3.138 \(\int \frac{c+d x^2+e x^4+f x^6}{x^2 \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=150 \[ -\frac{c}{a^3 x}-\frac{x \left (\frac{5 a^2 f}{b^2}+\frac{7 b c}{a}-\frac{a e}{b}-3 d\right )}{8 a^2 \left (a+b x^2\right )}-\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{4 a \left (a+b x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-3 a^3 f-a^2 b e-3 a b^2 d+15 b^3 c\right )}{8 a^{7/2} b^{5/2}} \]

[Out]

-(c/(a^3*x)) - (((b*c)/a - d + (a*e)/b - (a^2*f)/b^2)*x)/(4*a*(a + b*x^2)^2) - (
((7*b*c)/a - 3*d - (a*e)/b + (5*a^2*f)/b^2)*x)/(8*a^2*(a + b*x^2)) - ((15*b^3*c
- 3*a*b^2*d - a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*b^(5/2)
)

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Rubi [A]  time = 0.464153, antiderivative size = 150, 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 \[ -\frac{c}{a^3 x}-\frac{x \left (\frac{5 a^2 f}{b^2}+\frac{7 b c}{a}-\frac{a e}{b}-3 d\right )}{8 a^2 \left (a+b x^2\right )}-\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{4 a \left (a+b x^2\right )^2}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-3 a^3 f-a^2 b e-3 a b^2 d+15 b^3 c\right )}{8 a^{7/2} b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^3),x]

[Out]

-(c/(a^3*x)) - (((b*c)/a - d + (a*e)/b - (a^2*f)/b^2)*x)/(4*a*(a + b*x^2)^2) - (
((7*b*c)/a - 3*d - (a*e)/b + (5*a^2*f)/b^2)*x)/(8*a^2*(a + b*x^2)) - ((15*b^3*c
- 3*a*b^2*d - a^2*b*e - 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/2)*b^(5/2)
)

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Rubi in Sympy [A]  time = 151.552, size = 160, normalized size = 1.07 \[ - \frac{x \left (\frac{a^{3} f}{x^{2}} - \frac{a^{2} b e}{x^{2}} + \frac{a b^{2} d}{x^{2}} - \frac{b^{3} c}{x^{2}}\right )}{4 a b^{3} \left (a + b x^{2}\right )^{2}} - \frac{x \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{2 a^{2} b^{2} \left (a + b x^{2}\right )} - \frac{a^{2} f - a b e + b^{2} d}{a^{2} b^{3} x} - \frac{\left (3 a^{2} f - 4 a b e + 3 b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} b^{\frac{5}{2}}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.236805, size = 155, normalized size = 1.03 \[ -\frac{c}{a^3 x}-\frac{x \left (5 a^3 f-a^2 b e-3 a b^2 d+7 b^3 c\right )}{8 a^3 b^2 \left (a+b x^2\right )}+\frac{x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 a^2 b^2 \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f+a^2 b e+3 a b^2 d-15 b^3 c\right )}{8 a^{7/2} b^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^3),x]

[Out]

-(c/(a^3*x)) + ((-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x)/(4*a^2*b^2*(a + b*x^2)
^2) - ((7*b^3*c - 3*a*b^2*d - a^2*b*e + 5*a^3*f)*x)/(8*a^3*b^2*(a + b*x^2)) + ((
-15*b^3*c + 3*a*b^2*d + a^2*b*e + 3*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(7/
2)*b^(5/2))

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Maple [A]  time = 0.02, size = 237, normalized size = 1.6 \[ -{\frac{c}{{a}^{3}x}}-{\frac{5\,{x}^{3}f}{8\, \left ( b{x}^{2}+a \right ) ^{2}b}}+{\frac{{x}^{3}e}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,{x}^{3}bd}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{7\,{x}^{3}{b}^{2}c}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,axf}{8\, \left ( b{x}^{2}+a \right ) ^{2}{b}^{2}}}-{\frac{ex}{8\, \left ( b{x}^{2}+a \right ) ^{2}b}}+{\frac{5\,dx}{8\,a \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bxc}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,f}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,d}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,bc}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^2/(b*x^2+a)^3,x)

[Out]

-c/a^3/x-5/8/(b*x^2+a)^2/b*x^3*f+1/8/a/(b*x^2+a)^2*x^3*e+3/8/a^2/(b*x^2+a)^2*x^3
*b*d-7/8/a^3/(b*x^2+a)^2*x^3*b^2*c-3/8*a/(b*x^2+a)^2/b^2*x*f-1/8/(b*x^2+a)^2/b*x
*e+5/8/a/(b*x^2+a)^2*x*d-9/8/a^2/(b*x^2+a)^2*x*b*c+3/8/b^2/(a*b)^(1/2)*arctan(x*
b/(a*b)^(1/2))*f+1/8/a/b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*e+3/8/a^2/(a*b)^(1/
2)*arctan(x*b/(a*b)^(1/2))*d-15/8/a^3*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)^3*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238737, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (15 \, b^{5} c - 3 \, a b^{4} d - a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{5} + 2 \,{\left (15 \, a b^{4} c - 3 \, a^{2} b^{3} d - a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{3} +{\left (15 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d - a^{4} b e - 3 \, a^{5} f\right )} x\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (8 \, a^{2} b^{2} c +{\left (15 \, b^{4} c - 3 \, a b^{3} d - a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} +{\left (25 \, a b^{3} c - 5 \, a^{2} b^{2} d + a^{3} b e + 3 \, a^{4} f\right )} x^{2}\right )} \sqrt{-a b}}{16 \,{\left (a^{3} b^{4} x^{5} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2} x\right )} \sqrt{-a b}}, -\frac{{\left ({\left (15 \, b^{5} c - 3 \, a b^{4} d - a^{2} b^{3} e - 3 \, a^{3} b^{2} f\right )} x^{5} + 2 \,{\left (15 \, a b^{4} c - 3 \, a^{2} b^{3} d - a^{3} b^{2} e - 3 \, a^{4} b f\right )} x^{3} +{\left (15 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d - a^{4} b e - 3 \, a^{5} f\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (8 \, a^{2} b^{2} c +{\left (15 \, b^{4} c - 3 \, a b^{3} d - a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{4} +{\left (25 \, a b^{3} c - 5 \, a^{2} b^{2} d + a^{3} b e + 3 \, a^{4} f\right )} x^{2}\right )} \sqrt{a b}}{8 \,{\left (a^{3} b^{4} x^{5} + 2 \, a^{4} b^{3} x^{3} + a^{5} b^{2} x\right )} \sqrt{a b}}\right ] \]

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)^3*x^2),x, algorithm="fricas")

[Out]

[-1/16*(((15*b^5*c - 3*a*b^4*d - a^2*b^3*e - 3*a^3*b^2*f)*x^5 + 2*(15*a*b^4*c -
3*a^2*b^3*d - a^3*b^2*e - 3*a^4*b*f)*x^3 + (15*a^2*b^3*c - 3*a^3*b^2*d - a^4*b*e
- 3*a^5*f)*x)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(8*a^2*b^
2*c + (15*b^4*c - 3*a*b^3*d - a^2*b^2*e + 5*a^3*b*f)*x^4 + (25*a*b^3*c - 5*a^2*b
^2*d + a^3*b*e + 3*a^4*f)*x^2)*sqrt(-a*b))/((a^3*b^4*x^5 + 2*a^4*b^3*x^3 + a^5*b
^2*x)*sqrt(-a*b)), -1/8*(((15*b^5*c - 3*a*b^4*d - a^2*b^3*e - 3*a^3*b^2*f)*x^5 +
2*(15*a*b^4*c - 3*a^2*b^3*d - a^3*b^2*e - 3*a^4*b*f)*x^3 + (15*a^2*b^3*c - 3*a^
3*b^2*d - a^4*b*e - 3*a^5*f)*x)*arctan(sqrt(a*b)*x/a) + (8*a^2*b^2*c + (15*b^4*c
- 3*a*b^3*d - a^2*b^2*e + 5*a^3*b*f)*x^4 + (25*a*b^3*c - 5*a^2*b^2*d + a^3*b*e
+ 3*a^4*f)*x^2)*sqrt(a*b))/((a^3*b^4*x^5 + 2*a^4*b^3*x^3 + a^5*b^2*x)*sqrt(a*b))
]

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Sympy [A]  time = 40.3625, size = 250, normalized size = 1.67 \[ - \frac{\sqrt{- \frac{1}{a^{7} b^{5}}} \left (3 a^{3} f + a^{2} b e + 3 a b^{2} d - 15 b^{3} c\right ) \log{\left (- a^{4} b^{2} \sqrt{- \frac{1}{a^{7} b^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{7} b^{5}}} \left (3 a^{3} f + a^{2} b e + 3 a b^{2} d - 15 b^{3} c\right ) \log{\left (a^{4} b^{2} \sqrt{- \frac{1}{a^{7} b^{5}}} + x \right )}}{16} - \frac{8 a^{2} b^{2} c + x^{4} \left (5 a^{3} b f - a^{2} b^{2} e - 3 a b^{3} d + 15 b^{4} c\right ) + x^{2} \left (3 a^{4} f + a^{3} b e - 5 a^{2} b^{2} d + 25 a b^{3} c\right )}{8 a^{5} b^{2} x + 16 a^{4} b^{3} x^{3} + 8 a^{3} b^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**6+e*x**4+d*x**2+c)/x**2/(b*x**2+a)**3,x)

[Out]

-sqrt(-1/(a**7*b**5))*(3*a**3*f + a**2*b*e + 3*a*b**2*d - 15*b**3*c)*log(-a**4*b
**2*sqrt(-1/(a**7*b**5)) + x)/16 + sqrt(-1/(a**7*b**5))*(3*a**3*f + a**2*b*e + 3
*a*b**2*d - 15*b**3*c)*log(a**4*b**2*sqrt(-1/(a**7*b**5)) + x)/16 - (8*a**2*b**2
*c + x**4*(5*a**3*b*f - a**2*b**2*e - 3*a*b**3*d + 15*b**4*c) + x**2*(3*a**4*f +
a**3*b*e - 5*a**2*b**2*d + 25*a*b**3*c))/(8*a**5*b**2*x + 16*a**4*b**3*x**3 + 8
*a**3*b**4*x**5)

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GIAC/XCAS [A]  time = 0.217715, size = 207, normalized size = 1.38 \[ -\frac{c}{a^{3} x} - \frac{{\left (15 \, b^{3} c - 3 \, a b^{2} d - 3 \, a^{3} f - a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3} b^{2}} - \frac{7 \, b^{4} c x^{3} - 3 \, a b^{3} d x^{3} + 5 \, a^{3} b f x^{3} - a^{2} b^{2} x^{3} e + 9 \, a b^{3} c x - 5 \, a^{2} b^{2} d x + 3 \, a^{4} f x + a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{3} b^{2}} \]

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)^3*x^2),x, algorithm="giac")

[Out]

-c/(a^3*x) - 1/8*(15*b^3*c - 3*a*b^2*d - 3*a^3*f - a^2*b*e)*arctan(b*x/sqrt(a*b)
)/(sqrt(a*b)*a^3*b^2) - 1/8*(7*b^4*c*x^3 - 3*a*b^3*d*x^3 + 5*a^3*b*f*x^3 - a^2*b
^2*x^3*e + 9*a*b^3*c*x - 5*a^2*b^2*d*x + 3*a^4*f*x + a^3*b*x*e)/((b*x^2 + a)^2*a
^3*b^2)