3.2201 \(\int \frac{x^6}{\left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=146 \[ \frac{40 a^3 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}+\frac{10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

[Out]

(x^5*(2*a + b*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (5*a*x^3*(2*a + b*x))/
(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (10*a^2*x*(2*a + b*x))/((b^2 - 4*a*c)^
3*(a + b*x + c*x^2)) + (40*a^3*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*
a*c)^(7/2)

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Rubi [A]  time = 0.208879, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{40 a^3 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}+\frac{10 a^2 x (2 a+b x)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{x^5 (2 a+b x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac{5 a x^3 (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + b*x + c*x^2)^4,x]

[Out]

(x^5*(2*a + b*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) - (5*a*x^3*(2*a + b*x))/
(3*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) + (10*a^2*x*(2*a + b*x))/((b^2 - 4*a*c)^
3*(a + b*x + c*x^2)) + (40*a^3*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*
a*c)^(7/2)

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Rubi in Sympy [A]  time = 32.191, size = 139, normalized size = 0.95 \[ \frac{40 a^{3} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} + \frac{10 a^{2} x \left (2 a + b x\right )}{\left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )} - \frac{5 a x^{3} \left (2 a + b x\right )}{3 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{2}} + \frac{x^{5} \left (2 a + b x\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(c*x**2+b*x+a)**4,x)

[Out]

40*a**3*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(7/2) + 10*a**2*
x*(2*a + b*x)/((-4*a*c + b**2)**3*(a + b*x + c*x**2)) - 5*a*x**3*(2*a + b*x)/(3*
(-4*a*c + b**2)**2*(a + b*x + c*x**2)**2) + x**5*(2*a + b*x)/(3*(-4*a*c + b**2)*
(a + b*x + c*x**2)**3)

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Mathematica [B]  time = 0.374274, size = 314, normalized size = 2.15 \[ \frac{40 a^3 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{7/2}}+\frac{a^3 c^2 (5 b-2 c x)+a^2 b^2 c (9 c x-5 b)+a b^4 (b-6 c x)+b^6 x}{3 c^5 \left (4 a c-b^2\right ) (a+x (b+c x))^3}+\frac{74 a^3 b c^3-44 a^3 c^4 x-48 a^2 b^3 c^2+48 a^2 b^2 c^3 x+12 a b^5 c-12 a b^4 c^2 x-b^7+b^6 c x}{c^4 \left (4 a c-b^2\right )^3 (a+x (b+c x))}+\frac{-59 a^3 b c^3+26 a^3 c^4 x+48 a^2 b^3 c^2-72 a^2 b^2 c^3 x-12 a b^5 c+33 a b^4 c^2 x+b^7-4 b^6 c x}{3 c^5 \left (b^2-4 a c\right )^2 (a+x (b+c x))^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + b*x + c*x^2)^4,x]

[Out]

(b^7 - 12*a*b^5*c + 48*a^2*b^3*c^2 - 59*a^3*b*c^3 - 4*b^6*c*x + 33*a*b^4*c^2*x -
72*a^2*b^2*c^3*x + 26*a^3*c^4*x)/(3*c^5*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) +
(-b^7 + 12*a*b^5*c - 48*a^2*b^3*c^2 + 74*a^3*b*c^3 + b^6*c*x - 12*a*b^4*c^2*x +
48*a^2*b^2*c^3*x - 44*a^3*c^4*x)/(c^4*(-b^2 + 4*a*c)^3*(a + x*(b + c*x))) + (b^6
*x + a*b^4*(b - 6*c*x) + a^3*c^2*(5*b - 2*c*x) + a^2*b^2*c*(-5*b + 9*c*x))/(3*c^
5*(-b^2 + 4*a*c)*(a + x*(b + c*x))^3) + (40*a^3*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4
*a*c]])/(-b^2 + 4*a*c)^(7/2)

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Maple [B]  time = 0.024, size = 531, normalized size = 3.6 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{3}} \left ( -{\frac{ \left ( 44\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){x}^{5}}{c \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{b \left ( 14\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){x}^{4}}{{c}^{2} \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) }}-{\frac{ \left ( 160\,{a}^{4}{c}^{4}-286\,{a}^{3}{b}^{2}{c}^{3}+12\,{a}^{2}{b}^{4}{c}^{2}+7\,a{b}^{6}c-{b}^{8} \right ){x}^{3}}{3\, \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){c}^{3}}}+{\frac{ab \left ( 16\,{a}^{3}{c}^{3}+53\,{a}^{2}{b}^{2}{c}^{2}-12\,a{b}^{4}c+{b}^{6} \right ){x}^{2}}{ \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){c}^{3}}}-{\frac{{a}^{2} \left ( 20\,{a}^{3}{c}^{3}-66\,{a}^{2}{b}^{2}{c}^{2}+13\,a{b}^{4}c-{b}^{6} \right ) x}{ \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){c}^{3}}}+{\frac{ \left ( 66\,{a}^{2}{c}^{2}-13\,ac{b}^{2}+{b}^{4} \right ){a}^{3}b}{3\, \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ){c}^{3}}} \right ) }+40\,{\frac{{a}^{3}}{ \left ( 64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(c*x^2+b*x+a)^4,x)

[Out]

(-(44*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b
^4*c-b^6)*x^5-b*(14*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c^2/(64*a^3*c^3-48*a^
2*b^2*c^2+12*a*b^4*c-b^6)*x^4-1/3/c^3*(160*a^4*c^4-286*a^3*b^2*c^3+12*a^2*b^4*c^
2+7*a*b^6*c-b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3+b/c^3*a*(16*a^3*
c^3+53*a^2*b^2*c^2-12*a*b^4*c+b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^
2-a^2*(20*a^3*c^3-66*a^2*b^2*c^2+13*a*b^4*c-b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a
*b^4*c-b^6)/c^3*x+1/3*(66*a^2*c^2-13*a*b^2*c+b^4)*a^3*b/c^3/(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+40*a^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b
^4*c-b^6)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))

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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(x^6/(c*x^2 + b*x + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235967, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^2 + b*x + a)^4,x, algorithm="fricas")

[Out]

[-1/3*(60*(a^3*c^6*x^6 + 3*a^3*b*c^5*x^5 + 3*a^5*b*c^3*x + a^6*c^3 + 3*(a^3*b^2*
c^4 + a^4*c^5)*x^4 + (a^3*b^3*c^3 + 6*a^4*b*c^4)*x^3 + 3*(a^4*b^2*c^3 + a^5*c^4)
*x^2)*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)) + (a^3*b^5 - 13*a^4*b^3*c + 66*a^5
*b*c^2 + 3*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 44*a^3*c^5)*x^5 + 3*(b^7*c
- 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 14*a^3*b*c^4)*x^4 + (b^8 - 7*a*b^6*c - 12*a^2
*b^4*c^2 + 286*a^3*b^2*c^3 - 160*a^4*c^4)*x^3 + 3*(a*b^7 - 12*a^2*b^5*c + 53*a^3
*b^3*c^2 + 16*a^4*b*c^3)*x^2 + 3*(a^2*b^6 - 13*a^3*b^4*c + 66*a^4*b^2*c^2 - 20*a
^5*c^3)*x)*sqrt(b^2 - 4*a*c))/((a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2*c^5 -
64*a^6*c^6 + (b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*x^6 + 3*(b^7
*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x^5 + 3*(b^8*c^4 - 11*a*b^6
*c^5 + 36*a^2*b^4*c^6 - 16*a^3*b^2*c^7 - 64*a^4*c^8)*x^4 + (b^9*c^3 - 6*a*b^7*c^
4 - 24*a^2*b^5*c^5 + 224*a^3*b^3*c^6 - 384*a^4*b*c^7)*x^3 + 3*(a*b^8*c^3 - 11*a^
2*b^6*c^4 + 36*a^3*b^4*c^5 - 16*a^4*b^2*c^6 - 64*a^5*c^7)*x^2 + 3*(a^2*b^7*c^3 -
12*a^3*b^5*c^4 + 48*a^4*b^3*c^5 - 64*a^5*b*c^6)*x)*sqrt(b^2 - 4*a*c)), -1/3*(12
0*(a^3*c^6*x^6 + 3*a^3*b*c^5*x^5 + 3*a^5*b*c^3*x + a^6*c^3 + 3*(a^3*b^2*c^4 + a^
4*c^5)*x^4 + (a^3*b^3*c^3 + 6*a^4*b*c^4)*x^3 + 3*(a^4*b^2*c^3 + a^5*c^4)*x^2)*ar
ctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (a^3*b^5 - 13*a^4*b^3*c +
66*a^5*b*c^2 + 3*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 44*a^3*c^5)*x^5 + 3*
(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 14*a^3*b*c^4)*x^4 + (b^8 - 7*a*b^6*c -
12*a^2*b^4*c^2 + 286*a^3*b^2*c^3 - 160*a^4*c^4)*x^3 + 3*(a*b^7 - 12*a^2*b^5*c +
53*a^3*b^3*c^2 + 16*a^4*b*c^3)*x^2 + 3*(a^2*b^6 - 13*a^3*b^4*c + 66*a^4*b^2*c^2
- 20*a^5*c^3)*x)*sqrt(-b^2 + 4*a*c))/((a^3*b^6*c^3 - 12*a^4*b^4*c^4 + 48*a^5*b^2
*c^5 - 64*a^6*c^6 + (b^6*c^6 - 12*a*b^4*c^7 + 48*a^2*b^2*c^8 - 64*a^3*c^9)*x^6 +
3*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^7 - 64*a^3*b*c^8)*x^5 + 3*(b^8*c^4 - 1
1*a*b^6*c^5 + 36*a^2*b^4*c^6 - 16*a^3*b^2*c^7 - 64*a^4*c^8)*x^4 + (b^9*c^3 - 6*a
*b^7*c^4 - 24*a^2*b^5*c^5 + 224*a^3*b^3*c^6 - 384*a^4*b*c^7)*x^3 + 3*(a*b^8*c^3
- 11*a^2*b^6*c^4 + 36*a^3*b^4*c^5 - 16*a^4*b^2*c^6 - 64*a^5*c^7)*x^2 + 3*(a^2*b^
7*c^3 - 12*a^3*b^5*c^4 + 48*a^4*b^3*c^5 - 64*a^5*b*c^6)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 16.2877, size = 938, normalized size = 6.42 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(c*x**2+b*x+a)**4,x)

[Out]

-20*a**3*sqrt(-1/(4*a*c - b**2)**7)*log(x + (-5120*a**7*c**4*sqrt(-1/(4*a*c - b*
*2)**7) + 5120*a**6*b**2*c**3*sqrt(-1/(4*a*c - b**2)**7) - 1920*a**5*b**4*c**2*s
qrt(-1/(4*a*c - b**2)**7) + 320*a**4*b**6*c*sqrt(-1/(4*a*c - b**2)**7) - 20*a**3
*b**8*sqrt(-1/(4*a*c - b**2)**7) + 20*a**3*b)/(40*a**3*c)) + 20*a**3*sqrt(-1/(4*
a*c - b**2)**7)*log(x + (5120*a**7*c**4*sqrt(-1/(4*a*c - b**2)**7) - 5120*a**6*b
**2*c**3*sqrt(-1/(4*a*c - b**2)**7) + 1920*a**5*b**4*c**2*sqrt(-1/(4*a*c - b**2)
**7) - 320*a**4*b**6*c*sqrt(-1/(4*a*c - b**2)**7) + 20*a**3*b**8*sqrt(-1/(4*a*c
- b**2)**7) + 20*a**3*b)/(40*a**3*c)) - (-66*a**5*b*c**2 + 13*a**4*b**3*c - a**3
*b**5 + x**5*(132*a**3*c**5 - 144*a**2*b**2*c**4 + 36*a*b**4*c**3 - 3*b**6*c**2)
+ x**4*(42*a**3*b*c**4 - 144*a**2*b**3*c**3 + 36*a*b**5*c**2 - 3*b**7*c) + x**3
*(160*a**4*c**4 - 286*a**3*b**2*c**3 + 12*a**2*b**4*c**2 + 7*a*b**6*c - b**8) +
x**2*(-48*a**4*b*c**3 - 159*a**3*b**3*c**2 + 36*a**2*b**5*c - 3*a*b**7) + x*(60*
a**5*c**3 - 198*a**4*b**2*c**2 + 39*a**3*b**4*c - 3*a**2*b**6))/(192*a**6*c**6 -
144*a**5*b**2*c**5 + 36*a**4*b**4*c**4 - 3*a**3*b**6*c**3 + x**6*(192*a**3*c**9
- 144*a**2*b**2*c**8 + 36*a*b**4*c**7 - 3*b**6*c**6) + x**5*(576*a**3*b*c**8 -
432*a**2*b**3*c**7 + 108*a*b**5*c**6 - 9*b**7*c**5) + x**4*(576*a**4*c**8 + 144*
a**3*b**2*c**7 - 324*a**2*b**4*c**6 + 99*a*b**6*c**5 - 9*b**8*c**4) + x**3*(1152
*a**4*b*c**7 - 672*a**3*b**3*c**6 + 72*a**2*b**5*c**5 + 18*a*b**7*c**4 - 3*b**9*
c**3) + x**2*(576*a**5*c**7 + 144*a**4*b**2*c**6 - 324*a**3*b**4*c**5 + 99*a**2*
b**6*c**4 - 9*a*b**8*c**3) + x*(576*a**5*b*c**6 - 432*a**4*b**3*c**5 + 108*a**3*
b**5*c**4 - 9*a**2*b**7*c**3))

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GIAC/XCAS [A]  time = 0.208461, size = 521, normalized size = 3.57 \[ -\frac{40 \, a^{3} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{3 \, b^{6} c^{2} x^{5} - 36 \, a b^{4} c^{3} x^{5} + 144 \, a^{2} b^{2} c^{4} x^{5} - 132 \, a^{3} c^{5} x^{5} + 3 \, b^{7} c x^{4} - 36 \, a b^{5} c^{2} x^{4} + 144 \, a^{2} b^{3} c^{3} x^{4} - 42 \, a^{3} b c^{4} x^{4} + b^{8} x^{3} - 7 \, a b^{6} c x^{3} - 12 \, a^{2} b^{4} c^{2} x^{3} + 286 \, a^{3} b^{2} c^{3} x^{3} - 160 \, a^{4} c^{4} x^{3} + 3 \, a b^{7} x^{2} - 36 \, a^{2} b^{5} c x^{2} + 159 \, a^{3} b^{3} c^{2} x^{2} + 48 \, a^{4} b c^{3} x^{2} + 3 \, a^{2} b^{6} x - 39 \, a^{3} b^{4} c x + 198 \, a^{4} b^{2} c^{2} x - 60 \, a^{5} c^{3} x + a^{3} b^{5} - 13 \, a^{4} b^{3} c + 66 \, a^{5} b c^{2}}{3 \,{\left (b^{6} c^{3} - 12 \, a b^{4} c^{4} + 48 \, a^{2} b^{2} c^{5} - 64 \, a^{3} c^{6}\right )}{\left (c x^{2} + b x + a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^2 + b*x + a)^4,x, algorithm="giac")

[Out]

-40*a^3*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*c + 48*a^2*b^2*c
^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/3*(3*b^6*c^2*x^5 - 36*a*b^4*c^3*x^5 + 1
44*a^2*b^2*c^4*x^5 - 132*a^3*c^5*x^5 + 3*b^7*c*x^4 - 36*a*b^5*c^2*x^4 + 144*a^2*
b^3*c^3*x^4 - 42*a^3*b*c^4*x^4 + b^8*x^3 - 7*a*b^6*c*x^3 - 12*a^2*b^4*c^2*x^3 +
286*a^3*b^2*c^3*x^3 - 160*a^4*c^4*x^3 + 3*a*b^7*x^2 - 36*a^2*b^5*c*x^2 + 159*a^3
*b^3*c^2*x^2 + 48*a^4*b*c^3*x^2 + 3*a^2*b^6*x - 39*a^3*b^4*c*x + 198*a^4*b^2*c^2
*x - 60*a^5*c^3*x + a^3*b^5 - 13*a^4*b^3*c + 66*a^5*b*c^2)/((b^6*c^3 - 12*a*b^4*
c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*(c*x^2 + b*x + a)^3)