3.141 \(\int \frac{c+d x^2+e x^4+f x^6}{x^8 \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=234 \[ \frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{7 a^3 x^7}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^3 f+35 a^2 b e-63 a b^2 d+99 b^3 c\right )}{8 a^{13/2}}+\frac{b x \left (-7 a^3 f+11 a^2 b e-15 a b^2 d+19 b^3 c\right )}{8 a^6 \left (a+b x^2\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^5 \left (a+b x^2\right )^2} \]

[Out]

-c/(7*a^3*x^7) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*
x^3) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x)/(4*a^5*(a + b*x^2)^2) + (b*(19*b^3*c - 15*a*b^2*d + 11*a^2
*b*e - 7*a^3*f)*x)/(8*a^6*(a + b*x^2)) + (Sqrt[b]*(99*b^3*c - 63*a*b^2*d + 35*a^
2*b*e - 15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))

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Rubi [A]  time = 0.912121, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{7 a^3 x^7}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^3 f+35 a^2 b e-63 a b^2 d+99 b^3 c\right )}{8 a^{13/2}}+\frac{b x \left (-7 a^3 f+11 a^2 b e-15 a b^2 d+19 b^3 c\right )}{8 a^6 \left (a+b x^2\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^5 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^3),x]

[Out]

-c/(7*a^3*x^7) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*
x^3) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x)/(4*a^5*(a + b*x^2)^2) + (b*(19*b^3*c - 15*a*b^2*d + 11*a^2
*b*e - 7*a^3*f)*x)/(8*a^6*(a + b*x^2)) + (Sqrt[b]*(99*b^3*c - 63*a*b^2*d + 35*a^
2*b*e - 15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))

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

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)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 0.253282, size = 234, normalized size = 1. \[ \frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{7 a^3 x^7}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^3 f+35 a^2 b e-63 a b^2 d+99 b^3 c\right )}{8 a^{13/2}}+\frac{b x \left (-7 a^3 f+11 a^2 b e-15 a b^2 d+19 b^3 c\right )}{8 a^6 \left (a+b x^2\right )}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{b x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^5 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^8*(a + b*x^2)^3),x]

[Out]

-c/(7*a^3*x^7) + (3*b*c - a*d)/(5*a^4*x^5) - (6*b^2*c - 3*a*b*d + a^2*e)/(3*a^5*
x^3) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(a^6*x) + (b*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*x)/(4*a^5*(a + b*x^2)^2) + (b*(19*b^3*c - 15*a*b^2*d + 11*a^2
*b*e - 7*a^3*f)*x)/(8*a^6*(a + b*x^2)) + (Sqrt[b]*(99*b^3*c - 63*a*b^2*d + 35*a^
2*b*e - 15*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(13/2))

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Maple [A]  time = 0.026, size = 351, normalized size = 1.5 \[ -{\frac{c}{7\,{a}^{3}{x}^{7}}}-{\frac{d}{5\,{a}^{3}{x}^{5}}}+{\frac{3\,bc}{5\,{a}^{4}{x}^{5}}}-{\frac{e}{3\,{a}^{3}{x}^{3}}}+{\frac{bd}{{a}^{4}{x}^{3}}}-2\,{\frac{{b}^{2}c}{{a}^{5}{x}^{3}}}-{\frac{f}{{a}^{3}x}}+3\,{\frac{be}{{a}^{4}x}}-6\,{\frac{d{b}^{2}}{{a}^{5}x}}+10\,{\frac{{b}^{3}c}{{a}^{6}x}}-{\frac{7\,{b}^{2}{x}^{3}f}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{11\,{b}^{3}{x}^{3}e}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,{b}^{4}{x}^{3}d}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{19\,{b}^{5}{x}^{3}c}{8\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,fbx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,{b}^{2}ex}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{17\,d{b}^{3}x}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{21\,{b}^{4}cx}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,fb}{8\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{b}^{2}e}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,d{b}^{3}}{8\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{99\,{b}^{4}c}{8\,{a}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((f*x^6+e*x^4+d*x^2+c)/x^8/(b*x^2+a)^3,x)

[Out]

-1/7*c/a^3/x^7-1/5/a^3/x^5*d+3/5/a^4/x^5*b*c-1/3/a^3/x^3*e+1/a^4/x^3*b*d-2/a^5/x
^3*b^2*c-1/a^3/x*f+3/a^4/x*b*e-6/a^5/x*b^2*d+10/a^6/x*b^3*c-7/8*b^2/a^3/(b*x^2+a
)^2*x^3*f+11/8*b^3/a^4/(b*x^2+a)^2*x^3*e-15/8*b^4/a^5/(b*x^2+a)^2*x^3*d+19/8*b^5
/a^6/(b*x^2+a)^2*x^3*c-9/8*b/a^2/(b*x^2+a)^2*f*x+13/8*b^2/a^3/(b*x^2+a)^2*e*x-17
/8*b^3/a^4/(b*x^2+a)^2*d*x+21/8*b^4/a^5/(b*x^2+a)^2*c*x-15/8*b/a^3/(a*b)^(1/2)*a
rctan(x*b/(a*b)^(1/2))*f+35/8*b^2/a^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*e-63/8
*b^3/a^5/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d+99/8*b^4/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)^3*x^8),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24664, size = 1, normalized size = 0. \[ \left [\frac{210 \,{\left (99 \, b^{5} c - 63 \, a b^{4} d + 35 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{10} + 350 \,{\left (99 \, a b^{4} c - 63 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{8} + 112 \,{\left (99 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d + 35 \, a^{4} b e - 15 \, a^{5} f\right )} x^{6} - 240 \, a^{5} c - 16 \,{\left (99 \, a^{3} b^{2} c - 63 \, a^{4} b d + 35 \, a^{5} e\right )} x^{4} + 48 \,{\left (11 \, a^{4} b c - 7 \, a^{5} d\right )} x^{2} - 105 \,{\left ({\left (99 \, b^{5} c - 63 \, a b^{4} d + 35 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{11} + 2 \,{\left (99 \, a b^{4} c - 63 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{9} +{\left (99 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d + 35 \, a^{4} b e - 15 \, a^{5} f\right )} x^{7}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{1680 \,{\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}}, \frac{105 \,{\left (99 \, b^{5} c - 63 \, a b^{4} d + 35 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{10} + 175 \,{\left (99 \, a b^{4} c - 63 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{8} + 56 \,{\left (99 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d + 35 \, a^{4} b e - 15 \, a^{5} f\right )} x^{6} - 120 \, a^{5} c - 8 \,{\left (99 \, a^{3} b^{2} c - 63 \, a^{4} b d + 35 \, a^{5} e\right )} x^{4} + 24 \,{\left (11 \, a^{4} b c - 7 \, a^{5} d\right )} x^{2} + 105 \,{\left ({\left (99 \, b^{5} c - 63 \, a b^{4} d + 35 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{11} + 2 \,{\left (99 \, a b^{4} c - 63 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{9} +{\left (99 \, a^{2} b^{3} c - 63 \, a^{3} b^{2} d + 35 \, a^{4} b e - 15 \, a^{5} f\right )} x^{7}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{840 \,{\left (a^{6} b^{2} x^{11} + 2 \, a^{7} b x^{9} + a^{8} x^{7}\right )}}\right ] \]

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)^3*x^8),x, algorithm="fricas")

[Out]

[1/1680*(210*(99*b^5*c - 63*a*b^4*d + 35*a^2*b^3*e - 15*a^3*b^2*f)*x^10 + 350*(9
9*a*b^4*c - 63*a^2*b^3*d + 35*a^3*b^2*e - 15*a^4*b*f)*x^8 + 112*(99*a^2*b^3*c -
63*a^3*b^2*d + 35*a^4*b*e - 15*a^5*f)*x^6 - 240*a^5*c - 16*(99*a^3*b^2*c - 63*a^
4*b*d + 35*a^5*e)*x^4 + 48*(11*a^4*b*c - 7*a^5*d)*x^2 - 105*((99*b^5*c - 63*a*b^
4*d + 35*a^2*b^3*e - 15*a^3*b^2*f)*x^11 + 2*(99*a*b^4*c - 63*a^2*b^3*d + 35*a^3*
b^2*e - 15*a^4*b*f)*x^9 + (99*a^2*b^3*c - 63*a^3*b^2*d + 35*a^4*b*e - 15*a^5*f)*
x^7)*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^6*b^2*x^11 +
2*a^7*b*x^9 + a^8*x^7), 1/840*(105*(99*b^5*c - 63*a*b^4*d + 35*a^2*b^3*e - 15*a
^3*b^2*f)*x^10 + 175*(99*a*b^4*c - 63*a^2*b^3*d + 35*a^3*b^2*e - 15*a^4*b*f)*x^8
+ 56*(99*a^2*b^3*c - 63*a^3*b^2*d + 35*a^4*b*e - 15*a^5*f)*x^6 - 120*a^5*c - 8*
(99*a^3*b^2*c - 63*a^4*b*d + 35*a^5*e)*x^4 + 24*(11*a^4*b*c - 7*a^5*d)*x^2 + 105
*((99*b^5*c - 63*a*b^4*d + 35*a^2*b^3*e - 15*a^3*b^2*f)*x^11 + 2*(99*a*b^4*c - 6
3*a^2*b^3*d + 35*a^3*b^2*e - 15*a^4*b*f)*x^9 + (99*a^2*b^3*c - 63*a^3*b^2*d + 35
*a^4*b*e - 15*a^5*f)*x^7)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(a^6*b^2*x^11 + 2
*a^7*b*x^9 + a^8*x^7)]

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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**8/(b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218883, size = 338, normalized size = 1.44 \[ \frac{{\left (99 \, b^{4} c - 63 \, a b^{3} d - 15 \, a^{3} b f + 35 \, a^{2} b^{2} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{6}} + \frac{19 \, b^{5} c x^{3} - 15 \, a b^{4} d x^{3} - 7 \, a^{3} b^{2} f x^{3} + 11 \, a^{2} b^{3} x^{3} e + 21 \, a b^{4} c x - 17 \, a^{2} b^{3} d x - 9 \, a^{4} b f x + 13 \, a^{3} b^{2} x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{6}} + \frac{1050 \, b^{3} c x^{6} - 630 \, a b^{2} d x^{6} - 105 \, a^{3} f x^{6} + 315 \, a^{2} b x^{6} e - 210 \, a b^{2} c x^{4} + 105 \, a^{2} b d x^{4} - 35 \, a^{3} x^{4} e + 63 \, a^{2} b c x^{2} - 21 \, a^{3} d x^{2} - 15 \, a^{3} c}{105 \, a^{6} x^{7}} \]

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)^3*x^8),x, algorithm="giac")

[Out]

1/8*(99*b^4*c - 63*a*b^3*d - 15*a^3*b*f + 35*a^2*b^2*e)*arctan(b*x/sqrt(a*b))/(s
qrt(a*b)*a^6) + 1/8*(19*b^5*c*x^3 - 15*a*b^4*d*x^3 - 7*a^3*b^2*f*x^3 + 11*a^2*b^
3*x^3*e + 21*a*b^4*c*x - 17*a^2*b^3*d*x - 9*a^4*b*f*x + 13*a^3*b^2*x*e)/((b*x^2
+ a)^2*a^6) + 1/105*(1050*b^3*c*x^6 - 630*a*b^2*d*x^6 - 105*a^3*f*x^6 + 315*a^2*
b*x^6*e - 210*a*b^2*c*x^4 + 105*a^2*b*d*x^4 - 35*a^3*x^4*e + 63*a^2*b*c*x^2 - 21
*a^3*d*x^2 - 15*a^3*c)/(a^6*x^7)