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3.2188 \(\int \frac{x^7}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=294 \[ \frac{3 x^2 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 c^3 \left (b^2-4 a c\right )^2}+\frac{3 b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{5/2}}+\frac{3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{3 b x \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^4 \left (b^2-4 a c\right )^2}-\frac{b x^3 \left (2 b^2-11 a c\right )}{c^2 \left (b^2-4 a c\right )^2}+\frac{x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

[Out]

(-3*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c)*x)/(c^4*(b^2 - 4*a*c)^2) + (3*(2*b^4 - 13*a*
b^2*c + 16*a^2*c^2)*x^2)/(2*c^3*(b^2 - 4*a*c)^2) - (b*(2*b^2 - 11*a*c)*x^3)/(c^2
*(b^2 - 4*a*c)^2) + (x^6*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3
*x^4*(a*(b^2 - 8*a*c) + b*(b^2 - 6*a*c)*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^
2)) + (3*b*(2*b^6 - 21*a*b^4*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x
)/Sqrt[b^2 - 4*a*c]])/(c^5*(b^2 - 4*a*c)^(5/2)) + (3*(2*b^2 - a*c)*Log[a + b*x +
 c*x^2])/(2*c^5)

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Rubi [A]  time = 0.944299, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ \frac{3 x^2 \left (16 a^2 c^2-13 a b^2 c+2 b^4\right )}{2 c^3 \left (b^2-4 a c\right )^2}+\frac{3 b \left (-70 a^3 c^3+70 a^2 b^2 c^2-21 a b^4 c+2 b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \left (b^2-4 a c\right )^{5/2}}+\frac{3 \left (2 b^2-a c\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{3 b x \left (2 b^2-9 a c\right ) \left (b^2-3 a c\right )}{c^4 \left (b^2-4 a c\right )^2}-\frac{b x^3 \left (2 b^2-11 a c\right )}{c^2 \left (b^2-4 a c\right )^2}+\frac{x^6 (2 a+b x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 x^4 \left (b x \left (b^2-6 a c\right )+a \left (b^2-8 a c\right )\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x + c*x^2)^3,x]

[Out]

(-3*b*(2*b^2 - 9*a*c)*(b^2 - 3*a*c)*x)/(c^4*(b^2 - 4*a*c)^2) + (3*(2*b^4 - 13*a*
b^2*c + 16*a^2*c^2)*x^2)/(2*c^3*(b^2 - 4*a*c)^2) - (b*(2*b^2 - 11*a*c)*x^3)/(c^2
*(b^2 - 4*a*c)^2) + (x^6*(2*a + b*x))/(2*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (3
*x^4*(a*(b^2 - 8*a*c) + b*(b^2 - 6*a*c)*x))/(2*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^
2)) + (3*b*(2*b^6 - 21*a*b^4*c + 70*a^2*b^2*c^2 - 70*a^3*c^3)*ArcTanh[(b + 2*c*x
)/Sqrt[b^2 - 4*a*c]])/(c^5*(b^2 - 4*a*c)^(5/2)) + (3*(2*b^2 - a*c)*Log[a + b*x +
 c*x^2])/(2*c^5)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b x^{3} \left (- 11 a c + 2 b^{2}\right )}{c^{2} \left (- 4 a c + b^{2}\right )^{2}} - \frac{3 b x \left (- 9 a c + 2 b^{2}\right ) \left (- 3 a c + b^{2}\right )}{c^{4} \left (- 4 a c + b^{2}\right )^{2}} + \frac{3 b \left (- 70 a^{3} c^{3} + 70 a^{2} b^{2} c^{2} - 21 a b^{4} c + 2 b^{6}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{5} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{x^{6} \left (2 a + b x\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} + \frac{x^{4} \left (3 a \left (- 8 a c + b^{2}\right ) + 3 b x \left (- 6 a c + b^{2}\right )\right )}{2 c \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} + \frac{6 \left (8 a^{2} c^{2} - \frac{13 a b^{2} c}{2} + b^{4}\right ) \int x\, dx}{c^{3} \left (- 4 a c + b^{2}\right )^{2}} + \frac{3 \left (- \frac{a c}{2} + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(c*x**2+b*x+a)**3,x)

[Out]

-b*x**3*(-11*a*c + 2*b**2)/(c**2*(-4*a*c + b**2)**2) - 3*b*x*(-9*a*c + 2*b**2)*(
-3*a*c + b**2)/(c**4*(-4*a*c + b**2)**2) + 3*b*(-70*a**3*c**3 + 70*a**2*b**2*c**
2 - 21*a*b**4*c + 2*b**6)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(c**5*(-4*a*c +
 b**2)**(5/2)) + x**6*(2*a + b*x)/(2*(-4*a*c + b**2)*(a + b*x + c*x**2)**2) + x*
*4*(3*a*(-8*a*c + b**2) + 3*b*x*(-6*a*c + b**2))/(2*c*(-4*a*c + b**2)**2*(a + b*
x + c*x**2)) + 6*(8*a**2*c**2 - 13*a*b**2*c/2 + b**4)*Integral(x, x)/(c**3*(-4*a
*c + b**2)**2) + 3*(-a*c/2 + b**2)*log(a + b*x + c*x**2)/c**5

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Mathematica [A]  time = 0.839034, size = 299, normalized size = 1.02 \[ \frac{\frac{6 b c \left (70 a^3 c^3-70 a^2 b^2 c^2+21 a b^4 c-2 b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{-2 a^4 c^3+a^3 b c^2 (9 b-7 c x)+2 a^2 b^3 c (7 c x-3 b)+a b^5 (b-7 c x)+b^7 x}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac{48 a^4 c^4-153 a^3 b^2 c^3+126 a^3 b c^4 x+88 a^2 b^4 c^2-182 a^2 b^3 c^3 x-17 a b^6 c+70 a b^5 c^2 x+b^8-8 b^7 c x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}-3 c \left (a c-2 b^2\right ) \log (a+x (b+c x))-6 b c^2 x+c^3 x^2}{2 c^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x + c*x^2)^3,x]

[Out]

(-6*b*c^2*x + c^3*x^2 - (b^8 - 17*a*b^6*c + 88*a^2*b^4*c^2 - 153*a^3*b^2*c^3 + 4
8*a^4*c^4 - 8*b^7*c*x + 70*a*b^5*c^2*x - 182*a^2*b^3*c^3*x + 126*a^3*b*c^4*x)/((
b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (-2*a^4*c^3 + b^7*x + a*b^5*(b - 7*c*x) + a^
3*b*c^2*(9*b - 7*c*x) + 2*a^2*b^3*c*(-3*b + 7*c*x))/((b^2 - 4*a*c)*(a + x*(b + c
*x))^2) + (6*b*c*(-2*b^6 + 21*a*b^4*c - 70*a^2*b^2*c^2 + 70*a^3*c^3)*ArcTan[(b +
 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) - 3*c*(-2*b^2 + a*c)*Log[a + x
*(b + c*x)])/(2*c^6)

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Maple [B]  time = 0.039, size = 1691, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^2+b*x+a)^3,x)

[Out]

-20/c^2/(c*x^2+b*x+a)^2*a^5/(16*a^2*c^2-8*a*b^2*c+b^4)+210/c^2/(1024*a^5*c^5-128
0*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c
*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a
^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*a^3*b-210/c^3
/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)
^(1/2)*arctan((2*c*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1
024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1
/2))*a^2*b^3+63/c^4/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c
^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*(16*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8
*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+
20*a*b^8*c-b^10)^(1/2))*a*b^5+1/2/c^3*x^2-53/2/c^4/(c*x^2+b*x+a)^2/(16*a^2*c^2-8
*a*b^2*c+b^4)*x^2*a*b^6-58/c^4/(c*x^2+b*x+a)^2*a^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4
)*x-63/c/(c*x^2+b*x+a)^2*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a^3+91/c^2/(c*x^2+b*x+
a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3*a^2-35/c^3/(c*x^2+b*x+a)^2*b^5/(16*a^2*c
^2-8*a*b^2*c+b^4)*x^3*a+27/2/c^2/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*
a^3*b^2-3/c^4*x*b-24/c/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^4+7/2/c^
5/(c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^8-6/c^5/(1024*a^5*c^5-1280*a^
4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*(16
*a^2*c^2-8*a*b^2*c+b^4)*x+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b
^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b^7-24/c^2/(16*a^
2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*a^3+3/c^5/(16*
a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*b^6+47/c^3/(
c*x^2+b*x+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*a^2*b^4-73/c^2/(c*x^2+b*x+a)^2*a^4
*b/(16*a^2*c^2-8*a*b^2*c+b^4)*x+136/c^3/(c*x^2+b*x+a)^2*a^3*b^3/(16*a^2*c^2-8*a*
b^2*c+b^4)*x+7/c^5/(c*x^2+b*x+a)^2*a*b^7/(16*a^2*c^2-8*a*b^2*c+b^4)*x+115/2/c^3/
(c*x^2+b*x+a)^2*a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*b^2+7/2/c^5/(c*x^2+b*x+a)^2*a^2/(
16*a^2*c^2-8*a*b^2*c+b^4)*b^6+4/c^4/(c*x^2+b*x+a)^2*b^7/(16*a^2*c^2-8*a*b^2*c+b^
4)*x^3-55/2/c^4/(c*x^2+b*x+a)^2*a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*b^4+60/c^3/(16*a^
2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*a^2*b^2-51/2/c
^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^2+b*x+a))*a*b^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(x^7/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245113, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[-1/2*(3*(2*a^2*b^7 - 21*a^3*b^5*c + 70*a^4*b^3*c^2 - 70*a^5*b*c^3 + (2*b^7*c^2
- 21*a*b^5*c^3 + 70*a^2*b^3*c^4 - 70*a^3*b*c^5)*x^4 + 2*(2*b^8*c - 21*a*b^6*c^2
+ 70*a^2*b^4*c^3 - 70*a^3*b^2*c^4)*x^3 + (2*b^9 - 17*a*b^7*c + 28*a^2*b^5*c^2 +
70*a^3*b^3*c^3 - 140*a^4*b*c^4)*x^2 + 2*(2*a*b^8 - 21*a^2*b^6*c + 70*a^3*b^4*c^2
 - 70*a^4*b^2*c^3)*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)) - (7*a^2*b^6 - 55*
a^3*b^4*c + 115*a^4*b^2*c^2 - 40*a^5*c^3 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*
x^6 - 4*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^5 - (11*b^6*c^2 - 90*a*b^4*c^3
+ 192*a^2*b^2*c^4 - 32*a^3*c^5)*x^4 + 2*(b^7*c - 16*a*b^5*c^2 + 83*a^2*b^3*c^3 -
 143*a^3*b*c^4)*x^3 + (7*b^8 - 65*a*b^6*c + 191*a^2*b^4*c^2 - 173*a^3*b^2*c^3 -
32*a^4*c^4)*x^2 + 2*(7*a*b^7 - 61*a^2*b^5*c + 160*a^3*b^3*c^2 - 121*a^4*b*c^3)*x
 + 3*(2*a^2*b^6 - 17*a^3*b^4*c + 40*a^4*b^2*c^2 - 16*a^5*c^3 + (2*b^6*c^2 - 17*a
*b^4*c^3 + 40*a^2*b^2*c^4 - 16*a^3*c^5)*x^4 + 2*(2*b^7*c - 17*a*b^5*c^2 + 40*a^2
*b^3*c^3 - 16*a^3*b*c^4)*x^3 + (2*b^8 - 13*a*b^6*c + 6*a^2*b^4*c^2 + 64*a^3*b^2*
c^3 - 32*a^4*c^4)*x^2 + 2*(2*a*b^7 - 17*a^2*b^5*c + 40*a^3*b^3*c^2 - 16*a^4*b*c^
3)*x)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 1
6*a^4*c^7 + (b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*x^4 + 2*(b^5*c^6 - 8*a*b^3*c^7
+ 16*a^2*b*c^8)*x^3 + (b^6*c^5 - 6*a*b^4*c^6 + 32*a^3*c^8)*x^2 + 2*(a*b^5*c^5 -
8*a^2*b^3*c^6 + 16*a^3*b*c^7)*x)*sqrt(b^2 - 4*a*c)), -1/2*(6*(2*a^2*b^7 - 21*a^3
*b^5*c + 70*a^4*b^3*c^2 - 70*a^5*b*c^3 + (2*b^7*c^2 - 21*a*b^5*c^3 + 70*a^2*b^3*
c^4 - 70*a^3*b*c^5)*x^4 + 2*(2*b^8*c - 21*a*b^6*c^2 + 70*a^2*b^4*c^3 - 70*a^3*b^
2*c^4)*x^3 + (2*b^9 - 17*a*b^7*c + 28*a^2*b^5*c^2 + 70*a^3*b^3*c^3 - 140*a^4*b*c
^4)*x^2 + 2*(2*a*b^8 - 21*a^2*b^6*c + 70*a^3*b^4*c^2 - 70*a^4*b^2*c^3)*x)*arctan
(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (7*a^2*b^6 - 55*a^3*b^4*c + 11
5*a^4*b^2*c^2 - 40*a^5*c^3 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^6 - 4*(b^5*c
^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^5 - (11*b^6*c^2 - 90*a*b^4*c^3 + 192*a^2*b^2*
c^4 - 32*a^3*c^5)*x^4 + 2*(b^7*c - 16*a*b^5*c^2 + 83*a^2*b^3*c^3 - 143*a^3*b*c^4
)*x^3 + (7*b^8 - 65*a*b^6*c + 191*a^2*b^4*c^2 - 173*a^3*b^2*c^3 - 32*a^4*c^4)*x^
2 + 2*(7*a*b^7 - 61*a^2*b^5*c + 160*a^3*b^3*c^2 - 121*a^4*b*c^3)*x + 3*(2*a^2*b^
6 - 17*a^3*b^4*c + 40*a^4*b^2*c^2 - 16*a^5*c^3 + (2*b^6*c^2 - 17*a*b^4*c^3 + 40*
a^2*b^2*c^4 - 16*a^3*c^5)*x^4 + 2*(2*b^7*c - 17*a*b^5*c^2 + 40*a^2*b^3*c^3 - 16*
a^3*b*c^4)*x^3 + (2*b^8 - 13*a*b^6*c + 6*a^2*b^4*c^2 + 64*a^3*b^2*c^3 - 32*a^4*c
^4)*x^2 + 2*(2*a*b^7 - 17*a^2*b^5*c + 40*a^3*b^3*c^2 - 16*a^4*b*c^3)*x)*log(c*x^
2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a^2*b^4*c^5 - 8*a^3*b^2*c^6 + 16*a^4*c^7 + (
b^4*c^7 - 8*a*b^2*c^8 + 16*a^2*c^9)*x^4 + 2*(b^5*c^6 - 8*a*b^3*c^7 + 16*a^2*b*c^
8)*x^3 + (b^6*c^5 - 6*a*b^4*c^6 + 32*a^3*c^8)*x^2 + 2*(a*b^5*c^5 - 8*a^2*b^3*c^6
 + 16*a^3*b*c^7)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 20.5, size = 1875, normalized size = 6.38 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**2+b*x+a)**3,x)

[Out]

-3*b*x/c**4 + (-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 +
 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*
b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c**
5))*log(x + (96*a**4*c**3 - 159*a**3*b**2*c**2 + 64*a**3*c**7*(-3*b*sqrt(-(4*a*c
 - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(
1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 +
 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c**5)) + 57*a**2*b**4*c - 48*a**2*b
**2*c**6*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a
*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*
c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c**5)) -
 6*a*b**6 + 12*a*b**4*c**5*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**
2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**
4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2
*b**2)/(2*c**5)) - b**6*c**4*(-3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a
**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c
**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c -
 2*b**2)/(2*c**5)))/(210*a**3*b*c**3 - 210*a**2*b**3*c**2 + 63*a*b**5*c - 6*b**7
)) + (3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4
*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3
- 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c**5))*log(x
+ (96*a**4*c**3 - 159*a**3*b**2*c**2 + 64*a**3*c**7*(3*b*sqrt(-(4*a*c - b**2)**5
)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c
**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*
c - b**10)) - 3*(a*c - 2*b**2)/(2*c**5)) + 57*a**2*b**4*c - 48*a**2*b**2*c**6*(3
*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 + 21*a*b**4*c - 2*
b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a
**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c**5)) - 6*a*b**6 +
12*a*b**4*c**5*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2 +
 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*
b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c**
5)) - b**6*c**4*(3*b*sqrt(-(4*a*c - b**2)**5)*(70*a**3*c**3 - 70*a**2*b**2*c**2
+ 21*a*b**4*c - 2*b**6)/(2*c**5*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3
*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) - 3*(a*c - 2*b**2)/(2*c*
*5)))/(210*a**3*b*c**3 - 210*a**2*b**3*c**2 + 63*a*b**5*c - 6*b**7)) - (40*a**5*
c**3 - 115*a**4*b**2*c**2 + 55*a**3*b**4*c - 7*a**2*b**6 + x**3*(126*a**3*b*c**4
 - 182*a**2*b**3*c**3 + 70*a*b**5*c**2 - 8*b**7*c) + x**2*(48*a**4*c**4 - 27*a**
3*b**2*c**3 - 94*a**2*b**4*c**2 + 53*a*b**6*c - 7*b**8) + x*(146*a**4*b*c**3 - 2
72*a**3*b**3*c**2 + 116*a**2*b**5*c - 14*a*b**7))/(32*a**4*c**7 - 16*a**3*b**2*c
**6 + 2*a**2*b**4*c**5 + x**4*(32*a**2*c**9 - 16*a*b**2*c**8 + 2*b**4*c**7) + x*
*3*(64*a**2*b*c**8 - 32*a*b**3*c**7 + 4*b**5*c**6) + x**2*(64*a**3*c**8 - 12*a*b
**4*c**6 + 2*b**6*c**5) + x*(64*a**3*b*c**7 - 32*a**2*b**3*c**6 + 4*a*b**5*c**5)
) + x**2/(2*c**3)

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GIAC/XCAS [A]  time = 0.21239, size = 448, normalized size = 1.52 \[ -\frac{3 \,{\left (2 \, b^{7} - 21 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 70 \, a^{3} b c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{5} - 8 \, a b^{2} c^{6} + 16 \, a^{2} c^{7}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{3 \,{\left (2 \, b^{2} - a c\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac{c^{3} x^{2} - 6 \, b c^{2} x}{2 \, c^{6}} + \frac{7 \, a^{2} b^{6} - 55 \, a^{3} b^{4} c + 115 \, a^{4} b^{2} c^{2} - 40 \, a^{5} c^{3} + 2 \,{\left (4 \, b^{7} c - 35 \, a b^{5} c^{2} + 91 \, a^{2} b^{3} c^{3} - 63 \, a^{3} b c^{4}\right )} x^{3} +{\left (7 \, b^{8} - 53 \, a b^{6} c + 94 \, a^{2} b^{4} c^{2} + 27 \, a^{3} b^{2} c^{3} - 48 \, a^{4} c^{4}\right )} x^{2} + 2 \,{\left (7 \, a b^{7} - 58 \, a^{2} b^{5} c + 136 \, a^{3} b^{3} c^{2} - 73 \, a^{4} b c^{3}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )}^{2} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

-3*(2*b^7 - 21*a*b^5*c + 70*a^2*b^3*c^2 - 70*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(
-b^2 + 4*a*c))/((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*sqrt(-b^2 + 4*a*c)) + 3/2*(
2*b^2 - a*c)*ln(c*x^2 + b*x + a)/c^5 + 1/2*(c^3*x^2 - 6*b*c^2*x)/c^6 + 1/2*(7*a^
2*b^6 - 55*a^3*b^4*c + 115*a^4*b^2*c^2 - 40*a^5*c^3 + 2*(4*b^7*c - 35*a*b^5*c^2
+ 91*a^2*b^3*c^3 - 63*a^3*b*c^4)*x^3 + (7*b^8 - 53*a*b^6*c + 94*a^2*b^4*c^2 + 27
*a^3*b^2*c^3 - 48*a^4*c^4)*x^2 + 2*(7*a*b^7 - 58*a^2*b^5*c + 136*a^3*b^3*c^2 - 7
3*a^4*b*c^3)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*c^5)

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