3.2200 \(\int \frac{x^7}{\left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=291 \[ \frac{x^2 \left (b x \left (140 a^2 c^2-32 a b^2 c+3 b^4\right )+3 a \left (64 a^2 c^2-10 a b^2 c+b^4\right )\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{b x \left (38 a^2 c^2-11 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^3}+\frac{b \left (-140 a^3 c^3+70 a^2 b^2 c^2-14 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{7/2}}+\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (b x \left (b^2-14 a c\right )+a \left (b^2-24 a c\right )\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{\log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

-((b*(b^4 - 11*a*b^2*c + 38*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^3)) + (x^6*(2*a + b*x
))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (x^4*(a*(b^2 - 24*a*c) + b*(b^2 - 14*
a*c)*x))/(6*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (x^2*(3*a*(b^4 - 10*a*b^2*c
+ 64*a^2*c^2) + b*(3*b^4 - 32*a*b^2*c + 140*a^2*c^2)*x))/(6*c^2*(b^2 - 4*a*c)^3
*(a + b*x + c*x^2)) + (b*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3*c^3)*ArcTa
nh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(7/2)) + Log[a + b*x + c*x
^2]/(2*c^4)

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Rubi [A]  time = 1.07522, antiderivative size = 291, 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 \[ \frac{x^2 \left (b x \left (140 a^2 c^2-32 a b^2 c+3 b^4\right )+3 a \left (64 a^2 c^2-10 a b^2 c+b^4\right )\right )}{6 c^2 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac{b x \left (38 a^2 c^2-11 a b^2 c+b^4\right )}{c^3 \left (b^2-4 a c\right )^3}+\frac{b \left (-140 a^3 c^3+70 a^2 b^2 c^2-14 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \left (b^2-4 a c\right )^{7/2}}+\frac{x^6 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}+\frac{x^4 \left (b x \left (b^2-14 a c\right )+a \left (b^2-24 a c\right )\right )}{6 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}+\frac{\log \left (a+b x+c x^2\right )}{2 c^4} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x + c*x^2)^4,x]

[Out]

-((b*(b^4 - 11*a*b^2*c + 38*a^2*c^2)*x)/(c^3*(b^2 - 4*a*c)^3)) + (x^6*(2*a + b*x
))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (x^4*(a*(b^2 - 24*a*c) + b*(b^2 - 14*
a*c)*x))/(6*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (x^2*(3*a*(b^4 - 10*a*b^2*c
+ 64*a^2*c^2) + b*(3*b^4 - 32*a*b^2*c + 140*a^2*c^2)*x))/(6*c^2*(b^2 - 4*a*c)^3
*(a + b*x + c*x^2)) + (b*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3*c^3)*ArcTa
nh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^4*(b^2 - 4*a*c)^(7/2)) + Log[a + b*x + c*x
^2]/(2*c^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \left (- 140 a^{3} c^{3} + 70 a^{2} b^{2} c^{2} - 14 a b^{4} c + b^{6}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{4} \left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} + \frac{x^{6} \left (2 a + b x\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{3}} + \frac{x^{4} \left (a \left (- 24 a c + b^{2}\right ) + b x \left (- 14 a c + b^{2}\right )\right )}{6 c \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{2}} + \frac{x^{2} \left (3 a \left (64 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right ) + b x \left (140 a^{2} c^{2} - 32 a b^{2} c + 3 b^{4}\right )\right )}{6 c^{2} \left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )} - \frac{\left (- a c \left (- 38 a c + 11 b^{2}\right ) + b^{4}\right ) \int b\, dx}{c^{3} \left (- 4 a c + b^{2}\right )^{3}} + \frac{\log{\left (a + b x + c x^{2} \right )}}{2 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(c*x**2+b*x+a)**4,x)

[Out]

b*(-140*a**3*c**3 + 70*a**2*b**2*c**2 - 14*a*b**4*c + b**6)*atanh((b + 2*c*x)/sq
rt(-4*a*c + b**2))/(c**4*(-4*a*c + b**2)**(7/2)) + x**6*(2*a + b*x)/(3*(-4*a*c +
b**2)*(a + b*x + c*x**2)**3) + x**4*(a*(-24*a*c + b**2) + b*x*(-14*a*c + b**2))
/(6*c*(-4*a*c + b**2)**2*(a + b*x + c*x**2)**2) + x**2*(3*a*(64*a**2*c**2 - 10*a
*b**2*c + b**4) + b*x*(140*a**2*c**2 - 32*a*b**2*c + 3*b**4))/(6*c**2*(-4*a*c +
b**2)**3*(a + b*x + c*x**2)) - (-a*c*(-38*a*c + 11*b**2) + b**4)*Integral(b, x)/
(c**3*(-4*a*c + b**2)**3) + log(a + b*x + c*x**2)/(2*c**4)

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Mathematica [A]  time = 1.17588, size = 386, normalized size = 1.33 \[ \frac{\frac{6 b c^2 \left (-140 a^3 c^3+70 a^2 b^2 c^2-14 a b^4 c+b^6\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}+\frac{2 \left (-2 a^4 c^3+a^3 b c^2 (9 b-7 c x)+2 a^2 b^3 c (7 c x-3 b)+a b^5 (b-7 c x)+b^7 x\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^3}-\frac{3 c \left (192 a^4 c^4-374 a^3 b^2 c^3+308 a^3 b c^4 x+191 a^2 b^4 c^2-266 a^2 b^3 c^3 x-40 a b^6 c+70 a b^5 c^2 x+3 b^8-6 b^7 c x\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}+\frac{-72 a^4 c^4+233 a^3 b^2 c^3-182 a^3 b c^4 x-139 a^2 b^4 c^2+259 a^2 b^3 c^3 x+29 a b^6 c-98 a b^5 c^2 x-2 b^8+11 b^7 c x}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+3 c^2 \log (a+x (b+c x))}{6 c^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x + c*x^2)^4,x]

[Out]

((-2*b^8 + 29*a*b^6*c - 139*a^2*b^4*c^2 + 233*a^3*b^2*c^3 - 72*a^4*c^4 + 11*b^7*
c*x - 98*a*b^5*c^2*x + 259*a^2*b^3*c^3*x - 182*a^3*b*c^4*x)/((b^2 - 4*a*c)^2*(a
+ x*(b + c*x))^2) - (3*c*(3*b^8 - 40*a*b^6*c + 191*a^2*b^4*c^2 - 374*a^3*b^2*c^3
+ 192*a^4*c^4 - 6*b^7*c*x + 70*a*b^5*c^2*x - 266*a^2*b^3*c^3*x + 308*a^3*b*c^4*
x))/((b^2 - 4*a*c)^3*(a + x*(b + c*x))) + (2*(-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))^3) + (6*b*c^2*(b^6 - 14*a*b^4*c + 70*a^2*b^2*c^2 - 140*a^3*c^3)*A
rcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) + 3*c^2*Log[a + x*(b
+ c*x)])/(6*c^6)

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Maple [B]  time = 0.05, size = 1556, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^2+b*x+a)^4,x)

[Out]

((154*a^3*c^3-133*a^2*b^2*c^2+35*a*b^4*c-3*b^6)*b/c^2/(64*a^3*c^3-48*a^2*b^2*c^2
+12*a*b^4*c-b^6)*x^5+1/2*(192*a^4*c^4+242*a^3*b^2*c^3-341*a^2*b^4*c^2+100*a*b^6*
c-9*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^3*x^4+1/6*b/c^4*(2272*a^4*
c^4-1698*a^3*b^2*c^3+117*a^2*b^4*c^2+76*a*b^6*c-11*b^8)/(64*a^3*c^3-48*a^2*b^2*c
^2+12*a*b^4*c-b^6)*x^3+1/2/c^4*a*(288*a^4*c^4+152*a^3*b^2*c^3-381*a^2*b^4*c^2+11
9*a*b^6*c-11*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^2+1/2*a^2*b*(428*
a^3*c^3-460*a^2*b^2*c^2+126*a*b^4*c-11*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*
c-b^6)/c^4*x+1/6*(352*a^3*c^3-438*a^2*b^2*c^2+124*a*b^4*c-11*b^6)*a^3/c^4/(64*a^
3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6))/(c*x^2+b*x+a)^3+1/2/c^4*ln((64*a^3*c^3-48*
a^2*b^2*c^2+12*a*b^4*c-b^6)*c^3*(c*x^2+b*x+a))-140/(16384*a^7*c^13-28672*a^6*b^2
*c^12+21504*a^5*b^4*c^11-8960*a^4*b^6*c^10+2240*a^3*b^8*c^9-336*a^2*b^10*c^8+28*
a*b^12*c^7-b^14*c^6)^(1/2)*arctan((2*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*
c^4*x+(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^3*b)/(16384*a^7*c^13-28672*a^
6*b^2*c^12+21504*a^5*b^4*c^11-8960*a^4*b^6*c^10+2240*a^3*b^8*c^9-336*a^2*b^10*c^
8+28*a*b^12*c^7-b^14*c^6)^(1/2))*a^3*b*c^2+70/(16384*a^7*c^13-28672*a^6*b^2*c^12
+21504*a^5*b^4*c^11-8960*a^4*b^6*c^10+2240*a^3*b^8*c^9-336*a^2*b^10*c^8+28*a*b^1
2*c^7-b^14*c^6)^(1/2)*arctan((2*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^4*x
+(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^3*b)/(16384*a^7*c^13-28672*a^6*b^2
*c^12+21504*a^5*b^4*c^11-8960*a^4*b^6*c^10+2240*a^3*b^8*c^9-336*a^2*b^10*c^8+28*
a*b^12*c^7-b^14*c^6)^(1/2))*a^2*b^3*c-14/(16384*a^7*c^13-28672*a^6*b^2*c^12+2150
4*a^5*b^4*c^11-8960*a^4*b^6*c^10+2240*a^3*b^8*c^9-336*a^2*b^10*c^8+28*a*b^12*c^7
-b^14*c^6)^(1/2)*arctan((2*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^4*x+(64*
a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^3*b)/(16384*a^7*c^13-28672*a^6*b^2*c^12
+21504*a^5*b^4*c^11-8960*a^4*b^6*c^10+2240*a^3*b^8*c^9-336*a^2*b^10*c^8+28*a*b^1
2*c^7-b^14*c^6)^(1/2))*a*b^5+1/(16384*a^7*c^13-28672*a^6*b^2*c^12+21504*a^5*b^4*
c^11-8960*a^4*b^6*c^10+2240*a^3*b^8*c^9-336*a^2*b^10*c^8+28*a*b^12*c^7-b^14*c^6)
^(1/2)*arctan((2*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^4*x+(64*a^3*c^3-48
*a^2*b^2*c^2+12*a*b^4*c-b^6)*c^3*b)/(16384*a^7*c^13-28672*a^6*b^2*c^12+21504*a^5
*b^4*c^11-8960*a^4*b^6*c^10+2240*a^3*b^8*c^9-336*a^2*b^10*c^8+28*a*b^12*c^7-b^14
*c^6)^(1/2))*b^7/c

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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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.2529, 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)^4,x, algorithm="fricas")

[Out]

[1/6*(3*(a^3*b^7 - 14*a^4*b^5*c + 70*a^5*b^3*c^2 - 140*a^6*b*c^3 + (b^7*c^3 - 14
*a*b^5*c^4 + 70*a^2*b^3*c^5 - 140*a^3*b*c^6)*x^6 + 3*(b^8*c^2 - 14*a*b^6*c^3 + 7
0*a^2*b^4*c^4 - 140*a^3*b^2*c^5)*x^5 + 3*(b^9*c - 13*a*b^7*c^2 + 56*a^2*b^5*c^3
- 70*a^3*b^3*c^4 - 140*a^4*b*c^5)*x^4 + (b^10 - 8*a*b^8*c - 14*a^2*b^6*c^2 + 280
*a^3*b^4*c^3 - 840*a^4*b^2*c^4)*x^3 + 3*(a*b^9 - 13*a^2*b^7*c + 56*a^3*b^5*c^2 -
70*a^4*b^3*c^3 - 140*a^5*b*c^4)*x^2 + 3*(a^2*b^8 - 14*a^3*b^6*c + 70*a^4*b^4*c^
2 - 140*a^5*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)) + (11*a^3*b^6 - 1
24*a^4*b^4*c + 438*a^5*b^2*c^2 - 352*a^6*c^3 + 6*(3*b^7*c^2 - 35*a*b^5*c^3 + 133
*a^2*b^3*c^4 - 154*a^3*b*c^5)*x^5 + 3*(9*b^8*c - 100*a*b^6*c^2 + 341*a^2*b^4*c^3
- 242*a^3*b^2*c^4 - 192*a^4*c^5)*x^4 + (11*b^9 - 76*a*b^7*c - 117*a^2*b^5*c^2 +
1698*a^3*b^3*c^3 - 2272*a^4*b*c^4)*x^3 + 3*(11*a*b^8 - 119*a^2*b^6*c + 381*a^3*
b^4*c^2 - 152*a^4*b^2*c^3 - 288*a^5*c^4)*x^2 + 3*(11*a^2*b^7 - 126*a^3*b^5*c + 4
60*a^4*b^3*c^2 - 428*a^5*b*c^3)*x + 3*(a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 -
64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^6 + 3*(b^
7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*a*b^6*
c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*x^4 + (b^9 - 6*a*b^7*c - 24*
a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8 - 11*a^2*b^6*c + 3
6*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5*c + 4
8*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((a^3*
b^6*c^4 - 12*a^4*b^4*c^5 + 48*a^5*b^2*c^6 - 64*a^6*c^7 + (b^6*c^7 - 12*a*b^4*c^8
+ 48*a^2*b^2*c^9 - 64*a^3*c^10)*x^6 + 3*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^
8 - 64*a^3*b*c^9)*x^5 + 3*(b^8*c^5 - 11*a*b^6*c^6 + 36*a^2*b^4*c^7 - 16*a^3*b^2*
c^8 - 64*a^4*c^9)*x^4 + (b^9*c^4 - 6*a*b^7*c^5 - 24*a^2*b^5*c^6 + 224*a^3*b^3*c^
7 - 384*a^4*b*c^8)*x^3 + 3*(a*b^8*c^4 - 11*a^2*b^6*c^5 + 36*a^3*b^4*c^6 - 16*a^4
*b^2*c^7 - 64*a^5*c^8)*x^2 + 3*(a^2*b^7*c^4 - 12*a^3*b^5*c^5 + 48*a^4*b^3*c^6 -
64*a^5*b*c^7)*x)*sqrt(b^2 - 4*a*c)), -1/6*(6*(a^3*b^7 - 14*a^4*b^5*c + 70*a^5*b^
3*c^2 - 140*a^6*b*c^3 + (b^7*c^3 - 14*a*b^5*c^4 + 70*a^2*b^3*c^5 - 140*a^3*b*c^6
)*x^6 + 3*(b^8*c^2 - 14*a*b^6*c^3 + 70*a^2*b^4*c^4 - 140*a^3*b^2*c^5)*x^5 + 3*(b
^9*c - 13*a*b^7*c^2 + 56*a^2*b^5*c^3 - 70*a^3*b^3*c^4 - 140*a^4*b*c^5)*x^4 + (b^
10 - 8*a*b^8*c - 14*a^2*b^6*c^2 + 280*a^3*b^4*c^3 - 840*a^4*b^2*c^4)*x^3 + 3*(a*
b^9 - 13*a^2*b^7*c + 56*a^3*b^5*c^2 - 70*a^4*b^3*c^3 - 140*a^5*b*c^4)*x^2 + 3*(a
^2*b^8 - 14*a^3*b^6*c + 70*a^4*b^4*c^2 - 140*a^5*b^2*c^3)*x)*arctan(-sqrt(-b^2 +
4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (11*a^3*b^6 - 124*a^4*b^4*c + 438*a^5*b^2*c
^2 - 352*a^6*c^3 + 6*(3*b^7*c^2 - 35*a*b^5*c^3 + 133*a^2*b^3*c^4 - 154*a^3*b*c^5
)*x^5 + 3*(9*b^8*c - 100*a*b^6*c^2 + 341*a^2*b^4*c^3 - 242*a^3*b^2*c^4 - 192*a^4
*c^5)*x^4 + (11*b^9 - 76*a*b^7*c - 117*a^2*b^5*c^2 + 1698*a^3*b^3*c^3 - 2272*a^4
*b*c^4)*x^3 + 3*(11*a*b^8 - 119*a^2*b^6*c + 381*a^3*b^4*c^2 - 152*a^4*b^2*c^3 -
288*a^5*c^4)*x^2 + 3*(11*a^2*b^7 - 126*a^3*b^5*c + 460*a^4*b^3*c^2 - 428*a^5*b*c
^3)*x + 3*(a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*
a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^
2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^
3*b^2*c^4 - 64*a^4*c^5)*x^4 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^
3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c
^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3
)*x)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a^3*b^6*c^4 - 12*a^4*b^4*c^5 +
48*a^5*b^2*c^6 - 64*a^6*c^7 + (b^6*c^7 - 12*a*b^4*c^8 + 48*a^2*b^2*c^9 - 64*a^3*
c^10)*x^6 + 3*(b^7*c^6 - 12*a*b^5*c^7 + 48*a^2*b^3*c^8 - 64*a^3*b*c^9)*x^5 + 3*(
b^8*c^5 - 11*a*b^6*c^6 + 36*a^2*b^4*c^7 - 16*a^3*b^2*c^8 - 64*a^4*c^9)*x^4 + (b^
9*c^4 - 6*a*b^7*c^5 - 24*a^2*b^5*c^6 + 224*a^3*b^3*c^7 - 384*a^4*b*c^8)*x^3 + 3*
(a*b^8*c^4 - 11*a^2*b^6*c^5 + 36*a^3*b^4*c^6 - 16*a^4*b^2*c^7 - 64*a^5*c^8)*x^2
+ 3*(a^2*b^7*c^4 - 12*a^3*b^5*c^5 + 48*a^4*b^3*c^6 - 64*a^5*b*c^7)*x)*sqrt(-b^2
+ 4*a*c))]

_______________________________________________________________________________________

Sympy [A]  time = 26.8201, size = 2565, normalized size = 8.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**2+b*x+a)**4,x)

[Out]

(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c -
b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8
960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c -
b**14)) + 1/(2*c**4))*log(x + (-256*a**4*c**7*(-b*sqrt(-(4*a*c - b**2)**7)*(140*
a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 2
8672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**
8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) + 128*a**4*c
**3 + 256*a**3*b**2*c**6*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b
**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 +
21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**
10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) - 58*a**3*b**2*c**2 - 96*a**2*b**
4*c**5*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b*
*4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c
**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**
12*c - b**14)) + 1/(2*c**4)) + 13*a**2*b**4*c + 16*a*b**6*c**4*(-b*sqrt(-(4*a*c
- b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16
384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**
4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c*
*4)) - a*b**6 - b**8*c**3*(-b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*
b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6
+ 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b*
*10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)))/(140*a**3*b*c**3 - 70*a**2*b**3
*c**2 + 14*a*b**5*c - b**7)) + (b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a
**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c
**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**
2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4))*log(x + (-256*a**4*c**7*(b*s
qrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)
/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a
**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14
)) + 1/(2*c**4)) + 128*a**4*c**3 + 256*a**3*b**2*c**6*(b*sqrt(-(4*a*c - b**2)**7
)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c
**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a
**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) - 58*
a**3*b**2*c**2 - 96*a**2*b**4*c**5*(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 -
70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b*
*2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336
*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)) + 13*a**2*b**4*c + 16*a*
b**6*c**4*(b*sqrt(-(4*a*c - b**2)**7)*(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*
b**4*c - b**6)/(2*c**4*(16384*a**7*c**7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4
*c**5 - 8960*a**4*b**6*c**4 + 2240*a**3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b
**12*c - b**14)) + 1/(2*c**4)) - a*b**6 - b**8*c**3*(b*sqrt(-(4*a*c - b**2)**7)*
(140*a**3*c**3 - 70*a**2*b**2*c**2 + 14*a*b**4*c - b**6)/(2*c**4*(16384*a**7*c**
7 - 28672*a**6*b**2*c**6 + 21504*a**5*b**4*c**5 - 8960*a**4*b**6*c**4 + 2240*a**
3*b**8*c**3 - 336*a**2*b**10*c**2 + 28*a*b**12*c - b**14)) + 1/(2*c**4)))/(140*a
**3*b*c**3 - 70*a**2*b**3*c**2 + 14*a*b**5*c - b**7)) + (352*a**6*c**3 - 438*a**
5*b**2*c**2 + 124*a**4*b**4*c - 11*a**3*b**6 + x**5*(924*a**3*b*c**5 - 798*a**2*
b**3*c**4 + 210*a*b**5*c**3 - 18*b**7*c**2) + x**4*(576*a**4*c**5 + 726*a**3*b**
2*c**4 - 1023*a**2*b**4*c**3 + 300*a*b**6*c**2 - 27*b**8*c) + x**3*(2272*a**4*b*
c**4 - 1698*a**3*b**3*c**3 + 117*a**2*b**5*c**2 + 76*a*b**7*c - 11*b**9) + x**2*
(864*a**5*c**4 + 456*a**4*b**2*c**3 - 1143*a**3*b**4*c**2 + 357*a**2*b**6*c - 33
*a*b**8) + x*(1284*a**5*b*c**3 - 1380*a**4*b**3*c**2 + 378*a**3*b**5*c - 33*a**2
*b**7))/(384*a**6*c**7 - 288*a**5*b**2*c**6 + 72*a**4*b**4*c**5 - 6*a**3*b**6*c*
*4 + x**6*(384*a**3*c**10 - 288*a**2*b**2*c**9 + 72*a*b**4*c**8 - 6*b**6*c**7) +
x**5*(1152*a**3*b*c**9 - 864*a**2*b**3*c**8 + 216*a*b**5*c**7 - 18*b**7*c**6) +
x**4*(1152*a**4*c**9 + 288*a**3*b**2*c**8 - 648*a**2*b**4*c**7 + 198*a*b**6*c**
6 - 18*b**8*c**5) + x**3*(2304*a**4*b*c**8 - 1344*a**3*b**3*c**7 + 144*a**2*b**5
*c**6 + 36*a*b**7*c**5 - 6*b**9*c**4) + x**2*(1152*a**5*c**8 + 288*a**4*b**2*c**
7 - 648*a**3*b**4*c**6 + 198*a**2*b**6*c**5 - 18*a*b**8*c**4) + x*(1152*a**5*b*c
**7 - 864*a**4*b**3*c**6 + 216*a**3*b**5*c**5 - 18*a**2*b**7*c**4))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212298, size = 564, normalized size = 1.94 \[ -\frac{{\left (b^{7} - 14 \, a b^{5} c + 70 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac{11 \, a^{3} b^{6} - 124 \, a^{4} b^{4} c + 438 \, a^{5} b^{2} c^{2} - 352 \, a^{6} c^{3} + 6 \,{\left (3 \, b^{7} c^{2} - 35 \, a b^{5} c^{3} + 133 \, a^{2} b^{3} c^{4} - 154 \, a^{3} b c^{5}\right )} x^{5} + 3 \,{\left (9 \, b^{8} c - 100 \, a b^{6} c^{2} + 341 \, a^{2} b^{4} c^{3} - 242 \, a^{3} b^{2} c^{4} - 192 \, a^{4} c^{5}\right )} x^{4} +{\left (11 \, b^{9} - 76 \, a b^{7} c - 117 \, a^{2} b^{5} c^{2} + 1698 \, a^{3} b^{3} c^{3} - 2272 \, a^{4} b c^{4}\right )} x^{3} + 3 \,{\left (11 \, a b^{8} - 119 \, a^{2} b^{6} c + 381 \, a^{3} b^{4} c^{2} - 152 \, a^{4} b^{2} c^{3} - 288 \, a^{5} c^{4}\right )} x^{2} + 3 \,{\left (11 \, a^{2} b^{7} - 126 \, a^{3} b^{5} c + 460 \, a^{4} b^{3} c^{2} - 428 \, a^{5} b c^{3}\right )} x}{6 \,{\left (c x^{2} + b x + a\right )}^{3}{\left (b^{2} - 4 \, a c\right )}^{3} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^2 + b*x + a)^4,x, algorithm="giac")

[Out]

-(b^7 - 14*a*b^5*c + 70*a^2*b^3*c^2 - 140*a^3*b*c^3)*arctan((2*c*x + b)/sqrt(-b^
2 + 4*a*c))/((b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*b^2*c^6 - 64*a^3*c^7)*sqrt(-b^2 +
4*a*c)) + 1/2*ln(c*x^2 + b*x + a)/c^4 + 1/6*(11*a^3*b^6 - 124*a^4*b^4*c + 438*a^
5*b^2*c^2 - 352*a^6*c^3 + 6*(3*b^7*c^2 - 35*a*b^5*c^3 + 133*a^2*b^3*c^4 - 154*a^
3*b*c^5)*x^5 + 3*(9*b^8*c - 100*a*b^6*c^2 + 341*a^2*b^4*c^3 - 242*a^3*b^2*c^4 -
192*a^4*c^5)*x^4 + (11*b^9 - 76*a*b^7*c - 117*a^2*b^5*c^2 + 1698*a^3*b^3*c^3 - 2
272*a^4*b*c^4)*x^3 + 3*(11*a*b^8 - 119*a^2*b^6*c + 381*a^3*b^4*c^2 - 152*a^4*b^2
*c^3 - 288*a^5*c^4)*x^2 + 3*(11*a^2*b^7 - 126*a^3*b^5*c + 460*a^4*b^3*c^2 - 428*
a^5*b*c^3)*x)/((c*x^2 + b*x + a)^3*(b^2 - 4*a*c)^3*c^4)