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

Optimal. Leaf size=113 \[ -\frac{6 c^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.172935, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{6 c^2 \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 c \left (b+2 c x^2\right )}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{b+2 c x^2}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^2 + c*x^4)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 18.9878, size = 105, normalized size = 0.93 \[ - \frac{6 c^{2} \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{3 c \left (b + 2 c x^{2}\right )}{2 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} - \frac{b + 2 c x^{2}}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.193789, size = 106, normalized size = 0.94 \[ \frac{\frac{24 c^2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{\left (b+2 c x^2\right ) \left (-2 c \left (5 a+3 c x^4\right )+b^2-6 b c x^2\right )}{\left (a+b x^2+c x^4\right )^2}}{4 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(-(((b + 2*c*x^2)*(b^2 - 6*b*c*x^2 - 2*c*(5*a + 3*c*x^4)))/(a + b*x^2 + c*x^4)^2
) + (24*c^2*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(4*(b^
2 - 4*a*c)^2)

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Maple [A]  time = 0.009, size = 141, normalized size = 1.3 \[{\frac{2\,c{x}^{2}+b}{ \left ( 16\,ac-4\,{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{c}^{2}{x}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}+{\frac{3\,bc}{2\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}+6\,{\frac{{c}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^4+b*x^2+a)^3,x)

[Out]

1/4*(2*c*x^2+b)/(4*a*c-b^2)/(c*x^4+b*x^2+a)^2+3*c^2/(4*a*c-b^2)^2/(c*x^4+b*x^2+a
)*x^2+3/2*c/(4*a*c-b^2)^2/(c*x^4+b*x^2+a)*b+6*c^2/(4*a*c-b^2)^(5/2)*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/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.269147, size = 1, normalized size = 0.01 \[ \left [\frac{12 \,{\left (c^{4} x^{8} + 2 \, b c^{3} x^{6} + 2 \, a b c^{2} x^{2} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{4} + a^{2} c^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (12 \, c^{3} x^{6} + 18 \, b c^{2} x^{4} - b^{3} + 10 \, a b c + 4 \,{\left (b^{2} c + 5 \, a c^{2}\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{4 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{8} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{4} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{24 \,{\left (c^{4} x^{8} + 2 \, b c^{3} x^{6} + 2 \, a b c^{2} x^{2} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} x^{4} + a^{2} c^{2}\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (12 \, c^{3} x^{6} + 18 \, b c^{2} x^{4} - b^{3} + 10 \, a b c + 4 \,{\left (b^{2} c + 5 \, a c^{2}\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{8} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{6} + a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{4} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(12*(c^4*x^8 + 2*b*c^3*x^6 + 2*a*b*c^2*x^2 + (b^2*c^2 + 2*a*c^3)*x^4 + a^2*
c^2)*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)) + (12*c^3*x^6 + 18*b*c^2*x^4
- b^3 + 10*a*b*c + 4*(b^2*c + 5*a*c^2)*x^2)*sqrt(b^2 - 4*a*c))/(((b^4*c^2 - 8*a*
b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4
- 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*
a^2*b^3*c + 16*a^3*b*c^2)*x^2)*sqrt(b^2 - 4*a*c)), 1/4*(24*(c^4*x^8 + 2*b*c^3*x^
6 + 2*a*b*c^2*x^2 + (b^2*c^2 + 2*a*c^3)*x^4 + a^2*c^2)*arctan(-(2*c*x^2 + b)*sqr
t(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (12*c^3*x^6 + 18*b*c^2*x^4 - b^3 + 10*a*b*c + 4
*(b^2*c + 5*a*c^2)*x^2)*sqrt(-b^2 + 4*a*c))/(((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^
4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16
*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*
b*c^2)*x^2)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 33.359, size = 481, normalized size = 4.26 \[ - 3 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x^{2} + \frac{- 192 a^{3} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 144 a^{2} b^{2} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 36 a b^{4} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b^{6} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b c^{2}}{6 c^{3}} \right )} + 3 c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x^{2} + \frac{192 a^{3} c^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 144 a^{2} b^{2} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 36 a b^{4} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 3 b^{6} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 3 b c^{2}}{6 c^{3}} \right )} + \frac{10 a b c - b^{3} + 18 b c^{2} x^{4} + 12 c^{3} x^{6} + x^{2} \left (20 a c^{2} + 4 b^{2} c\right )}{64 a^{4} c^{2} - 32 a^{3} b^{2} c + 4 a^{2} b^{4} + x^{8} \left (64 a^{2} c^{4} - 32 a b^{2} c^{3} + 4 b^{4} c^{2}\right ) + x^{6} \left (128 a^{2} b c^{3} - 64 a b^{3} c^{2} + 8 b^{5} c\right ) + x^{4} \left (128 a^{3} c^{3} - 24 a b^{4} c + 4 b^{6}\right ) + x^{2} \left (128 a^{3} b c^{2} - 64 a^{2} b^{3} c + 8 a b^{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*c**2*sqrt(-1/(4*a*c - b**2)**5)*log(x**2 + (-192*a**3*c**5*sqrt(-1/(4*a*c - b
**2)**5) + 144*a**2*b**2*c**4*sqrt(-1/(4*a*c - b**2)**5) - 36*a*b**4*c**3*sqrt(-
1/(4*a*c - b**2)**5) + 3*b**6*c**2*sqrt(-1/(4*a*c - b**2)**5) + 3*b*c**2)/(6*c**
3)) + 3*c**2*sqrt(-1/(4*a*c - b**2)**5)*log(x**2 + (192*a**3*c**5*sqrt(-1/(4*a*c
- b**2)**5) - 144*a**2*b**2*c**4*sqrt(-1/(4*a*c - b**2)**5) + 36*a*b**4*c**3*sq
rt(-1/(4*a*c - b**2)**5) - 3*b**6*c**2*sqrt(-1/(4*a*c - b**2)**5) + 3*b*c**2)/(6
*c**3)) + (10*a*b*c - b**3 + 18*b*c**2*x**4 + 12*c**3*x**6 + x**2*(20*a*c**2 + 4
*b**2*c))/(64*a**4*c**2 - 32*a**3*b**2*c + 4*a**2*b**4 + x**8*(64*a**2*c**4 - 32
*a*b**2*c**3 + 4*b**4*c**2) + x**6*(128*a**2*b*c**3 - 64*a*b**3*c**2 + 8*b**5*c)
+ x**4*(128*a**3*c**3 - 24*a*b**4*c + 4*b**6) + x**2*(128*a**3*b*c**2 - 64*a**2
*b**3*c + 8*a*b**5))

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GIAC/XCAS [A]  time = 15.728, size = 194, normalized size = 1.72 \[ \frac{6 \, c^{2} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{3} x^{6} + 18 \, b c^{2} x^{4} + 4 \, b^{2} c x^{2} + 20 \, a c^{2} x^{2} - b^{3} + 10 \, a b c}{4 \,{\left (c x^{4} + b x^{2} + a\right )}^{2}{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*c^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*(12*c^3*x^6 + 18*b*c^2*x^4 + 4*b^2*c*x^2 + 20*a*c^2*x^2
- b^3 + 10*a*b*c)/((c*x^4 + b*x^2 + a)^2*(b^4 - 8*a*b^2*c + 16*a^2*c^2))