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

Optimal. Leaf size=112 \[ -\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{2 a \left (a+b x^2\right )}-\frac{c}{a^2 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f-a^2 b e-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}+\frac{f x}{b^2} \]

[Out]

-(c/(a^2*x)) + (f*x)/b^2 - (((b*c)/a - d + (a*e)/b - (a^2*f)/b^2)*x)/(2*a*(a + b
*x^2)) - ((3*b^3*c - a*b^2*d - a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
2*a^(5/2)*b^(5/2))

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Rubi [A]  time = 0.322123, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{2 a \left (a+b x^2\right )}-\frac{c}{a^2 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f-a^2 b e-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}+\frac{f x}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^2),x]

[Out]

-(c/(a^2*x)) + (f*x)/b^2 - (((b*c)/a - d + (a*e)/b - (a^2*f)/b^2)*x)/(2*a*(a + b
*x^2)) - ((3*b^3*c - a*b^2*d - a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
2*a^(5/2)*b^(5/2))

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Rubi in Sympy [A]  time = 140.885, size = 122, normalized size = 1.09 \[ \frac{f x}{b^{2}} - \frac{x \left (\frac{a^{3} f}{x^{2}} - \frac{a^{2} b e}{x^{2}} + \frac{a b^{2} d}{x^{2}} - \frac{b^{3} c}{x^{2}}\right )}{2 a b^{3} \left (a + b x^{2}\right )} - \frac{a^{2} f - a b e + b^{2} d}{a b^{3} x} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x/b**2 - x*(a**3*f/x**2 - a**2*b*e/x**2 + a*b**2*d/x**2 - b**3*c/x**2)/(2*a*b*
*3*(a + b*x**2)) - (a**2*f - a*b*e + b**2*d)/(a*b**3*x) - (3*a**2*f - 2*a*b*e +
b**2*d)*atan(sqrt(b)*x/sqrt(a))/(a**(3/2)*b**(5/2))

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Mathematica [A]  time = 0.114736, size = 115, normalized size = 1.03 \[ -\frac{c}{a^2 x}+\frac{x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a^2 b^2 \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f-a^2 b e-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}+\frac{f x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

-(c/(a^2*x)) + (f*x)/b^2 + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(2*a^2*b^2
*(a + b*x^2)) - ((3*b^3*c - a*b^2*d - a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt
[a]])/(2*a^(5/2)*b^(5/2))

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Maple [A]  time = 0.017, size = 165, normalized size = 1.5 \[{\frac{fx}{{b}^{2}}}-{\frac{c}{x{a}^{2}}}+{\frac{axf}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{ex}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{dx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bxc}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,af}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{d}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,bc}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^2/(b*x^2+a)^2,x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234765, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, b^{4} c - a b^{3} d - a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} +{\left (3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f\right )} x\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (2 \, a^{2} b f x^{4} - 2 \, a b^{2} c -{\left (3 \, b^{3} c - a b^{2} d + a^{2} b e - 3 \, a^{3} f\right )} x^{2}\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b^{3} x^{3} + a^{3} b^{2} x\right )} \sqrt{-a b}}, -\frac{{\left ({\left (3 \, b^{4} c - a b^{3} d - a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} +{\left (3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, a^{2} b f x^{4} - 2 \, a b^{2} c -{\left (3 \, b^{3} c - a b^{2} d + a^{2} b e - 3 \, a^{3} f\right )} x^{2}\right )} \sqrt{a b}}{2 \,{\left (a^{2} b^{3} x^{3} + a^{3} b^{2} x\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.2556, size = 197, normalized size = 1.76 \[ \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{3} f - a^{2} b e - a b^{2} d + 3 b^{3} c\right ) \log{\left (- a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{3} f - a^{2} b e - a b^{2} d + 3 b^{3} c\right ) \log{\left (a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{4} + \frac{- 2 a b^{2} c + x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - 3 b^{3} c\right )}{2 a^{3} b^{2} x + 2 a^{2} b^{3} x^{3}} + \frac{f x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(a**5*b**5))*(3*a**3*f - a**2*b*e - a*b**2*d + 3*b**3*c)*log(-a**3*b**2*
sqrt(-1/(a**5*b**5)) + x)/4 - sqrt(-1/(a**5*b**5))*(3*a**3*f - a**2*b*e - a*b**2
*d + 3*b**3*c)*log(a**3*b**2*sqrt(-1/(a**5*b**5)) + x)/4 + (-2*a*b**2*c + x**2*(
a**3*f - a**2*b*e + a*b**2*d - 3*b**3*c))/(2*a**3*b**2*x + 2*a**2*b**3*x**3) + f
*x/b**2

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GIAC/XCAS [A]  time = 0.216176, size = 165, normalized size = 1.47 \[ \frac{f x}{b^{2}} - \frac{{\left (3 \, b^{3} c - a b^{2} d + 3 \, a^{3} f - a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2} b^{2}} - \frac{3 \, b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e + 2 \, a b^{2} c}{2 \,{\left (b x^{3} + a x\right )} a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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