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

Optimal. Leaf size=118 \[ \frac{x \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^3 f-3 a^2 b e+a b^2 d+b^3 c\right )}{2 a^{3/2} b^{7/2}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.276915, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^3 f-3 a^2 b e+a b^2 d+b^3 c\right )}{2 a^{3/2} b^{7/2}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 81.0225, size = 114, normalized size = 0.97 \[ \frac{f x^{3}}{3 b^{2}} - \frac{x \left (2 a f - b e\right )}{b^{3}} - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{2 a b^{3} \left (a + b x^{2}\right )} + \frac{\left (5 a^{3} f - 3 a^{2} b e + a b^{2} d + b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.165728, size = 122, normalized size = 1.03 \[ -\frac{x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a b^3 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^3 f-3 a^2 b e+a b^2 d+b^3 c\right )}{2 a^{3/2} b^{7/2}}+\frac{x (b e-2 a f)}{b^3}+\frac{f x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 177, normalized size = 1.5 \[{\frac{f{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{afx}{{b}^{3}}}+{\frac{ex}{{b}^{2}}}-{\frac{x{a}^{2}f}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{axe}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{dx}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{cx}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{2}f}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,ae}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{d}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{c}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239996, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a b^{3} c + a^{2} b^{2} d - 3 \, a^{3} b e + 5 \, a^{4} f +{\left (b^{4} c + a b^{3} d - 3 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (2 \, a b^{2} f x^{5} + 2 \,{\left (3 \, a b^{2} e - 5 \, a^{2} b f\right )} x^{3} + 3 \,{\left (b^{3} c - a b^{2} d + 3 \, a^{2} b e - 5 \, a^{3} f\right )} x\right )} \sqrt{-a b}}{12 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{-a b}}, \frac{3 \,{\left (a b^{3} c + a^{2} b^{2} d - 3 \, a^{3} b e + 5 \, a^{4} f +{\left (b^{4} c + a b^{3} d - 3 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (2 \, a b^{2} f x^{5} + 2 \,{\left (3 \, a b^{2} e - 5 \, a^{2} b f\right )} x^{3} + 3 \,{\left (b^{3} c - a b^{2} d + 3 \, a^{2} b e - 5 \, a^{3} f\right )} x\right )} \sqrt{a b}}{6 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{a b}}\right ] \]

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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.68975, size = 199, normalized size = 1.69 \[ - \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} b^{7}}} \left (5 a^{3} f - 3 a^{2} b e + a b^{2} d + b^{3} c\right ) \log{\left (- a^{2} b^{3} \sqrt{- \frac{1}{a^{3} b^{7}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b^{7}}} \left (5 a^{3} f - 3 a^{2} b e + a b^{2} d + b^{3} c\right ) \log{\left (a^{2} b^{3} \sqrt{- \frac{1}{a^{3} b^{7}}} + x \right )}}{4} + \frac{f x^{3}}{3 b^{2}} - \frac{x \left (2 a f - b e\right )}{b^{3}} \]

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)

[Out]

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

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GIAC/XCAS [A]  time = 0.215635, size = 170, normalized size = 1.44 \[ \frac{{\left (b^{3} c + a b^{2} d + 5 \, a^{3} f - 3 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b^{3}} + \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{2 \,{\left (b x^{2} + a\right )} a b^{3}} + \frac{b^{4} f x^{3} - 6 \, a b^{3} f x + 3 \, b^{4} x e}{3 \, b^{6}} \]

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, algorithm="giac")

[Out]

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