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

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

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.486456, 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, Rules used = {1805, 1802, 205} \[ \frac{b x \left (11 a^2 b e-7 a^3 f-15 a b^2 d+19 b^3 c\right )}{8 a^6 \left (a+b x^2\right )}+\frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 a^5 \left (a+b x^2\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^2 b e-15 a^3 f-63 a b^2 d+99 b^3 c\right )}{8 a^{13/2}}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{7 a^3 x^7} \]

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))

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^8 \left (a+b x^2\right )^3} \, dx &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}-\frac{\int \frac{-4 c+4 \left (\frac{b c}{a}-d\right ) x^2-\frac{4 \left (b^2 c-a b d+a^2 e\right ) x^4}{a^2}+\frac{4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^6}{a^3}-\frac{3 b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^8}{a^4}}{x^8 \left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac{b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac{\int \frac{8 c-8 \left (\frac{2 b c}{a}-d\right ) x^2+8 \left (\frac{3 b^2 c}{a^2}-\frac{2 b d}{a}+e\right ) x^4-\frac{8 \left (4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f\right ) x^6}{a^3}+\frac{b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x^8}{a^4}}{x^8 \left (a+b x^2\right )} \, dx}{8 a^2}\\ &=\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac{b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac{\int \left (\frac{8 c}{a x^8}+\frac{8 (-3 b c+a d)}{a^2 x^6}+\frac{8 \left (6 b^2 c-3 a b d+a^2 e\right )}{a^3 x^4}+\frac{8 \left (-10 b^3 c+6 a b^2 d-3 a^2 b e+a^3 f\right )}{a^4 x^2}-\frac{b \left (-99 b^3 c+63 a b^2 d-35 a^2 b e+15 a^3 f\right )}{a^4 \left (a+b x^2\right )}\right ) \, dx}{8 a^2}\\ &=-\frac{c}{7 a^3 x^7}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{6 b^2 c-3 a b d+a^2 e}{3 a^5 x^3}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac{b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac{\left (b \left (99 b^3 c-63 a b^2 d+35 a^2 b e-15 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^6}\\ &=-\frac{c}{7 a^3 x^7}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{6 b^2 c-3 a b d+a^2 e}{3 a^5 x^3}+\frac{10 b^3 c-6 a b^2 d+3 a^2 b e-a^3 f}{a^6 x}+\frac{b \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{4 a^5 \left (a+b x^2\right )^2}+\frac{b \left (19 b^3 c-15 a b^2 d+11 a^2 b e-7 a^3 f\right ) x}{8 a^6 \left (a+b x^2\right )}+\frac{\sqrt{b} \left (99 b^3 c-63 a b^2 d+35 a^2 b e-15 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.13587, size = 234, normalized size = 1. \[ \frac{b x \left (11 a^2 b e-7 a^3 f-15 a b^2 d+19 b^3 c\right )}{8 a^6 \left (a+b x^2\right )}+\frac{b x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 a^5 \left (a+b x^2\right )^2}+\frac{3 a^2 b e+a^3 (-f)-6 a b^2 d+10 b^3 c}{a^6 x}+\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^2 b e-15 a^3 f-63 a b^2 d+99 b^3 c\right )}{8 a^{13/2}}-\frac{a^2 e-3 a b d+6 b^2 c}{3 a^5 x^3}+\frac{3 b c-a d}{5 a^4 x^5}-\frac{c}{7 a^3 x^7} \]

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))

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 351, normalized size = 1.5 \begin{align*} -{\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{{b}^{2}d}{{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\,bfx}{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\,{b}^{3}dx}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{21\,c{b}^{4}x}{8\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{15\,bf}{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\,{b}^{3}d}{8\,{a}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{99\,c{b}^{4}}{8\,{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^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/a^3*b^2/(b*x^2+a)^2*x^3*f+11/8/a^4*b^3/(b*x^2+a)^2*x^3*e-15/8/a^5*b^4/(b*x^2
+a)^2*x^3*d+19/8/a^6*b^5/(b*x^2+a)^2*x^3*c-9/8/a^2*b/(b*x^2+a)^2*f*x+13/8/a^3*b^2/(b*x^2+a)^2*e*x-17/8/a^4*b^3
/(b*x^2+a)^2*d*x+21/8/a^5*b^4/(b*x^2+a)^2*c*x-15/8/a^3*b/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*f+35/8/a^4*b^2/(a
*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*e-63/8/a^5*b^3/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))*d+99/8/a^6*b^4/(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^8/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.46752, size = 1539, normalized size = 6.58 \begin{align*} \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 (x \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 ] \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^8/(b*x^2+a)^3,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*(99*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 - 63*a^2*b^3*d + 35*a^3*b^2*e - 1
5*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(x*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. \begin{align*} \text{Timed out} \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**8/(b*x**2+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 1.21301, size = 338, normalized size = 1.44 \begin{align*} \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}} \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^8/(b*x^2+a)^3,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))/(sqrt(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)