3.19 \(\int \frac{x^8}{\left (a x^2+b x^3+c x^4\right )^2} \, dx\)

Optimal. Leaf size=150 \[ -\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{2 x \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}-\frac{b x^2}{c \left (b^2-4 a c\right )}+\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b \log \left (a+b x+c x^2\right )}{c^3} \]

[Out]

(2*(b^2 - 3*a*c)*x)/(c^2*(b^2 - 4*a*c)) - (b*x^2)/(c*(b^2 - 4*a*c)) + (x^3*(2*a
+ b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*Arc
Tanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) - (b*Log[a + b*x
+ c*x^2])/c^3

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Rubi [A]  time = 0.331493, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ -\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{2 x \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}-\frac{b x^2}{c \left (b^2-4 a c\right )}+\frac{x^3 (2 a+b x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{b \log \left (a+b x+c x^2\right )}{c^3} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

(2*(b^2 - 3*a*c)*x)/(c^2*(b^2 - 4*a*c)) - (b*x^2)/(c*(b^2 - 4*a*c)) + (x^3*(2*a
+ b*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*Arc
Tanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) - (b*Log[a + b*x
+ c*x^2])/c^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(c*x**4+b*x**3+a*x**2)**2,x)

[Out]

-2*b*Integral(x, x)/(c*(-4*a*c + b**2)) - b*log(a + b*x + c*x**2)/c**3 + x**3*(2
*a + b*x)/((-4*a*c + b**2)*(a + b*x + c*x**2)) + 2*x*(-3*a*c + b**2)/(c**2*(-4*a
*c + b**2)) - 2*(6*a**2*c**2 - 6*a*b**2*c + b**4)*atanh((b + 2*c*x)/sqrt(-4*a*c
+ b**2))/(c**3*(-4*a*c + b**2)**(3/2))

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Mathematica [A]  time = 0.311148, size = 132, normalized size = 0.88 \[ \frac{-\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{a^2 c (3 b-2 c x)-a b^2 (b-4 c x)+b^4 (-x)}{\left (b^2-4 a c\right ) (a+x (b+c x))}-b \log (a+x (b+c x))+c x}{c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a*x^2 + b*x^3 + c*x^4)^2,x]

[Out]

(c*x + (-(b^4*x) - a*b^2*(b - 4*c*x) + a^2*c*(3*b - 2*c*x))/((b^2 - 4*a*c)*(a +
x*(b + c*x))) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 +
4*a*c]])/(-b^2 + 4*a*c)^(3/2) - b*Log[a + x*(b + c*x)])/c^3

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Maple [B]  time = 0.015, size = 569, normalized size = 3.8 \[{\frac{x}{{c}^{2}}}+2\,{\frac{{a}^{2}x}{c \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{ax{b}^{2}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{x{b}^{4}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{{a}^{2}b}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a{b}^{3}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-4\,{\frac{\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ) ab}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}}+{\frac{\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ){b}^{3}}{{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-12\,{\frac{{a}^{2}}{c\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }+12\,{\frac{a{b}^{2}}{{c}^{2}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }-2\,{\frac{{b}^{4}}{{c}^{3}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) cx+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(c*x^4+b*x^3+a*x^2)^2,x)

[Out]

x/c^2+2/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a^2-4/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b^
2+1/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^4-3/c^2/(c*x^2+b*x+a)*a^2*b/(4*a*c-b^2)+1/
c^3/(c*x^2+b*x+a)*a*b^3/(4*a*c-b^2)-4/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+
a))*a*b+1/c^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^3-12/c/(64*a^3*c^3-48*
a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x+(4*a*c-b^2)*b)/(64*a
^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a^2+12/c^2/(64*a^3*c^3-48*a^2*b^2*c
^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48
*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b^2-2/c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*
b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)^(1/2))*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^8/(c*x^4 + b*x^3 + a*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305713, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} +{\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (a b^{3} - 3 \, a^{2} b c -{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{3} -{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2} +{\left (b^{4} - 5 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x +{\left (a b^{3} - 4 \, a^{2} b c +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2} +{\left (b^{4} - 4 \, a b^{2} c\right )} x\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} +{\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{2} +{\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (a b^{3} - 3 \, a^{2} b c -{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{3} -{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2} +{\left (b^{4} - 5 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x +{\left (a b^{3} - 4 \, a^{2} b c +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2} +{\left (b^{4} - 4 \, a b^{2} c\right )} x\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{2} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^4 + b*x^3 + a*x^2)^2,x, algorithm="fricas")

[Out]

[-((a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^2 + (b
^5 - 6*a*b^3*c + 6*a^2*b*c^2)*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)) + (a*b^3
- 3*a^2*b*c - (b^2*c^2 - 4*a*c^3)*x^3 - (b^3*c - 4*a*b*c^2)*x^2 + (b^4 - 5*a*b^
2*c + 6*a^2*c^2)*x + (a*b^3 - 4*a^2*b*c + (b^3*c - 4*a*b*c^2)*x^2 + (b^4 - 4*a*b
^2*c)*x)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2
*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(b^2 - 4*a*c)), (2*(a*b^4 - 6
*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^2 + (b^5 - 6*a*b^3*
c + 6*a^2*b*c^2)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (a*b
^3 - 3*a^2*b*c - (b^2*c^2 - 4*a*c^3)*x^3 - (b^3*c - 4*a*b*c^2)*x^2 + (b^4 - 5*a*
b^2*c + 6*a^2*c^2)*x + (a*b^3 - 4*a^2*b*c + (b^3*c - 4*a*b*c^2)*x^2 + (b^4 - 4*a
*b^2*c)*x)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a*b^2*c^3 - 4*a^2*c^4 + (
b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 5.99094, size = 842, normalized size = 5.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(c*x**4+b*x**3+a*x**2)**2,x)

[Out]

(-b/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(c**3*(64*
a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-10*a**2*b*c - 16
*a**2*c**4*(-b/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)
/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 2*a*b**3 + 8*
a*b**2*c**3*(-b/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4
)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - b**4*c**2*(-
b/c**3 - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(c**3*(64*a*
*3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))))/(12*a**2*c**2 - 12*a*b**2*c
+ 2*b**4)) + (-b/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b*
*4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-10
*a**2*b*c - 16*a**2*c**4*(-b/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*
b**2*c + b**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) +
2*a*b**3 + 8*a*b**2*c**3*(-b/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a
*b**2*c + b**4)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))
- b**4*c**2*(-b/c**3 + sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4
)/(c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))))/(12*a**2*c**2
- 12*a*b**2*c + 2*b**4)) + (-3*a**2*b*c + a*b**3 + x*(2*a**2*c**2 - 4*a*b**2*c
+ b**4))/(4*a**2*c**4 - a*b**2*c**3 + x**2*(4*a*c**5 - b**2*c**4) + x*(4*a*b*c**
4 - b**3*c**3)) + x/c**2

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GIAC/XCAS [A]  time = 0.268717, size = 217, normalized size = 1.45 \[ \frac{2 \,{\left (b^{4} - 6 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x}{c^{2}} - \frac{b{\rm ln}\left (c x^{2} + b x + a\right )}{c^{3}} - \frac{\frac{{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x}{c} + \frac{a b^{3} - 3 \, a^{2} b c}{c}}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^4 + b*x^3 + a*x^2)^2,x, algorithm="giac")

[Out]

2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3
- 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + x/c^2 - b*ln(c*x^2 + b*x + a)/c^3 - ((b^4 - 4*
a*b^2*c + 2*a^2*c^2)*x/c + (a*b^3 - 3*a^2*b*c)/c)/((c*x^2 + b*x + a)*(b^2 - 4*a*
c)*c^2)