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

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

[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[(S
qrt[b]*x)/Sqrt[a]])/(2*a^(13/2))

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Rubi [A]  time = 0.376065, 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, Rules used = {1805, 1802, 205} \[ -\frac{b^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 a^6 \left (a+b x^2\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{3 a^5 x^3}-\frac{b \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{a^6 x}-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (7 a^2 b e-5 a^3 f-9 a b^2 d+11 b^3 c\right )}{2 a^{13/2}}-\frac{a^2 e-2 a b d+3 b^2 c}{5 a^4 x^5}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{9 a^2 x^9} \]

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[(S
qrt[b]*x)/Sqrt[a]])/(2*a^(13/2))

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{x^{10} \left (a+b x^2\right )^2} \, dx &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}-\frac{\int \frac{-2 c+2 \left (\frac{b c}{a}-d\right ) x^2-\frac{2 \left (b^2 c-a b d+a^2 e\right ) x^4}{a^2}+\frac{2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}-\frac{2 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}+\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{10}}{a^5}}{x^{10} \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 c}{a x^{10}}-\frac{2 (-2 b c+a d)}{a^2 x^8}-\frac{2 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^6}-\frac{2 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^4}+\frac{2 b \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^5 x^2}-\frac{b^2 \left (-11 b^3 c+9 a b^2 d-7 a^2 b e+5 a^3 f\right )}{a^5 \left (a+b x^2\right )}\right ) \, dx}{2 a}\\ &=-\frac{c}{9 a^2 x^9}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}-\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}-\frac{\left (b^2 \left (11 b^3 c-9 a b^2 d+7 a^2 b e-5 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^6}\\ &=-\frac{c}{9 a^2 x^9}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{3 b^2 c-2 a b d+a^2 e}{5 a^4 x^5}+\frac{4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{3 a^5 x^3}-\frac{b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac{b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{2 a^6 \left (a+b x^2\right )}-\frac{b^{3/2} \left (11 b^3 c-9 a b^2 d+7 a^2 b e-5 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.113852, size = 230, normalized size = 1. \[ \frac{b^2 x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{2 a^6 \left (a+b x^2\right )}+\frac{2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{3 a^5 x^3}+\frac{b \left (-3 a^2 b e+2 a^3 f+4 a b^2 d-5 b^3 c\right )}{a^6 x}+\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-7 a^2 b e+5 a^3 f+9 a b^2 d-11 b^3 c\right )}{2 a^{13/2}}+\frac{a^2 (-e)+2 a b d-3 b^2 c}{5 a^4 x^5}+\frac{2 b c-a d}{7 a^3 x^7}-\frac{c}{9 a^2 x^9} \]

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)*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(13/2))

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Maple [A]  time = 0.016, size = 318, normalized size = 1.4 \begin{align*} -{\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{{b}^{2}d}{{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\,{b}^{2}f}{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\,{b}^{4}d}{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}}}} \end{align*}

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(b*x/(a*b)^(1/2))*f-7/2*b^3/a^4/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*e+9/2*b^4/a^5/(a*b)^(1/2)*a
rctan(b*x/(a*b)^(1/2))*d-11/2*b^5/a^6/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*c

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.30327, size = 1284, normalized size = 5.58 \begin{align*} \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 (x \sqrt{\frac{b}{a}}\right )}{630 \,{\left (a^{6} b x^{11} + a^{7} x^{9}\right )}}\right ] \end{align*}

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, 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(x*sqrt(b/a)))/(a^6*b*x^11 + a^7*x^9)]

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Sympy [A]  time = 144.033, size = 449, normalized size = 1.95 \begin{align*} - \frac{\sqrt{- \frac{b^{3}}{a^{13}}} \left (5 a^{3} f - 7 a^{2} b e + 9 a b^{2} d - 11 b^{3} c\right ) \log{\left (- \frac{a^{7} \sqrt{- \frac{b^{3}}{a^{13}}} \left (5 a^{3} f - 7 a^{2} b e + 9 a b^{2} d - 11 b^{3} c\right )}{5 a^{3} b^{2} f - 7 a^{2} b^{3} e + 9 a b^{4} d - 11 b^{5} c} + x \right )}}{4} + \frac{\sqrt{- \frac{b^{3}}{a^{13}}} \left (5 a^{3} f - 7 a^{2} b e + 9 a b^{2} d - 11 b^{3} c\right ) \log{\left (\frac{a^{7} \sqrt{- \frac{b^{3}}{a^{13}}} \left (5 a^{3} f - 7 a^{2} b e + 9 a b^{2} d - 11 b^{3} c\right )}{5 a^{3} b^{2} f - 7 a^{2} b^{3} e + 9 a b^{4} d - 11 b^{5} c} + x \right )}}{4} + \frac{- 70 a^{5} c + x^{10} \left (1575 a^{3} b^{2} f - 2205 a^{2} b^{3} e + 2835 a b^{4} d - 3465 b^{5} c\right ) + x^{8} \left (1050 a^{4} b f - 1470 a^{3} b^{2} e + 1890 a^{2} b^{3} d - 2310 a b^{4} c\right ) + x^{6} \left (- 210 a^{5} f + 294 a^{4} b e - 378 a^{3} b^{2} d + 462 a^{2} b^{3} c\right ) + x^{4} \left (- 126 a^{5} e + 162 a^{4} b d - 198 a^{3} b^{2} c\right ) + x^{2} \left (- 90 a^{5} d + 110 a^{4} b c\right )}{630 a^{7} x^{9} + 630 a^{6} b x^{11}} \end{align*}

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]

-sqrt(-b**3/a**13)*(5*a**3*f - 7*a**2*b*e + 9*a*b**2*d - 11*b**3*c)*log(-a**7*sqrt(-b**3/a**13)*(5*a**3*f - 7*
a**2*b*e + 9*a*b**2*d - 11*b**3*c)/(5*a**3*b**2*f - 7*a**2*b**3*e + 9*a*b**4*d - 11*b**5*c) + x)/4 + sqrt(-b**
3/a**13)*(5*a**3*f - 7*a**2*b*e + 9*a*b**2*d - 11*b**3*c)*log(a**7*sqrt(-b**3/a**13)*(5*a**3*f - 7*a**2*b*e +
9*a*b**2*d - 11*b**3*c)/(5*a**3*b**2*f - 7*a**2*b**3*e + 9*a*b**4*d - 11*b**5*c) + x)/4 + (-70*a**5*c + x**10*
(1575*a**3*b**2*f - 2205*a**2*b**3*e + 2835*a*b**4*d - 3465*b**5*c) + x**8*(1050*a**4*b*f - 1470*a**3*b**2*e +
1890*a**2*b**3*d - 2310*a*b**4*c) + x**6*(-210*a**5*f + 294*a**4*b*e - 378*a**3*b**2*d + 462*a**2*b**3*c) + x
**4*(-126*a**5*e + 162*a**4*b*d - 198*a**3*b**2*c) + x**2*(-90*a**5*d + 110*a**4*b*c))/(630*a**7*x**9 + 630*a*
*6*b*x**11)

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Giac [A]  time = 1.24262, size = 340, normalized size = 1.48 \begin{align*} -\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}} \end{align*}

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, 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))/(sqrt(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)