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

Optimal. Leaf size=121 \[ \frac{2 b c-a d}{a^3 x}+\frac{x \left (\frac{b^2 c}{a^2}-\frac{b d}{a}-\frac{a f}{b}+e\right )}{2 a \left (a+b x^2\right )}-\frac{c}{3 a^2 x^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f+a^2 b e-3 a b^2 d+5 b^3 c\right )}{2 a^{7/2} b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.364802, antiderivative size = 121, 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 \[ \frac{2 b c-a d}{a^3 x}+\frac{x \left (\frac{b^2 c}{a^2}-\frac{b d}{a}-\frac{a f}{b}+e\right )}{2 a \left (a+b x^2\right )}-\frac{c}{3 a^2 x^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f+a^2 b e-3 a b^2 d+5 b^3 c\right )}{2 a^{7/2} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 143.952, size = 146, normalized size = 1.21 \[ - \frac{x \left (\frac{a^{3} f}{x^{4}} - \frac{a^{2} b e}{x^{4}} + \frac{a b^{2} d}{x^{4}} - \frac{b^{3} c}{x^{4}}\right )}{2 a b^{3} \left (a + b x^{2}\right )} - \frac{a^{2} f - a b e + b^{2} d}{3 a b^{3} x^{3}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{a^{2} b^{2} x} + \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} b^{\frac{3}{2}}} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.137094, size = 125, normalized size = 1.03 \[ \frac{2 b c-a d}{a^3 x}-\frac{c}{3 a^2 x^3}-\frac{x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a^3 b \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f+a^2 b e-3 a b^2 d+5 b^3 c\right )}{2 a^{7/2} b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 182, normalized size = 1.5 \[ -{\frac{c}{3\,{x}^{3}{a}^{2}}}-{\frac{d}{x{a}^{2}}}+2\,{\frac{bc}{{a}^{3}x}}-{\frac{fx}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{ex}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bxd}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}xc}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{f}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,bd}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}c}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*c/a^2/x^3-1/a^2/x*d+2/a^3/x*b*c-1/2/b*x/(b*x^2+a)*f+1/2/a*x/(b*x^2+a)*e-1/2
/a^2*b*x/(b*x^2+a)*d+1/2/a^3*b^2*x/(b*x^2+a)*c+1/2/b/(a*b)^(1/2)*arctan(x*b/(a*b
)^(1/2))*f+1/2/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*e-3/2/a^2*b/(a*b)^(1/2)*arc
tan(x*b/(a*b)^(1/2))*d+5/2/a^3*b^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)^2*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241104, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (5 \, b^{4} c - 3 \, a b^{3} d + a^{2} b^{2} e + a^{3} b f\right )} x^{5} +{\left (5 \, a b^{3} c - 3 \, a^{2} b^{2} d + a^{3} b e + a^{4} f\right )} x^{3}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (3 \,{\left (5 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e - a^{3} f\right )} x^{4} - 2 \, a^{2} b c + 2 \,{\left (5 \, a b^{2} c - 3 \, a^{2} b d\right )} x^{2}\right )} \sqrt{-a b}}{12 \,{\left (a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt{-a b}}, \frac{3 \,{\left ({\left (5 \, b^{4} c - 3 \, a b^{3} d + a^{2} b^{2} e + a^{3} b f\right )} x^{5} +{\left (5 \, a b^{3} c - 3 \, a^{2} b^{2} d + a^{3} b e + a^{4} f\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \,{\left (5 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e - a^{3} f\right )} x^{4} - 2 \, a^{2} b c + 2 \,{\left (5 \, a b^{2} c - 3 \, a^{2} b d\right )} x^{2}\right )} \sqrt{a b}}{6 \,{\left (a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt{a b}}\right ] \]

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^4),x, algorithm="fricas")

[Out]

[1/12*(3*((5*b^4*c - 3*a*b^3*d + a^2*b^2*e + a^3*b*f)*x^5 + (5*a*b^3*c - 3*a^2*b
^2*d + a^3*b*e + a^4*f)*x^3)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a))
+ 2*(3*(5*b^3*c - 3*a*b^2*d + a^2*b*e - a^3*f)*x^4 - 2*a^2*b*c + 2*(5*a*b^2*c -
3*a^2*b*d)*x^2)*sqrt(-a*b))/((a^3*b^2*x^5 + a^4*b*x^3)*sqrt(-a*b)), 1/6*(3*((5*
b^4*c - 3*a*b^3*d + a^2*b^2*e + a^3*b*f)*x^5 + (5*a*b^3*c - 3*a^2*b^2*d + a^3*b*
e + a^4*f)*x^3)*arctan(sqrt(a*b)*x/a) + (3*(5*b^3*c - 3*a*b^2*d + a^2*b*e - a^3*
f)*x^4 - 2*a^2*b*c + 2*(5*a*b^2*c - 3*a^2*b*d)*x^2)*sqrt(a*b))/((a^3*b^2*x^5 + a
^4*b*x^3)*sqrt(a*b))]

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Sympy [A]  time = 23.8513, size = 212, normalized size = 1.75 \[ - \frac{\sqrt{- \frac{1}{a^{7} b^{3}}} \left (a^{3} f + a^{2} b e - 3 a b^{2} d + 5 b^{3} c\right ) \log{\left (- a^{4} b \sqrt{- \frac{1}{a^{7} b^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{7} b^{3}}} \left (a^{3} f + a^{2} b e - 3 a b^{2} d + 5 b^{3} c\right ) \log{\left (a^{4} b \sqrt{- \frac{1}{a^{7} b^{3}}} + x \right )}}{4} - \frac{2 a^{2} b c + x^{4} \left (3 a^{3} f - 3 a^{2} b e + 9 a b^{2} d - 15 b^{3} c\right ) + x^{2} \left (6 a^{2} b d - 10 a b^{2} c\right )}{6 a^{4} b x^{3} + 6 a^{3} b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**7*b**3))*(a**3*f + a**2*b*e - 3*a*b**2*d + 5*b**3*c)*log(-a**4*b*sq
rt(-1/(a**7*b**3)) + x)/4 + sqrt(-1/(a**7*b**3))*(a**3*f + a**2*b*e - 3*a*b**2*d
+ 5*b**3*c)*log(a**4*b*sqrt(-1/(a**7*b**3)) + x)/4 - (2*a**2*b*c + x**4*(3*a**3
*f - 3*a**2*b*e + 9*a*b**2*d - 15*b**3*c) + x**2*(6*a**2*b*d - 10*a*b**2*c))/(6*
a**4*b*x**3 + 6*a**3*b**2*x**5)

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GIAC/XCAS [A]  time = 0.216214, size = 166, normalized size = 1.37 \[ \frac{{\left (5 \, b^{3} c - 3 \, a b^{2} d + a^{3} f + a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3} b} + \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{2 \,{\left (b x^{2} + a\right )} a^{3} b} + \frac{6 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{3} x^{3}} \]

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)^2*x^4),x, algorithm="giac")

[Out]

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