3.131 \(\int \frac{c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=189 \[ \frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-3 a^3 f+5 a^2 b e-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \]

[Out]

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

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Rubi [A]  time = 0.577026, antiderivative size = 189, 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 \[ \frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7}-\frac{a^2 e-2 a b d+3 b^2 c}{3 a^4 x^3}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-3 a^3 f+5 a^2 b e-7 a b^2 d+9 b^3 c\right )}{2 a^{11/2}}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 160.894, size = 202, normalized size = 1.07 \[ - \frac{x \left (\frac{a^{3} f}{x^{8}} - \frac{a^{2} b e}{x^{8}} + \frac{a b^{2} d}{x^{8}} - \frac{b^{3} c}{x^{8}}\right )}{2 a b^{3} \left (a + b x^{2}\right )} - \frac{a^{2} f - a b e + b^{2} d}{7 a b^{3} x^{7}} + \frac{2 a^{2} f - 2 a b e + b^{2} d}{5 a^{2} b^{2} x^{5}} - \frac{3 a^{2} f - 2 a b e + b^{2} d}{3 a^{3} b x^{3}} + \frac{3 a^{2} f - 2 a b e + b^{2} d}{a^{4} x} + \frac{\sqrt{b} \left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((f*x**6+e*x**4+d*x**2+c)/x**8/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.194289, size = 190, normalized size = 1.01 \[ \frac{2 b c-a d}{5 a^3 x^5}-\frac{c}{7 a^2 x^7}+\frac{a^2 (-e)+2 a b d-3 b^2 c}{3 a^4 x^3}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f-5 a^2 b e+7 a b^2 d-9 b^3 c\right )}{2 a^{11/2}}-\frac{b x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a^5 \left (a+b x^2\right )}+\frac{a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{a^5 x} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^2),x]

[Out]

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

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Maple [A]  time = 0.023, size = 268, normalized size = 1.4 \[ -{\frac{c}{7\,{a}^{2}{x}^{7}}}-{\frac{d}{5\,{x}^{5}{a}^{2}}}+{\frac{2\,bc}{5\,{a}^{3}{x}^{5}}}-{\frac{e}{3\,{x}^{3}{a}^{2}}}+{\frac{2\,bd}{3\,{a}^{3}{x}^{3}}}-{\frac{{b}^{2}c}{{a}^{4}{x}^{3}}}-{\frac{f}{x{a}^{2}}}+2\,{\frac{be}{{a}^{3}x}}-3\,{\frac{d{b}^{2}}{{a}^{4}x}}+4\,{\frac{{b}^{3}c}{{a}^{5}x}}-{\frac{bxf}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}xe}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{b}^{3}xd}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{4}xc}{2\,{a}^{5} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,fb}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}e}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,d{b}^{3}}{2\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,{b}^{4}c}{2\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a)^2,x)

[Out]

-1/7*c/a^2/x^7-1/5/a^2/x^5*d+2/5/a^3/x^5*b*c-1/3/a^2/x^3*e+2/3/a^3/x^3*b*d-1/a^4
/x^3*b^2*c-1/a^2/x*f+2/a^3/x*b*e-3/a^4/x*b^2*d+4/a^5/x*b^3*c-1/2*b/a^2*x/(b*x^2+
a)*f+1/2*b^2/a^3*x/(b*x^2+a)*e-1/2*b^3/a^4*x/(b*x^2+a)*d+1/2*b^4/a^5*x/(b*x^2+a)
*c-3/2*b/a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*f+5/2*b^2/a^3/(a*b)^(1/2)*arcta
n(x*b/(a*b)^(1/2))*e-7/2*b^3/a^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d+9/2*b^4/a
^5/(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^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239315, size = 1, normalized size = 0.01 \[ \left [\frac{210 \,{\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 140 \,{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} - 60 \, a^{4} c - 28 \,{\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d + 5 \, a^{4} e\right )} x^{4} + 12 \,{\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2} - 105 \,{\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{9} +{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{7}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{420 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, \frac{105 \,{\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{8} + 70 \,{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{6} - 30 \, a^{4} c - 14 \,{\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d + 5 \, a^{4} e\right )} x^{4} + 6 \,{\left (9 \, a^{3} b c - 7 \, a^{4} d\right )} x^{2} + 105 \,{\left ({\left (9 \, b^{4} c - 7 \, a b^{3} d + 5 \, a^{2} b^{2} e - 3 \, a^{3} b f\right )} x^{9} +{\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d + 5 \, a^{3} b e - 3 \, a^{4} f\right )} x^{7}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{210 \,{\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}\right ] \]

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^8),x, algorithm="fricas")

[Out]

[1/420*(210*(9*b^4*c - 7*a*b^3*d + 5*a^2*b^2*e - 3*a^3*b*f)*x^8 + 140*(9*a*b^3*c
- 7*a^2*b^2*d + 5*a^3*b*e - 3*a^4*f)*x^6 - 60*a^4*c - 28*(9*a^2*b^2*c - 7*a^3*b
*d + 5*a^4*e)*x^4 + 12*(9*a^3*b*c - 7*a^4*d)*x^2 - 105*((9*b^4*c - 7*a*b^3*d + 5
*a^2*b^2*e - 3*a^3*b*f)*x^9 + (9*a*b^3*c - 7*a^2*b^2*d + 5*a^3*b*e - 3*a^4*f)*x^
7)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^5*b*x^9 + a^6*
x^7), 1/210*(105*(9*b^4*c - 7*a*b^3*d + 5*a^2*b^2*e - 3*a^3*b*f)*x^8 + 70*(9*a*b
^3*c - 7*a^2*b^2*d + 5*a^3*b*e - 3*a^4*f)*x^6 - 30*a^4*c - 14*(9*a^2*b^2*c - 7*a
^3*b*d + 5*a^4*e)*x^4 + 6*(9*a^3*b*c - 7*a^4*d)*x^2 + 105*((9*b^4*c - 7*a*b^3*d
+ 5*a^2*b^2*e - 3*a^3*b*f)*x^9 + (9*a*b^3*c - 7*a^2*b^2*d + 5*a^3*b*e - 3*a^4*f)
*x^7)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(a^5*b*x^9 + a^6*x^7)]

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Sympy [A]  time = 127.943, size = 394, normalized size = 2.08 \[ \frac{\sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log{\left (- \frac{a^{6} \sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} - \frac{\sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right ) \log{\left (\frac{a^{6} \sqrt{- \frac{b}{a^{11}}} \left (3 a^{3} f - 5 a^{2} b e + 7 a b^{2} d - 9 b^{3} c\right )}{3 a^{3} b f - 5 a^{2} b^{2} e + 7 a b^{3} d - 9 b^{4} c} + x \right )}}{4} - \frac{30 a^{4} c + x^{8} \left (315 a^{3} b f - 525 a^{2} b^{2} e + 735 a b^{3} d - 945 b^{4} c\right ) + x^{6} \left (210 a^{4} f - 350 a^{3} b e + 490 a^{2} b^{2} d - 630 a b^{3} c\right ) + x^{4} \left (70 a^{4} e - 98 a^{3} b d + 126 a^{2} b^{2} c\right ) + x^{2} \left (42 a^{4} d - 54 a^{3} b c\right )}{210 a^{6} x^{7} + 210 a^{5} b x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x**6+e*x**4+d*x**2+c)/x**8/(b*x**2+a)**2,x)

[Out]

sqrt(-b/a**11)*(3*a**3*f - 5*a**2*b*e + 7*a*b**2*d - 9*b**3*c)*log(-a**6*sqrt(-b
/a**11)*(3*a**3*f - 5*a**2*b*e + 7*a*b**2*d - 9*b**3*c)/(3*a**3*b*f - 5*a**2*b**
2*e + 7*a*b**3*d - 9*b**4*c) + x)/4 - sqrt(-b/a**11)*(3*a**3*f - 5*a**2*b*e + 7*
a*b**2*d - 9*b**3*c)*log(a**6*sqrt(-b/a**11)*(3*a**3*f - 5*a**2*b*e + 7*a*b**2*d
- 9*b**3*c)/(3*a**3*b*f - 5*a**2*b**2*e + 7*a*b**3*d - 9*b**4*c) + x)/4 - (30*a
**4*c + x**8*(315*a**3*b*f - 525*a**2*b**2*e + 735*a*b**3*d - 945*b**4*c) + x**6
*(210*a**4*f - 350*a**3*b*e + 490*a**2*b**2*d - 630*a*b**3*c) + x**4*(70*a**4*e
- 98*a**3*b*d + 126*a**2*b**2*c) + x**2*(42*a**4*d - 54*a**3*b*c))/(210*a**6*x**
7 + 210*a**5*b*x**9)

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GIAC/XCAS [A]  time = 0.217014, size = 271, normalized size = 1.43 \[ \frac{{\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f + 5 \, a^{2} b^{2} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{5}} + \frac{b^{4} c x - a b^{3} d x - a^{3} b f x + a^{2} b^{2} x e}{2 \,{\left (b x^{2} + a\right )} a^{5}} + \frac{420 \, b^{3} c x^{6} - 315 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 210 \, a^{2} b x^{6} e - 105 \, a b^{2} c x^{4} + 70 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 42 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{5} x^{7}} \]

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^8),x, algorithm="giac")

[Out]

1/2*(9*b^4*c - 7*a*b^3*d - 3*a^3*b*f + 5*a^2*b^2*e)*arctan(b*x/sqrt(a*b))/(sqrt(
a*b)*a^5) + 1/2*(b^4*c*x - a*b^3*d*x - a^3*b*f*x + a^2*b^2*x*e)/((b*x^2 + a)*a^5
) + 1/105*(420*b^3*c*x^6 - 315*a*b^2*d*x^6 - 105*a^3*f*x^6 + 210*a^2*b*x^6*e - 1
05*a*b^2*c*x^4 + 70*a^2*b*d*x^4 - 35*a^3*x^4*e + 42*a^2*b*c*x^2 - 21*a^3*d*x^2 -
15*a^3*c)/(a^5*x^7)