3.64 \(\int \frac{x \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=123 \[ \frac{x^2 \left (-\left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )-b (a f+c d)+2 a c e}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) (2 a f-b e+2 c d)}{\left (b^2-4 a c\right )^{3/2}} \]

[Out]

(2*a*c*e - b*(c*d + a*f) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x^2)/(2*c*(b^2 -
4*a*c)*(a + b*x^2 + c*x^4)) + ((2*c*d - b*e + 2*a*f)*ArcTanh[(b + 2*c*x^2)/Sqrt[
b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

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Rubi [A]  time = 0.351927, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{x^2 \left (-\left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )-b (a f+c d)+2 a c e}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) (2 a f-b e+2 c d)}{\left (b^2-4 a c\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]

[Out]

(2*a*c*e - b*(c*d + a*f) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x^2)/(2*c*(b^2 -
4*a*c)*(a + b*x^2 + c*x^4)) + ((2*c*d - b*e + 2*a*f)*ArcTanh[(b + 2*c*x^2)/Sqrt[
b^2 - 4*a*c]])/(b^2 - 4*a*c)^(3/2)

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Rubi in Sympy [A]  time = 19.5848, size = 114, normalized size = 0.93 \[ \frac{\left (2 a f - b e + 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{a b f - 2 a c e + b c d + x^{2} \left (- 2 a c f + b^{2} f - b c e + 2 c^{2} d\right )}{2 c \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a*f - b*e + 2*c*d)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)*
*(3/2) - (a*b*f - 2*a*c*e + b*c*d + x**2*(-2*a*c*f + b**2*f - b*c*e + 2*c**2*d))
/(2*c*(-4*a*c + b**2)*(a + b*x**2 + c*x**4))

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Mathematica [A]  time = 0.197209, size = 130, normalized size = 1.06 \[ \frac{a b f-2 a c \left (e+f x^2\right )+b^2 f x^2+b c \left (d-e x^2\right )+2 c^2 d x^2}{2 c \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )}-\frac{\tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) (-2 a f+b e-2 c d)}{\left (4 a c-b^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*b*f + 2*c^2*d*x^2 + b^2*f*x^2 + b*c*(d - e*x^2) - 2*a*c*(e + f*x^2))/(2*c*(-b
^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - ((-2*c*d + b*e - 2*a*f)*ArcTan[(b + 2*c*x^2)/
Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)

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Maple [A]  time = 0.014, size = 205, normalized size = 1.7 \[{\frac{1}{2\,c{x}^{4}+2\,b{x}^{2}+2\,a} \left ( -{\frac{ \left ( 2\,acf-{b}^{2}f+bce-2\,{c}^{2}d \right ){x}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) c}}+{\frac{abf-2\,ace+bcd}{ \left ( 4\,ac-{b}^{2} \right ) c}} \right ) }+2\,{\frac{fa}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{be\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{cd}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x)

[Out]

1/2*(-(2*a*c*f-b^2*f+b*c*e-2*c^2*d)/(4*a*c-b^2)/c*x^2+1/c*(a*b*f-2*a*c*e+b*c*d)/
(4*a*c-b^2))/(c*x^4+b*x^2+a)+2/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^
(1/2))*f*a-1/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*e+2/(4*a*
c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c*d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295108, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (2 \, c^{3} d - b c^{2} e + 2 \, a c^{2} f\right )} x^{4} + 2 \, a c^{2} d - a b c e + 2 \, a^{2} c f +{\left (2 \, b c^{2} d - b^{2} c e + 2 \, a b c f\right )} x^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (b c d - 2 \, a c e + a b f +{\left (2 \, c^{2} d - b c e +{\left (b^{2} - 2 \, a c\right )} f\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left ({\left (2 \, c^{3} d - b c^{2} e + 2 \, a c^{2} f\right )} x^{4} + 2 \, a c^{2} d - a b c e + 2 \, a^{2} c f +{\left (2 \, b c^{2} d - b^{2} c e + 2 \, a b c f\right )} x^{2}\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (b c d - 2 \, a c e + a b f +{\left (2 \, c^{2} d - b c e +{\left (b^{2} - 2 \, a c\right )} f\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(((2*c^3*d - b*c^2*e + 2*a*c^2*f)*x^4 + 2*a*c^2*d - a*b*c*e + 2*a^2*c*f +
(2*b*c^2*d - b^2*c*e + 2*a*b*c*f)*x^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)) + (b*c*d - 2*a*c*e + a*b*f + (2*c^2*d - b*c*e + (b^2 - 2*a*c)*f)*x^2)*sqrt(
b^2 - 4*a*c))/(((b^2*c^2 - 4*a*c^3)*x^4 + a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c
^2)*x^2)*sqrt(b^2 - 4*a*c)), -1/2*(2*((2*c^3*d - b*c^2*e + 2*a*c^2*f)*x^4 + 2*a*
c^2*d - a*b*c*e + 2*a^2*c*f + (2*b*c^2*d - b^2*c*e + 2*a*b*c*f)*x^2)*arctan(-(2*
c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (b*c*d - 2*a*c*e + a*b*f + (2*c^2
*d - b*c*e + (b^2 - 2*a*c)*f)*x^2)*sqrt(-b^2 + 4*a*c))/(((b^2*c^2 - 4*a*c^3)*x^4
+ a*b^2*c - 4*a^2*c^2 + (b^3*c - 4*a*b*c^2)*x^2)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 131.162, size = 474, normalized size = 3.85 \[ - \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) \log{\left (x^{2} + \frac{- 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) + 2 a b f - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) - b^{2} e + 2 b c d}{4 a c f - 2 b c e + 4 c^{2} d} \right )}}{2} + \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) \log{\left (x^{2} + \frac{16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) + 2 a b f + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (2 a f - b e + 2 c d\right ) - b^{2} e + 2 b c d}{4 a c f - 2 b c e + 4 c^{2} d} \right )}}{2} - \frac{- a b f + 2 a c e - b c d + x^{2} \left (2 a c f - b^{2} f + b c e - 2 c^{2} d\right )}{8 a^{2} c^{2} - 2 a b^{2} c + x^{4} \left (8 a c^{3} - 2 b^{2} c^{2}\right ) + x^{2} \left (8 a b c^{2} - 2 b^{3} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(4*a*c - b**2)**3)*(2*a*f - b*e + 2*c*d)*log(x**2 + (-16*a**2*c**2*sqrt
(-1/(4*a*c - b**2)**3)*(2*a*f - b*e + 2*c*d) + 8*a*b**2*c*sqrt(-1/(4*a*c - b**2)
**3)*(2*a*f - b*e + 2*c*d) + 2*a*b*f - b**4*sqrt(-1/(4*a*c - b**2)**3)*(2*a*f -
b*e + 2*c*d) - b**2*e + 2*b*c*d)/(4*a*c*f - 2*b*c*e + 4*c**2*d))/2 + sqrt(-1/(4*
a*c - b**2)**3)*(2*a*f - b*e + 2*c*d)*log(x**2 + (16*a**2*c**2*sqrt(-1/(4*a*c -
b**2)**3)*(2*a*f - b*e + 2*c*d) - 8*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(2*a*f -
b*e + 2*c*d) + 2*a*b*f + b**4*sqrt(-1/(4*a*c - b**2)**3)*(2*a*f - b*e + 2*c*d)
- b**2*e + 2*b*c*d)/(4*a*c*f - 2*b*c*e + 4*c**2*d))/2 - (-a*b*f + 2*a*c*e - b*c*
d + x**2*(2*a*c*f - b**2*f + b*c*e - 2*c**2*d))/(8*a**2*c**2 - 2*a*b**2*c + x**4
*(8*a*c**3 - 2*b**2*c**2) + x**2*(8*a*b*c**2 - 2*b**3*c))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError