3.132 \(\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=230 \[ \frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{9 a^2 x^9}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-5 a^3 f+7 a^2 b e-9 a b^2 d+11 b^3 c\right )}{2 a^{13/2}}-\frac{b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}-\frac{b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 x^3} \]

[Out]

-c/(9*a^2*x^9) + (2*b*c - a*d)/(7*a^3*x^7) - (3*b^2*c - 2*a*b*d + a^2*e)/(5*a^4*
x^5) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(3*a^5*x^3) - (b*(5*b^3*c - 4*a
*b^2*d + 3*a^2*b*e - 2*a^3*f))/(a^6*x) - (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f
)*x)/(2*a^6*(a + b*x^2)) - (b^(3/2)*(11*b^3*c - 9*a*b^2*d + 7*a^2*b*e - 5*a^3*f)
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(13/2))

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Rubi [A]  time = 0.73867, antiderivative size = 230, 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 \[ \frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{9 a^2 x^9}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-5 a^3 f+7 a^2 b e-9 a b^2 d+11 b^3 c\right )}{2 a^{13/2}}-\frac{b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}-\frac{b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2),x]

[Out]

-c/(9*a^2*x^9) + (2*b*c - a*d)/(7*a^3*x^7) - (3*b^2*c - 2*a*b*d + a^2*e)/(5*a^4*
x^5) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(3*a^5*x^3) - (b*(5*b^3*c - 4*a
*b^2*d + 3*a^2*b*e - 2*a^3*f))/(a^6*x) - (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f
)*x)/(2*a^6*(a + b*x^2)) - (b^(3/2)*(11*b^3*c - 9*a*b^2*d + 7*a^2*b*e - 5*a^3*f)
*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(13/2))

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Rubi in Sympy [A]  time = 168.751, size = 231, normalized size = 1. \[ - \frac{x \left (\frac{a^{3} f}{x^{10}} - \frac{a^{2} b e}{x^{10}} + \frac{a b^{2} d}{x^{10}} - \frac{b^{3} c}{x^{10}}\right )}{2 a b^{3} \left (a + b x^{2}\right )} - \frac{a^{2} f - a b e + b^{2} d}{9 a b^{3} x^{9}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{7 a^{2} b^{2} x^{7}} - \frac{3 a^{2} f - 2 a b e + b^{2} d}{5 a^{3} b x^{5}} + \frac{3 a^{2} f - 2 a b e + b^{2} d}{3 a^{4} x^{3}} - \frac{b \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{a^{5} x} - \frac{b^{\frac{3}{2}} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**6+e*x**4+d*x**2+c)/x**10/(b*x**2+a)**2,x)

[Out]

-x*(a**3*f/x**10 - a**2*b*e/x**10 + a*b**2*d/x**10 - b**3*c/x**10)/(2*a*b**3*(a
+ b*x**2)) - (a**2*f - a*b*e + b**2*d)/(9*a*b**3*x**9) + (2*a**2*f - 2*a*b*e + b
**2*d)/(7*a**2*b**2*x**7) - (3*a**2*f - 2*a*b*e + b**2*d)/(5*a**3*b*x**5) + (3*a
**2*f - 2*a*b*e + b**2*d)/(3*a**4*x**3) - b*(3*a**2*f - 2*a*b*e + b**2*d)/(a**5*
x) - b**(3/2)*(3*a**2*f - 2*a*b*e + b**2*d)*atan(sqrt(b)*x/sqrt(a))/a**(11/2)

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Mathematica [A]  time = 0.221184, size = 230, normalized size = 1. \[ \frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{9 a^2 x^9}+\frac{a^2 (-e)+2 a b d-3 b^2 c}{5 a^4 x^5}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^3 f-7 a^2 b e+9 a b^2 d-11 b^3 c\right )}{2 a^{13/2}}+\frac{b^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a^6 \left (a+b x^2\right )}+\frac{b \left (2 a^3 f-3 a^2 b e+4 a b^2 d-5 b^3 c\right )}{a^6 x}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{3 a^5 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^2),x]

[Out]

-c/(9*a^2*x^9) + (2*b*c - a*d)/(7*a^3*x^7) + (-3*b^2*c + 2*a*b*d - a^2*e)/(5*a^4
*x^5) + (4*b^3*c - 3*a*b^2*d + 2*a^2*b*e - a^3*f)/(3*a^5*x^3) + (b*(-5*b^3*c + 4
*a*b^2*d - 3*a^2*b*e + 2*a^3*f))/(a^6*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e +
a^3*f)*x)/(2*a^6*(a + b*x^2)) + (b^(3/2)*(-11*b^3*c + 9*a*b^2*d - 7*a^2*b*e + 5*
a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(13/2))

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Maple [A]  time = 0.024, size = 318, normalized size = 1.4 \[ -{\frac{c}{9\,{a}^{2}{x}^{9}}}-{\frac{d}{7\,{a}^{2}{x}^{7}}}+{\frac{2\,bc}{7\,{a}^{3}{x}^{7}}}-{\frac{e}{5\,{x}^{5}{a}^{2}}}+{\frac{2\,bd}{5\,{a}^{3}{x}^{5}}}-{\frac{3\,{b}^{2}c}{5\,{a}^{4}{x}^{5}}}-{\frac{f}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,be}{3\,{a}^{3}{x}^{3}}}-{\frac{d{b}^{2}}{{a}^{4}{x}^{3}}}+{\frac{4\,{b}^{3}c}{3\,{a}^{5}{x}^{3}}}+2\,{\frac{fb}{{a}^{3}x}}-3\,{\frac{e{b}^{2}}{{a}^{4}x}}+4\,{\frac{d{b}^{3}}{{a}^{5}x}}-5\,{\frac{c{b}^{4}}{{a}^{6}x}}+{\frac{{b}^{2}xf}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xe}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}xd}{2\,{a}^{5} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{5}xc}{2\,{a}^{6} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,f{b}^{2}}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{b}^{3}e}{2\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,d{b}^{4}}{2\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{11\,{b}^{5}c}{2\,{a}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^10/(b*x^2+a)^2,x)

[Out]

-1/9*c/a^2/x^9-1/7/a^2/x^7*d+2/7/a^3/x^7*b*c-1/5/a^2/x^5*e+2/5/a^3/x^5*b*d-3/5/a
^4/x^5*b^2*c-1/3/a^2/x^3*f+2/3/a^3/x^3*b*e-1/a^4/x^3*b^2*d+4/3/a^5/x^3*b^3*c+2*b
/a^3/x*f-3*b^2/a^4/x*e+4*b^3/a^5/x*d-5*b^4/a^6/x*c+1/2*b^2/a^3*x/(b*x^2+a)*f-1/2
*b^3/a^4*x/(b*x^2+a)*e+1/2*b^4/a^5*x/(b*x^2+a)*d-1/2*b^5/a^6*x/(b*x^2+a)*c+5/2*b
^2/a^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*f-7/2*b^3/a^4/(a*b)^(1/2)*arctan(x*b/
(a*b)^(1/2))*e+9/2*b^4/a^5/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d-11/2*b^5/a^6/(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^10),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241076, size = 1, normalized size = 0. \[ \left [-\frac{630 \,{\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{10} + 420 \,{\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{8} - 84 \,{\left (11 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 7 \, a^{4} b e - 5 \, a^{5} f\right )} x^{6} + 140 \, a^{5} c + 36 \,{\left (11 \, a^{3} b^{2} c - 9 \, a^{4} b d + 7 \, a^{5} e\right )} x^{4} - 20 \,{\left (11 \, a^{4} b c - 9 \, a^{5} d\right )} x^{2} + 315 \,{\left ({\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{11} +{\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{9}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{1260 \,{\left (a^{6} b x^{11} + a^{7} x^{9}\right )}}, -\frac{315 \,{\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{10} + 210 \,{\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{8} - 42 \,{\left (11 \, a^{2} b^{3} c - 9 \, a^{3} b^{2} d + 7 \, a^{4} b e - 5 \, a^{5} f\right )} x^{6} + 70 \, a^{5} c + 18 \,{\left (11 \, a^{3} b^{2} c - 9 \, a^{4} b d + 7 \, a^{5} e\right )} x^{4} - 10 \,{\left (11 \, a^{4} b c - 9 \, a^{5} d\right )} x^{2} + 315 \,{\left ({\left (11 \, b^{5} c - 9 \, a b^{4} d + 7 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{11} +{\left (11 \, a b^{4} c - 9 \, a^{2} b^{3} d + 7 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x^{9}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{630 \,{\left (a^{6} b x^{11} + a^{7} x^{9}\right )}}\right ] \]

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^10),x, algorithm="fricas")

[Out]

[-1/1260*(630*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^10 + 420*(11*
a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)*x^8 - 84*(11*a^2*b^3*c - 9*a^3*
b^2*d + 7*a^4*b*e - 5*a^5*f)*x^6 + 140*a^5*c + 36*(11*a^3*b^2*c - 9*a^4*b*d + 7*
a^5*e)*x^4 - 20*(11*a^4*b*c - 9*a^5*d)*x^2 + 315*((11*b^5*c - 9*a*b^4*d + 7*a^2*
b^3*e - 5*a^3*b^2*f)*x^11 + (11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)
*x^9)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^6*b*x^11 +
a^7*x^9), -1/630*(315*(11*b^5*c - 9*a*b^4*d + 7*a^2*b^3*e - 5*a^3*b^2*f)*x^10 +
210*(11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a^4*b*f)*x^8 - 42*(11*a^2*b^3*c
- 9*a^3*b^2*d + 7*a^4*b*e - 5*a^5*f)*x^6 + 70*a^5*c + 18*(11*a^3*b^2*c - 9*a^4*b
*d + 7*a^5*e)*x^4 - 10*(11*a^4*b*c - 9*a^5*d)*x^2 + 315*((11*b^5*c - 9*a*b^4*d +
7*a^2*b^3*e - 5*a^3*b^2*f)*x^11 + (11*a*b^4*c - 9*a^2*b^3*d + 7*a^3*b^2*e - 5*a
^4*b*f)*x^9)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(a^6*b*x^11 + a^7*x^9)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**6+e*x**4+d*x**2+c)/x**10/(b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213703, size = 340, normalized size = 1.48 \[ -\frac{{\left (11 \, b^{5} c - 9 \, a b^{4} d - 5 \, a^{3} b^{2} f + 7 \, a^{2} b^{3} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{6}} - \frac{b^{5} c x - a b^{4} d x - a^{3} b^{2} f x + a^{2} b^{3} x e}{2 \,{\left (b x^{2} + a\right )} a^{6}} - \frac{1575 \, b^{4} c x^{8} - 1260 \, a b^{3} d x^{8} - 630 \, a^{3} b f x^{8} + 945 \, a^{2} b^{2} x^{8} e - 420 \, a b^{3} c x^{6} + 315 \, a^{2} b^{2} d x^{6} + 105 \, a^{4} f x^{6} - 210 \, a^{3} b x^{6} e + 189 \, a^{2} b^{2} c x^{4} - 126 \, a^{3} b d x^{4} + 63 \, a^{4} x^{4} e - 90 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{6} x^{9}} \]

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^10),x, algorithm="giac")

[Out]

-1/2*(11*b^5*c - 9*a*b^4*d - 5*a^3*b^2*f + 7*a^2*b^3*e)*arctan(b*x/sqrt(a*b))/(s
qrt(a*b)*a^6) - 1/2*(b^5*c*x - a*b^4*d*x - a^3*b^2*f*x + a^2*b^3*x*e)/((b*x^2 +
a)*a^6) - 1/315*(1575*b^4*c*x^8 - 1260*a*b^3*d*x^8 - 630*a^3*b*f*x^8 + 945*a^2*b
^2*x^8*e - 420*a*b^3*c*x^6 + 315*a^2*b^2*d*x^6 + 105*a^4*f*x^6 - 210*a^3*b*x^6*e
+ 189*a^2*b^2*c*x^4 - 126*a^3*b*d*x^4 + 63*a^4*x^4*e - 90*a^3*b*c*x^2 + 45*a^4*
d*x^2 + 35*a^4*c)/(a^6*x^9)