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

Optimal. Leaf size=82 \[ \frac{b c-a d}{a^2 x}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{5/2} b^{3/2}}-\frac{c}{3 a x^3}+\frac{f x}{b} \]

[Out]

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

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Rubi [A]  time = 0.174724, antiderivative size = 82, 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 \[ \frac{b c-a d}{a^2 x}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{5/2} b^{3/2}}-\frac{c}{3 a x^3}+\frac{f x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int f\, dx}{b} - \frac{c}{3 a x^{3}} - \frac{a d - b c}{a^{2} x} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.176651, size = 83, normalized size = 1.01 \[ \frac{b c-a d}{a^2 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{5/2} b^{3/2}}-\frac{c}{3 a x^3}+\frac{f x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 115, normalized size = 1.4 \[{\frac{fx}{b}}-{\frac{c}{3\,a{x}^{3}}}-{\frac{d}{ax}}+{\frac{bc}{x{a}^{2}}}-{\frac{af}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{e\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{bd}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}c}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

f*x/b-1/3*c/a/x^3-1/a/x*d+1/a^2/x*b*c-1/b*a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*
f+1/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*e-b/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2)
)*d+b^2/a^2/(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)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233999, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (3 \, a^{2} f x^{4} - a b c + 3 \,{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-a b}}{6 \, \sqrt{-a b} a^{2} b x^{3}}, \frac{3 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, a^{2} f x^{4} - a b c + 3 \,{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{a b}}{3 \, \sqrt{a b} a^{2} b x^{3}}\right ] \]

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^4),x, algorithm="fricas")

[Out]

[-1/6*(3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3*log(-(2*a*b*x - (b*x^2 - a)*sqr
t(-a*b))/(b*x^2 + a)) - 2*(3*a^2*f*x^4 - a*b*c + 3*(b^2*c - a*b*d)*x^2)*sqrt(-a*
b))/(sqrt(-a*b)*a^2*b*x^3), 1/3*(3*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^3*arcta
n(sqrt(a*b)*x/a) + (3*a^2*f*x^4 - a*b*c + 3*(b^2*c - a*b*d)*x^2)*sqrt(a*b))/(sqr
t(a*b)*a^2*b*x^3)]

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Sympy [A]  time = 4.4074, size = 151, normalized size = 1.84 \[ \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{2} + \frac{f x}{b} - \frac{a c + x^{2} \left (3 a d - 3 b c\right )}{3 a^{2} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21612, size = 109, normalized size = 1.33 \[ \frac{f x}{b} + \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2} b} + \frac{3 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{2} x^{3}} \]

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^4),x, algorithm="giac")

[Out]

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