3.142 \(\int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=277 \[ \frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{9 a^3 x^9}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-35 a^3 f+63 a^2 b e-99 a b^2 d+143 b^3 c\right )}{8 a^{15/2}}-\frac{b^2 x \left (-11 a^3 f+15 a^2 b e-19 a b^2 d+23 b^3 c\right )}{8 a^7 \left (a+b x^2\right )}-\frac{b \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{3 a^6 x^3}-\frac{b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2} \]

[Out]

-c/(9*a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*
x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(3*a^6*x^3) - (b*(15*b^3*c - 1
0*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^
3*f)*x)/(4*a^6*(a + b*x^2)^2) - (b^2*(23*b^3*c - 19*a*b^2*d + 15*a^2*b*e - 11*a^
3*f)*x)/(8*a^7*(a + b*x^2)) - (b^(3/2)*(143*b^3*c - 99*a*b^2*d + 63*a^2*b*e - 35
*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(15/2))

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Rubi [A]  time = 1.16117, antiderivative size = 277, 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 \[ \frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{9 a^3 x^9}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-35 a^3 f+63 a^2 b e-99 a b^2 d+143 b^3 c\right )}{8 a^{15/2}}-\frac{b^2 x \left (-11 a^3 f+15 a^2 b e-19 a b^2 d+23 b^3 c\right )}{8 a^7 \left (a+b x^2\right )}-\frac{b \left (-3 a^3 f+6 a^2 b e-10 a b^2 d+15 b^3 c\right )}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{3 a^6 x^3}-\frac{b^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^3),x]

[Out]

-c/(9*a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*
x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(3*a^6*x^3) - (b*(15*b^3*c - 1
0*a*b^2*d + 6*a^2*b*e - 3*a^3*f))/(a^7*x) - (b^2*(b^3*c - a*b^2*d + a^2*b*e - a^
3*f)*x)/(4*a^6*(a + b*x^2)^2) - (b^2*(23*b^3*c - 19*a*b^2*d + 15*a^2*b*e - 11*a^
3*f)*x)/(8*a^7*(a + b*x^2)) - (b^(3/2)*(143*b^3*c - 99*a*b^2*d + 63*a^2*b*e - 35
*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(15/2))

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

[Out]

Timed out

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Mathematica [A]  time = 0.284251, size = 276, normalized size = 1. \[ \frac{3 b c-a d}{7 a^4 x^7}-\frac{c}{9 a^3 x^9}-\frac{a^2 e-3 a b d+6 b^2 c}{5 a^5 x^5}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^3 f-63 a^2 b e+99 a b^2 d-143 b^3 c\right )}{8 a^{15/2}}+\frac{b^2 x \left (11 a^3 f-15 a^2 b e+19 a b^2 d-23 b^3 c\right )}{8 a^7 \left (a+b x^2\right )}+\frac{b \left (3 a^3 f-6 a^2 b e+10 a b^2 d-15 b^3 c\right )}{a^7 x}+\frac{a^3 (-f)+3 a^2 b e-6 a b^2 d+10 b^3 c}{3 a^6 x^3}+\frac{b^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 a^6 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^10*(a + b*x^2)^3),x]

[Out]

-c/(9*a^3*x^9) + (3*b*c - a*d)/(7*a^4*x^7) - (6*b^2*c - 3*a*b*d + a^2*e)/(5*a^5*
x^5) + (10*b^3*c - 6*a*b^2*d + 3*a^2*b*e - a^3*f)/(3*a^6*x^3) + (b*(-15*b^3*c +
10*a*b^2*d - 6*a^2*b*e + 3*a^3*f))/(a^7*x) + (b^2*(-(b^3*c) + a*b^2*d - a^2*b*e
+ a^3*f)*x)/(4*a^6*(a + b*x^2)^2) + (b^2*(-23*b^3*c + 19*a*b^2*d - 15*a^2*b*e +
11*a^3*f)*x)/(8*a^7*(a + b*x^2)) + (b^(3/2)*(-143*b^3*c + 99*a*b^2*d - 63*a^2*b*
e + 35*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*a^(15/2))

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Maple [A]  time = 0.028, size = 401, normalized size = 1.5 \[{\frac{21\,d{b}^{4}x}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{25\,{b}^{5}cx}{8\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{d}{7\,{a}^{3}{x}^{7}}}-{\frac{e}{5\,{a}^{3}{x}^{5}}}-{\frac{f}{3\,{a}^{3}{x}^{3}}}-{\frac{c}{9\,{a}^{3}{x}^{9}}}+{\frac{11\,{b}^{3}{x}^{3}f}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,{b}^{4}{x}^{3}e}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{17\,{b}^{3}ex}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{fb}{{a}^{4}x}}-6\,{\frac{e{b}^{2}}{{a}^{5}x}}+10\,{\frac{d{b}^{3}}{{a}^{6}x}}-15\,{\frac{{b}^{4}c}{{a}^{7}x}}+{\frac{3\,bc}{7\,{a}^{4}{x}^{7}}}+{\frac{3\,bd}{5\,{a}^{4}{x}^{5}}}-{\frac{6\,{b}^{2}c}{5\,{a}^{5}{x}^{5}}}+{\frac{be}{{a}^{4}{x}^{3}}}-2\,{\frac{d{b}^{2}}{{a}^{5}{x}^{3}}}+{\frac{10\,{b}^{3}c}{3\,{a}^{6}{x}^{3}}}+{\frac{19\,{b}^{5}{x}^{3}d}{8\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{23\,{b}^{6}{x}^{3}c}{8\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,f{b}^{2}x}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,f{b}^{2}}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,{b}^{3}e}{8\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{99\,d{b}^{4}}{8\,{a}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{143\,{b}^{5}c}{8\,{a}^{7}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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)^3,x)

[Out]

21/8*b^4/a^5/(b*x^2+a)^2*d*x-25/8*b^5/a^6/(b*x^2+a)^2*c*x-1/7/a^3/x^7*d-1/5/a^3/
x^5*e-1/3/a^3/x^3*f-1/9*c/a^3/x^9+11/8*b^3/a^4/(b*x^2+a)^2*x^3*f-15/8*b^4/a^5/(b
*x^2+a)^2*x^3*e-17/8*b^3/a^4/(b*x^2+a)^2*e*x+3*b/a^4/x*f-6*b^2/a^5/x*e+10*b^3/a^
6/x*d-15*b^4/a^7/x*c+3/7/a^4/x^7*b*c+3/5/a^4/x^5*b*d-6/5/a^5/x^5*b^2*c+1/a^4/x^3
*b*e-2/a^5/x^3*b^2*d+10/3/a^6/x^3*b^3*c+19/8*b^5/a^6/(b*x^2+a)^2*x^3*d-23/8*b^6/
a^7/(b*x^2+a)^2*x^3*c+13/8*b^2/a^3/(b*x^2+a)^2*f*x+35/8*b^2/a^4/(a*b)^(1/2)*arct
an(x*b/(a*b)^(1/2))*f-63/8*b^3/a^5/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*e+99/8*b^
4/a^6/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d-143/8*b^5/a^7/(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^10),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23619, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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^10),x, algorithm="fricas")

[Out]

[-1/5040*(630*(143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^12 + 1050
*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^10 + 336*(143*a^2*
b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)*x^8 + 560*a^6*c - 48*(143*a^3*
b^3*c - 99*a^4*b^2*d + 63*a^5*b*e - 35*a^6*f)*x^6 + 16*(143*a^4*b^2*c - 99*a^5*b
*d + 63*a^6*e)*x^4 - 80*(13*a^5*b*c - 9*a^6*d)*x^2 + 315*((143*b^6*c - 99*a*b^5*
d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^13 + 2*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b
^3*e - 35*a^4*b^2*f)*x^11 + (143*a^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^
5*b*f)*x^9)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)))/(a^7*b^2
*x^13 + 2*a^8*b*x^11 + a^9*x^9), -1/2520*(315*(143*b^6*c - 99*a*b^5*d + 63*a^2*b
^4*e - 35*a^3*b^3*f)*x^12 + 525*(143*a*b^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*
a^4*b^2*f)*x^10 + 168*(143*a^2*b^4*c - 99*a^3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)
*x^8 + 280*a^6*c - 24*(143*a^3*b^3*c - 99*a^4*b^2*d + 63*a^5*b*e - 35*a^6*f)*x^6
+ 8*(143*a^4*b^2*c - 99*a^5*b*d + 63*a^6*e)*x^4 - 40*(13*a^5*b*c - 9*a^6*d)*x^2
+ 315*((143*b^6*c - 99*a*b^5*d + 63*a^2*b^4*e - 35*a^3*b^3*f)*x^13 + 2*(143*a*b
^5*c - 99*a^2*b^4*d + 63*a^3*b^3*e - 35*a^4*b^2*f)*x^11 + (143*a^2*b^4*c - 99*a^
3*b^3*d + 63*a^4*b^2*e - 35*a^5*b*f)*x^9)*sqrt(b/a)*arctan(b*x/(a*sqrt(b/a))))/(
a^7*b^2*x^13 + 2*a^8*b*x^11 + a^9*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)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.216141, size = 406, normalized size = 1.47 \[ -\frac{{\left (143 \, b^{5} c - 99 \, a b^{4} d - 35 \, a^{3} b^{2} f + 63 \, a^{2} b^{3} e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{7}} - \frac{23 \, b^{6} c x^{3} - 19 \, a b^{5} d x^{3} - 11 \, a^{3} b^{3} f x^{3} + 15 \, a^{2} b^{4} x^{3} e + 25 \, a b^{5} c x - 21 \, a^{2} b^{4} d x - 13 \, a^{4} b^{2} f x + 17 \, a^{3} b^{3} x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{7}} - \frac{4725 \, b^{4} c x^{8} - 3150 \, a b^{3} d x^{8} - 945 \, a^{3} b f x^{8} + 1890 \, a^{2} b^{2} x^{8} e - 1050 \, a b^{3} c x^{6} + 630 \, a^{2} b^{2} d x^{6} + 105 \, a^{4} f x^{6} - 315 \, a^{3} b x^{6} e + 378 \, a^{2} b^{2} c x^{4} - 189 \, a^{3} b d x^{4} + 63 \, a^{4} x^{4} e - 135 \, a^{3} b c x^{2} + 45 \, a^{4} d x^{2} + 35 \, a^{4} c}{315 \, a^{7} x^{9}} \]

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)^3*x^10),x, algorithm="giac")

[Out]

-1/8*(143*b^5*c - 99*a*b^4*d - 35*a^3*b^2*f + 63*a^2*b^3*e)*arctan(b*x/sqrt(a*b)
)/(sqrt(a*b)*a^7) - 1/8*(23*b^6*c*x^3 - 19*a*b^5*d*x^3 - 11*a^3*b^3*f*x^3 + 15*a
^2*b^4*x^3*e + 25*a*b^5*c*x - 21*a^2*b^4*d*x - 13*a^4*b^2*f*x + 17*a^3*b^3*x*e)/
((b*x^2 + a)^2*a^7) - 1/315*(4725*b^4*c*x^8 - 3150*a*b^3*d*x^8 - 945*a^3*b*f*x^8
+ 1890*a^2*b^2*x^8*e - 1050*a*b^3*c*x^6 + 630*a^2*b^2*d*x^6 + 105*a^4*f*x^6 - 3
15*a^3*b*x^6*e + 378*a^2*b^2*c*x^4 - 189*a^3*b*d*x^4 + 63*a^4*x^4*e - 135*a^3*b*
c*x^2 + 45*a^4*d*x^2 + 35*a^4*c)/(a^7*x^9)