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

Optimal. Leaf size=104 \[ \frac{b c-a d}{3 a^2 x^3}-\frac{a^2 e-a b d+b^2 c}{a^3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{7/2} \sqrt{b}}-\frac{c}{5 a x^5} \]

[Out]

-c/(5*a*x^5) + (b*c - a*d)/(3*a^2*x^3) - (b^2*c - a*b*d + a^2*e)/(a^3*x) - ((b^3
*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*Sqrt[b])

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Rubi [A]  time = 0.208074, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{b c-a d}{3 a^2 x^3}-\frac{a^2 e-a b d+b^2 c}{a^3 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{7/2} \sqrt{b}}-\frac{c}{5 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

-c/(5*a*x^5) + (b*c - a*d)/(3*a^2*x^3) - (b^2*c - a*b*d + a^2*e)/(a^3*x) - ((b^3
*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*Sqrt[b])

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Rubi in Sympy [A]  time = 41.9197, size = 90, normalized size = 0.87 \[ - \frac{c}{5 a x^{5}} - \frac{a d - b c}{3 a^{2} x^{3}} - \frac{a^{2} e - a b d + b^{2} c}{a^{3} x} + \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{7}{2}} \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/(5*a*x**5) - (a*d - b*c)/(3*a**2*x**3) - (a**2*e - a*b*d + b**2*c)/(a**3*x) +
(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*atan(sqrt(b)*x/sqrt(a))/(a**(7/2)*sqrt(
b))

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Mathematica [A]  time = 0.181875, size = 103, normalized size = 0.99 \[ \frac{b c-a d}{3 a^2 x^3}+\frac{a^2 (-e)+a b d-b^2 c}{a^3 x}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{7/2} \sqrt{b}}-\frac{c}{5 a x^5} \]

Antiderivative was successfully verified.

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

[Out]

-c/(5*a*x^5) + (b*c - a*d)/(3*a^2*x^3) + (-(b^2*c) + a*b*d - a^2*e)/(a^3*x) + ((
-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(7/2)*Sqrt
[b])

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Maple [A]  time = 0.01, size = 142, normalized size = 1.4 \[ -{\frac{c}{5\,a{x}^{5}}}-{\frac{d}{3\,a{x}^{3}}}+{\frac{bc}{3\,{x}^{3}{a}^{2}}}-{\frac{e}{ax}}+{\frac{bd}{x{a}^{2}}}-{\frac{{b}^{2}c}{{a}^{3}x}}+{f\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{be}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}d}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{{b}^{3}c}{{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234085, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{5} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (15 \,{\left (b^{2} c - a b d + a^{2} e\right )} x^{4} + 3 \, a^{2} c - 5 \,{\left (a b c - a^{2} d\right )} x^{2}\right )} \sqrt{-a b}}{30 \, \sqrt{-a b} a^{3} x^{5}}, -\frac{15 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{5} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (15 \,{\left (b^{2} c - a b d + a^{2} e\right )} x^{4} + 3 \, a^{2} c - 5 \,{\left (a b c - a^{2} d\right )} x^{2}\right )} \sqrt{a b}}{15 \, \sqrt{a b} a^{3} x^{5}}\right ] \]

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)*x^6),x, algorithm="fricas")

[Out]

[-1/30*(15*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^5*log((2*a*b*x + (b*x^2 - a)*sq
rt(-a*b))/(b*x^2 + a)) + 2*(15*(b^2*c - a*b*d + a^2*e)*x^4 + 3*a^2*c - 5*(a*b*c
- a^2*d)*x^2)*sqrt(-a*b))/(sqrt(-a*b)*a^3*x^5), -1/15*(15*(b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*x^5*arctan(sqrt(a*b)*x/a) + (15*(b^2*c - a*b*d + a^2*e)*x^4 + 3*a^
2*c - 5*(a*b*c - a^2*d)*x^2)*sqrt(a*b))/(sqrt(a*b)*a^3*x^5)]

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Sympy [A]  time = 10.5453, size = 167, normalized size = 1.61 \[ - \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- a^{4} \sqrt{- \frac{1}{a^{7} b}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{4} \sqrt{- \frac{1}{a^{7} b}} + x \right )}}{2} - \frac{3 a^{2} c + x^{4} \left (15 a^{2} e - 15 a b d + 15 b^{2} c\right ) + x^{2} \left (5 a^{2} d - 5 a b c\right )}{15 a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**7*b))*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(-a**4*sqrt(-1/(a*
*7*b)) + x)/2 + sqrt(-1/(a**7*b))*(a**3*f - a**2*b*e + a*b**2*d - b**3*c)*log(a*
*4*sqrt(-1/(a**7*b)) + x)/2 - (3*a**2*c + x**4*(15*a**2*e - 15*a*b*d + 15*b**2*c
) + x**2*(5*a**2*d - 5*a*b*c))/(15*a**3*x**5)

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GIAC/XCAS [A]  time = 0.218994, size = 142, normalized size = 1.37 \[ -\frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{3}} - \frac{15 \, b^{2} c x^{4} - 15 \, a b d x^{4} + 15 \, a^{2} x^{4} e - 5 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{3} x^{5}} \]

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)*x^6),x, algorithm="giac")

[Out]

-(b^3*c - a*b^2*d - a^3*f + a^2*b*e)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) - 1/1
5*(15*b^2*c*x^4 - 15*a*b*d*x^4 + 15*a^2*x^4*e - 5*a*b*c*x^2 + 5*a^2*d*x^2 + 3*a^
2*c)/(a^3*x^5)