*c^6
)*d - (13*b^3*c^4 + 11*a*b*c^5)*e)*x^5 + 175*(2*(5*b^3*c^4 + 22*a*b*c^5)*d - (5*
b^4*c^3 + 22*a*b^2*c^4)*e)*x^4 + 28*(2*(3*b^4*c^3 + 101*a*b^2*c^4 + 73*a^2*c^5)*
d - (3*b^5*c^2 + 101*a*b^3*c^3 + 73*a^2*b*c^4)*e)*x^3 - 14*(2*(b^5*c^2 - 28*a*b^
3*c^3 - 219*a^2*b*c^4)*d - (b^6*c - 28*a*b^4*c^2 - 219*a^2*b^2*c^3)*e)*x^2 - (3*
b^7 - 50*a*b^5*c + 326*a^2*b^3*c^2 - 1116*a^3*b*c^3)*d - (a*b^6 - 19*a^2*b^4*c +
174*a^3*b^2*c^2 + 384*a^4*c^3)*e + 4*(2*(b^6*c - 19*a*b^4*c^2 + 174*a^2*b^2*c^3
+ 279*a^3*c^4)*d - (b^7 - 19*a*b^5*c + 174*a^2*b^3*c^2 + 279*a^3*b*c^3)*e)*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*(840*(2*a^4*c^4*d - a^4*b*c^3*e + (2*c^8*d - b*c^7*e)*x^8 + 4*(2*b*c^7*d -
b^2*c^6*e)*x^7 + 2*(2*(3*b^2*c^6 + 2*a*c^7)*d - (3*b^3*c^5 + 2*a*b*c^6)*e)*x^6
+ 4*(2*(b^3*c^5 + 3*a*b*c^6)*d - (b^4*c^4 + 3*a*b^2*c^5)*e)*x^5 + (2*(b^4*c^4 +
12*a*b^2*c^5 + 6*a^2*c^6)*d - (b^5*c^3 + 12*a*b^3*c^4 + 6*a^2*b*c^5)*e)*x^4 + 4*
(2*(a*b^3*c^4 + 3*a^2*b*c^5)*d - (a*b^4*c^3 + 3*a^2*b^2*c^4)*e)*x^3 + 2*(2*(3*a^
2*b^2*c^4 + 2*a^3*c^5)*d - (3*a^2*b^3*c^3 + 2*a^3*b*c^4)*e)*x^2 + 4*(2*a^3*b*c^4
*d - a^3*b^2*c^3*e)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (
420*(2*c^7*d - b*c^6*e)*x^7 + 1470*(2*b*c^6*d - b^2*c^5*e)*x^6 + 140*(2*(13*b^2*
c^5 + 11*a*c^6)*d - (13*b^3*c^4 + 11*a*b*c^5)*e)*x^5 + 175*(2*(5*b^3*c^4 + 22*a*
b*c^5)*d - (5*b^4*c^3 + 22*a*b^2*c^4)*e)*x^4 + 28*(2*(3*b^4*c^3 + 101*a*b^2*c^4
+ 73*a^2*c^5)*d - (3*b^5*c^2 + 101*a*b^3*c^3 + 73*a^2*b*c^4)*e)*x^3 - 14*(2*(b^5
*c^2 - 28*a*b^3*c^3 - 219*a^2*b*c^4)*d - (b^6*c - 28*a*b^4*c^2 - 219*a^2*b^2*c^3
)*e)*x^2 - (3*b^7 - 50*a*b^5*c + 326*a^2*b^3*c^2 - 1116*a^3*b*c^3)*d - (a*b^6 -
19*a^2*b^4*c + 174*a^3*b^2*c^2 + 384*a^4*c^3)*e + 4*(2*(b^6*c - 19*a*b^4*c^2 + 1
74*a^2*b^2*c^3 + 279*a^3*c^4)*d - (b^7 - 19*a*b^5*c + 174*a^2*b^3*c^2 + 279*a^3*
b*c^3)*e)*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 [A]  time = 33.143, size = 1564, normalized size = 7.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

35*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*log(x + (-35840*a**5*c**8*sqrt(
-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) + 44800*a**4*b**2*c**7*sqrt(-1/(4*a*c - b**2
)**9)*(b*e - 2*c*d) - 22400*a**3*b**4*c**6*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c
*d) + 5600*a**2*b**6*c**5*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) - 700*a*b**8*
c**4*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) + 35*b**10*c**3*sqrt(-1/(4*a*c - b
**2)**9)*(b*e - 2*c*d) + 35*b**2*c**3*e - 70*b*c**4*d)/(70*b*c**4*e - 140*c**5*d
)) - 35*c**3*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d)*log(x + (35840*a**5*c**8*s
qrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) - 44800*a**4*b**2*c**7*sqrt(-1/(4*a*c -
b**2)**9)*(b*e - 2*c*d) + 22400*a**3*b**4*c**6*sqrt(-1/(4*a*c - b**2)**9)*(b*e -
2*c*d) - 5600*a**2*b**6*c**5*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) + 700*a*b
**8*c**4*sqrt(-1/(4*a*c - b**2)**9)*(b*e - 2*c*d) - 35*b**10*c**3*sqrt(-1/(4*a*c
- b**2)**9)*(b*e - 2*c*d) + 35*b**2*c**3*e - 70*b*c**4*d)/(70*b*c**4*e - 140*c*
*5*d)) - (384*a**4*c**3*e + 174*a**3*b**2*c**2*e - 1116*a**3*b*c**3*d - 19*a**2*
b**4*c*e + 326*a**2*b**3*c**2*d + a*b**6*e - 50*a*b**5*c*d + 3*b**7*d + x**7*(42
0*b*c**6*e - 840*c**7*d) + x**6*(1470*b**2*c**5*e - 2940*b*c**6*d) + x**5*(1540*
a*b*c**5*e - 3080*a*c**6*d + 1820*b**3*c**4*e - 3640*b**2*c**5*d) + x**4*(3850*a
*b**2*c**4*e - 7700*a*b*c**5*d + 875*b**4*c**3*e - 1750*b**3*c**4*d) + x**3*(204
4*a**2*b*c**4*e - 4088*a**2*c**5*d + 2828*a*b**3*c**3*e - 5656*a*b**2*c**4*d + 8
4*b**5*c**2*e - 168*b**4*c**3*d) + x**2*(3066*a**2*b**2*c**3*e - 6132*a**2*b*c**
4*d + 392*a*b**4*c**2*e - 784*a*b**3*c**3*d - 14*b**6*c*e + 28*b**5*c**2*d) + x*
(1116*a**3*b*c**3*e - 2232*a**3*c**4*d + 696*a**2*b**3*c**2*e - 1392*a**2*b**2*c
**3*d - 76*a*b**5*c*e + 152*a*b**4*c**2*d + 4*b**7*e - 8*b**6*c*d))/(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*(12288*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 + 7
2*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*(12288*a**7*b*c**4 - 122
88*a**6*b**3*c**3 + 4608*a**5*b**5*c**2 - 768*a**4*b**7*c + 48*a**3*b**9))

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GIAC/XCAS [A]  time = 0.211681, size = 826, normalized size = 3.77 \[ \frac{70 \,{\left (2 \, c^{4} d - b c^{3} e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{840 \, c^{7} d x^{7} - 420 \, b c^{6} x^{7} e + 2940 \, b c^{6} d x^{6} - 1470 \, b^{2} c^{5} x^{6} e + 3640 \, b^{2} c^{5} d x^{5} + 3080 \, a c^{6} d x^{5} - 1820 \, b^{3} c^{4} x^{5} e - 1540 \, a b c^{5} x^{5} e + 1750 \, b^{3} c^{4} d x^{4} + 7700 \, a b c^{5} d x^{4} - 875 \, b^{4} c^{3} x^{4} e - 3850 \, a b^{2} c^{4} x^{4} e + 168 \, b^{4} c^{3} d x^{3} + 5656 \, a b^{2} c^{4} d x^{3} + 4088 \, a^{2} c^{5} d x^{3} - 84 \, b^{5} c^{2} x^{3} e - 2828 \, a b^{3} c^{3} x^{3} e - 2044 \, a^{2} b c^{4} x^{3} e - 28 \, b^{5} c^{2} d x^{2} + 784 \, a b^{3} c^{3} d x^{2} + 6132 \, a^{2} b c^{4} d x^{2} + 14 \, b^{6} c x^{2} e - 392 \, a b^{4} c^{2} x^{2} e - 3066 \, a^{2} b^{2} c^{3} x^{2} e + 8 \, b^{6} c d x - 152 \, a b^{4} c^{2} d x + 1392 \, a^{2} b^{2} c^{3} d x + 2232 \, a^{3} c^{4} d x - 4 \, b^{7} x e + 76 \, a b^{5} c x e - 696 \, a^{2} b^{3} c^{2} x e - 1116 \, a^{3} b c^{3} x e - 3 \, b^{7} d + 50 \, a b^{5} c d - 326 \, a^{2} b^{3} c^{2} d + 1116 \, a^{3} b c^{3} d - a b^{6} e + 19 \, a^{2} b^{4} c e - 174 \, a^{3} b^{2} c^{2} e - 384 \, a^{4} c^{3} e}{12 \,{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )}{\left (c x^{2} + b x + a\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

70*(2*c^4*d - b*c^3*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^8 - 16*a*b^6*c
+ 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqrt(-b^2 + 4*a*c)) + 1/12*(8
40*c^7*d*x^7 - 420*b*c^6*x^7*e + 2940*b*c^6*d*x^6 - 1470*b^2*c^5*x^6*e + 3640*b^
2*c^5*d*x^5 + 3080*a*c^6*d*x^5 - 1820*b^3*c^4*x^5*e - 1540*a*b*c^5*x^5*e + 1750*
b^3*c^4*d*x^4 + 7700*a*b*c^5*d*x^4 - 875*b^4*c^3*x^4*e - 3850*a*b^2*c^4*x^4*e +
168*b^4*c^3*d*x^3 + 5656*a*b^2*c^4*d*x^3 + 4088*a^2*c^5*d*x^3 - 84*b^5*c^2*x^3*e
- 2828*a*b^3*c^3*x^3*e - 2044*a^2*b*c^4*x^3*e - 28*b^5*c^2*d*x^2 + 784*a*b^3*c^
3*d*x^2 + 6132*a^2*b*c^4*d*x^2 + 14*b^6*c*x^2*e - 392*a*b^4*c^2*x^2*e - 3066*a^2
*b^2*c^3*x^2*e + 8*b^6*c*d*x - 152*a*b^4*c^2*d*x + 1392*a^2*b^2*c^3*d*x + 2232*a
^3*c^4*d*x - 4*b^7*x*e + 76*a*b^5*c*x*e - 696*a^2*b^3*c^2*x*e - 1116*a^3*b*c^3*x
*e - 3*b^7*d + 50*a*b^5*c*d - 326*a^2*b^3*c^2*d + 1116*a^3*b*c^3*d - a*b^6*e + 1
9*a^2*b^4*c*e - 174*a^3*b^2*c^2*e - 384*a^4*c^3*e)/((b^8 - 16*a*b^6*c + 96*a^2*b
^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*(c*x^2 + b*x + a)^4)