3.874 \(\int \frac{x^9}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=121 \[ -\frac{6 a^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{3 a x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

(x^6*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*a*x^2*(2*a + b*
x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (6*a^2*ArcTanh[(b + 2*c*x^2)/Sqr
t[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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Rubi [A]  time = 0.225365, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -\frac{6 a^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{3 a x^2 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^6 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^2 + c*x^4)^3,x]

[Out]

(x^6*(2*a + b*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2) - (3*a*x^2*(2*a + b*
x^2))/(2*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - (6*a^2*ArcTanh[(b + 2*c*x^2)/Sqr
t[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2)

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Rubi in Sympy [A]  time = 26.4906, size = 112, normalized size = 0.93 \[ - \frac{6 a^{2} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{3 a x^{2} \left (2 a + b x^{2}\right )}{2 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} + \frac{x^{6} \left (2 a + b x^{2}\right )}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(c*x**4+b*x**2+a)**3,x)

[Out]

-6*a**2*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(5/2) - 3*a*x
**2*(2*a + b*x**2)/(2*(-4*a*c + b**2)**2*(a + b*x**2 + c*x**4)) + x**6*(2*a + b*
x**2)/(4*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)**2)

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Mathematica [A]  time = 0.303986, size = 194, normalized size = 1.6 \[ \frac{1}{4} \left (\frac{24 a^2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{a^2 c \left (2 c x^2-3 b\right )+a b^2 \left (b-4 c x^2\right )+b^4 x^2}{c^3 \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}+\frac{22 a^2 b c^2-20 a^2 c^3 x^2-8 a b^3 c+16 a b^2 c^2 x^2+b^5-2 b^4 c x^2}{c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^2 + c*x^4)^3,x]

[Out]

((b^5 - 8*a*b^3*c + 22*a^2*b*c^2 - 2*b^4*c*x^2 + 16*a*b^2*c^2*x^2 - 20*a^2*c^3*x
^2)/(c^3*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b^4*x^2 + a*b^2*(b - 4*c*x^2) +
a^2*c*(-3*b + 2*c*x^2))/(c^3*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)^2) + (24*a^2*Ar
cTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2))/4

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Maple [B]  time = 0.021, size = 267, normalized size = 2.2 \[{\frac{1}{2\, \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 10\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ){x}^{6}}{c \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{b \left ( 2\,{a}^{2}{c}^{2}+8\,a{b}^{2}c-{b}^{4} \right ){x}^{4}}{2\,{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}-{\frac{a \left ( 6\,{a}^{2}{c}^{2}-10\,a{b}^{2}c+{b}^{4} \right ){x}^{2}}{{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }}+{\frac{{a}^{2}b \left ( 10\,ac-{b}^{2} \right ) }{2\,{c}^{2} \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) }} \right ) }+6\,{\frac{{a}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(c*x^4+b*x^2+a)^3,x)

[Out]

1/2*(-1/c*(10*a^2*c^2-8*a*b^2*c+b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*b*(2*a^2
*c^2+8*a*b^2*c-b^4)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-a*(6*a^2*c^2-10*a*b^2*c+b
^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^2*x^2+1/2*a^2*b*(10*a*c-b^2)/c^2/(16*a^2*c^2-8*
a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+6*a^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/
2)*arctan((2*c*x^2+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^9/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.27215, 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^9/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")

[Out]

[1/4*(12*(a^2*c^4*x^8 + 2*a^2*b*c^3*x^6 + 2*a^3*b*c^2*x^2 + a^4*c^2 + (a^2*b^2*c
^2 + 2*a^3*c^3)*x^4)*log(-(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x^2 - (2*c^2*x^4
+ 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) - (2*(b^4*c -
8*a*b^2*c^2 + 10*a^2*c^3)*x^6 + a^2*b^3 - 10*a^3*b*c + (b^5 - 8*a*b^3*c - 2*a^2
*b*c^2)*x^4 + 2*(a*b^4 - 10*a^2*b^2*c + 6*a^3*c^2)*x^2)*sqrt(b^2 - 4*a*c))/(((b^
4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^8 + a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4
+ 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^6 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^
3*c^5)*x^4 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x^2)*sqrt(b^2 - 4*a*c)
), 1/4*(24*(a^2*c^4*x^8 + 2*a^2*b*c^3*x^6 + 2*a^3*b*c^2*x^2 + a^4*c^2 + (a^2*b^2
*c^2 + 2*a^3*c^3)*x^4)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) -
(2*(b^4*c - 8*a*b^2*c^2 + 10*a^2*c^3)*x^6 + a^2*b^3 - 10*a^3*b*c + (b^5 - 8*a*b
^3*c - 2*a^2*b*c^2)*x^4 + 2*(a*b^4 - 10*a^2*b^2*c + 6*a^3*c^2)*x^2)*sqrt(-b^2 +
4*a*c))/(((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^8 + a^2*b^4*c^2 - 8*a^3*b^2*c^3
+ 16*a^4*c^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^6 + (b^6*c^2 - 6*a*b^
4*c^3 + 32*a^3*c^5)*x^4 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x^2)*sqrt
(-b^2 + 4*a*c))]

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Sympy [A]  time = 39.2561, size = 554, normalized size = 4.58 \[ - 3 a^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x^{2} + \frac{- 192 a^{5} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{4} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a^{3} b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b}{6 a^{2} c} \right )} + 3 a^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x^{2} + \frac{192 a^{5} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{4} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a^{3} b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 3 a^{2} b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 a^{2} b}{6 a^{2} c} \right )} - \frac{- 10 a^{3} b c + a^{2} b^{3} + x^{6} \left (20 a^{2} c^{3} - 16 a b^{2} c^{2} + 2 b^{4} c\right ) + x^{4} \left (- 2 a^{2} b c^{2} - 8 a b^{3} c + b^{5}\right ) + x^{2} \left (12 a^{3} c^{2} - 20 a^{2} b^{2} c + 2 a b^{4}\right )}{64 a^{4} c^{4} - 32 a^{3} b^{2} c^{3} + 4 a^{2} b^{4} c^{2} + x^{8} \left (64 a^{2} c^{6} - 32 a b^{2} c^{5} + 4 b^{4} c^{4}\right ) + x^{6} \left (128 a^{2} b c^{5} - 64 a b^{3} c^{4} + 8 b^{5} c^{3}\right ) + x^{4} \left (128 a^{3} c^{5} - 24 a b^{4} c^{3} + 4 b^{6} c^{2}\right ) + x^{2} \left (128 a^{3} b c^{4} - 64 a^{2} b^{3} c^{3} + 8 a b^{5} c^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(c*x**4+b*x**2+a)**3,x)

[Out]

-3*a**2*sqrt(-1/(4*a*c - b**2)**5)*log(x**2 + (-192*a**5*c**3*sqrt(-1/(4*a*c - b
**2)**5) + 144*a**4*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) - 36*a**3*b**4*c*sqrt(-
1/(4*a*c - b**2)**5) + 3*a**2*b**6*sqrt(-1/(4*a*c - b**2)**5) + 3*a**2*b)/(6*a**
2*c)) + 3*a**2*sqrt(-1/(4*a*c - b**2)**5)*log(x**2 + (192*a**5*c**3*sqrt(-1/(4*a
*c - b**2)**5) - 144*a**4*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) + 36*a**3*b**4*c*
sqrt(-1/(4*a*c - b**2)**5) - 3*a**2*b**6*sqrt(-1/(4*a*c - b**2)**5) + 3*a**2*b)/
(6*a**2*c)) - (-10*a**3*b*c + a**2*b**3 + x**6*(20*a**2*c**3 - 16*a*b**2*c**2 +
2*b**4*c) + x**4*(-2*a**2*b*c**2 - 8*a*b**3*c + b**5) + x**2*(12*a**3*c**2 - 20*
a**2*b**2*c + 2*a*b**4))/(64*a**4*c**4 - 32*a**3*b**2*c**3 + 4*a**2*b**4*c**2 +
x**8*(64*a**2*c**6 - 32*a*b**2*c**5 + 4*b**4*c**4) + x**6*(128*a**2*b*c**5 - 64*
a*b**3*c**4 + 8*b**5*c**3) + x**4*(128*a**3*c**5 - 24*a*b**4*c**3 + 4*b**6*c**2)
+ x**2*(128*a**3*b*c**4 - 64*a**2*b**3*c**3 + 8*a*b**5*c**2))

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GIAC/XCAS [A]  time = 15.6508, size = 286, normalized size = 2.36 \[ \frac{6 \, a^{2} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{4} c x^{6} - 16 \, a b^{2} c^{2} x^{6} + 20 \, a^{2} c^{3} x^{6} + b^{5} x^{4} - 8 \, a b^{3} c x^{4} - 2 \, a^{2} b c^{2} x^{4} + 2 \, a b^{4} x^{2} - 20 \, a^{2} b^{2} c x^{2} + 12 \, a^{3} c^{2} x^{2} + a^{2} b^{3} - 10 \, a^{3} b c}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

6*a^2*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*s
qrt(-b^2 + 4*a*c)) - 1/4*(2*b^4*c*x^6 - 16*a*b^2*c^2*x^6 + 20*a^2*c^3*x^6 + b^5*
x^4 - 8*a*b^3*c*x^4 - 2*a^2*b*c^2*x^4 + 2*a*b^4*x^2 - 20*a^2*b^2*c*x^2 + 12*a^3*
c^2*x^2 + a^2*b^3 - 10*a^3*b*c)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*(c*x^4 + b
*x^2 + a)^2)