3.2.31 \(\int \frac {c+d x^2+e x^4+f x^6}{x^8 (a+b x^2)^2} \, dx\) [131]

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

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1819, 1816, 211} \begin {gather*} \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} \text {ArcTan}\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} \end {gather*}

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]

-1/7*c/(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))

Rule 211

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

Rule 1816

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 1819

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]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 190, normalized size = 1.01 \begin {gather*} -\frac {c}{7 a^2 x^7}+\frac {2 b c-a d}{5 a^3 x^5}+\frac {-3 b^2 c+2 a b d-a^2 e}{3 a^4 x^3}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{a^5 x}-\frac {b \left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{2 a^5 \left (a+b x^2\right )}-\frac {\sqrt {b} \left (-9 b^3 c+7 a b^2 d-5 a^2 b e+3 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{11/2}} \end {gather*}

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]

-1/7*c/(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 = 0.13, size = 174, normalized size = 0.92

method	result	size
default	\(-\frac {b \left (\frac {\left (\frac {1}{2} a^{3} f -\frac {1}{2} a^{2} b e +\frac {1}{2} a \,b^{2} d -\frac {1}{2} b^{3} c \right ) x}{b \,x^{2}+a}+\frac {\left (3 a^{3} f -5 a^{2} b e +7 a \,b^{2} d -9 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{5}}-\frac {c}{7 a^{2} x^{7}}-\frac {a d -2 b c}{5 a^{3} x^{5}}-\frac {a^{2} e -2 a b d +3 b^{2} c}{3 a^{4} x^{3}}-\frac {a^{3} f -2 a^{2} b e +3 a \,b^{2} d -4 b^{3} c}{a^{5} x}\)	\(174\)
risch	\(\frac {-\frac {b \left (3 a^{3} f -5 a^{2} b e +7 a \,b^{2} d -9 b^{3} c \right ) x^{8}}{2 a^{5}}-\frac {\left (3 a^{3} f -5 a^{2} b e +7 a \,b^{2} d -9 b^{3} c \right ) x^{6}}{3 a^{4}}-\frac {\left (5 a^{2} e -7 a b d +9 b^{2} c \right ) x^{4}}{15 a^{3}}-\frac {\left (7 a d -9 b c \right ) x^{2}}{35 a^{2}}-\frac {c}{7 a}}{x^{7} \left (b \,x^{2}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{11} \textit {\_Z}^{2}+9 a^{6} b \,f^{2}-30 a^{5} b^{2} e f +42 a^{4} b^{3} d f +25 a^{4} b^{3} e^{2}-54 a^{3} b^{4} c f -70 a^{3} b^{4} d e +90 a^{2} b^{5} c e +49 a^{2} b^{5} d^{2}-126 a \,b^{6} c d +81 b^{7} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{11}+18 a^{6} b \,f^{2}-60 a^{5} b^{2} e f +84 a^{4} b^{3} d f +50 a^{4} b^{3} e^{2}-108 a^{3} b^{4} c f -140 a^{3} b^{4} d e +180 a^{2} b^{5} c e +98 a^{2} b^{5} d^{2}-252 a \,b^{6} c d +162 b^{7} c^{2}\right ) x +\left (3 a^{9} f -5 a^{8} b e +7 a^{7} b^{2} d -9 a^{6} b^{3} c \right ) \textit {\_R} \right )\right )}{4}\)	\(393\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.53, size = 198, normalized size = 1.05 \begin {gather*} \frac {105 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f + 5 \, a^{2} b^{2} e\right )} x^{8} + 70 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f + 5 \, a^{3} b e\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}}{210 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}} + \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}} \end {gather*}

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, algorithm="maxima")

[Out]

1/210*(105*(9*b^4*c - 7*a*b^3*d - 3*a^3*b*f + 5*a^2*b^2*e)*x^8 + 70*(9*a*b^3*c - 7*a^2*b^2*d - 3*a^4*f + 5*a^3
*b*e)*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)/(a^5*b*x^9 +
a^6*x^7) + 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)

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Fricas [A]
time = 2.62, size = 524, normalized size = 2.77 \begin {gather*} \left [\frac {210 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f\right )} x^{8} + 140 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f\right )} x^{6} - 60 \, a^{4} c - 28 \, {\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d\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 - 3 \, a^{3} b f\right )} x^{9} + {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f\right )} x^{7} + 5 \, {\left (a^{2} b^{2} x^{9} + a^{3} b x^{7}\right )} e\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 70 \, {\left (15 \, a^{2} b^{2} x^{8} + 10 \, a^{3} b x^{6} - 2 \, a^{4} x^{4}\right )} e}{420 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}, \frac {105 \, {\left (9 \, b^{4} c - 7 \, a b^{3} d - 3 \, a^{3} b f\right )} x^{8} + 70 \, {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f\right )} x^{6} - 30 \, a^{4} c - 14 \, {\left (9 \, a^{2} b^{2} c - 7 \, a^{3} b d\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 - 3 \, a^{3} b f\right )} x^{9} + {\left (9 \, a b^{3} c - 7 \, a^{2} b^{2} d - 3 \, a^{4} f\right )} x^{7} + 5 \, {\left (a^{2} b^{2} x^{9} + a^{3} b x^{7}\right )} e\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 35 \, {\left (15 \, a^{2} b^{2} x^{8} + 10 \, a^{3} b x^{6} - 2 \, a^{4} x^{4}\right )} e}{210 \, {\left (a^{5} b x^{9} + a^{6} x^{7}\right )}}\right ] \end {gather*}

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, algorithm="fricas")

[Out]

[1/420*(210*(9*b^4*c - 7*a*b^3*d - 3*a^3*b*f)*x^8 + 140*(9*a*b^3*c - 7*a^2*b^2*d - 3*a^4*f)*x^6 - 60*a^4*c - 2
8*(9*a^2*b^2*c - 7*a^3*b*d)*x^4 + 12*(9*a^3*b*c - 7*a^4*d)*x^2 + 105*((9*b^4*c - 7*a*b^3*d - 3*a^3*b*f)*x^9 +
(9*a*b^3*c - 7*a^2*b^2*d - 3*a^4*f)*x^7 + 5*(a^2*b^2*x^9 + a^3*b*x^7)*e)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b
/a) - a)/(b*x^2 + a)) + 70*(15*a^2*b^2*x^8 + 10*a^3*b*x^6 - 2*a^4*x^4)*e)/(a^5*b*x^9 + a^6*x^7), 1/210*(105*(9
*b^4*c - 7*a*b^3*d - 3*a^3*b*f)*x^8 + 70*(9*a*b^3*c - 7*a^2*b^2*d - 3*a^4*f)*x^6 - 30*a^4*c - 14*(9*a^2*b^2*c
- 7*a^3*b*d)*x^4 + 6*(9*a^3*b*c - 7*a^4*d)*x^2 + 105*((9*b^4*c - 7*a*b^3*d - 3*a^3*b*f)*x^9 + (9*a*b^3*c - 7*a
^2*b^2*d - 3*a^4*f)*x^7 + 5*(a^2*b^2*x^9 + a^3*b*x^7)*e)*sqrt(b/a)*arctan(x*sqrt(b/a)) + 35*(15*a^2*b^2*x^8 +
10*a^3*b*x^6 - 2*a^4*x^4)*e)/(a^5*b*x^9 + a^6*x^7)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

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Giac [A]
time = 1.41, size = 201, normalized size = 1.06 \begin {gather*} \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}} \end {gather*}

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, 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 - 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)
/(a^5*x^7)

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Mupad [B]
time = 0.99, size = 181, normalized size = 0.96 \begin {gather*} \frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-3\,f\,a^3+5\,e\,a^2\,b-7\,d\,a\,b^2+9\,c\,b^3\right )}{2\,a^{11/2}}-\frac {\frac {c}{7\,a}-\frac {x^6\,\left (-3\,f\,a^3+5\,e\,a^2\,b-7\,d\,a\,b^2+9\,c\,b^3\right )}{3\,a^4}+\frac {x^2\,\left (7\,a\,d-9\,b\,c\right )}{35\,a^2}+\frac {x^4\,\left (5\,e\,a^2-7\,d\,a\,b+9\,c\,b^2\right )}{15\,a^3}-\frac {b\,x^8\,\left (-3\,f\,a^3+5\,e\,a^2\,b-7\,d\,a\,b^2+9\,c\,b^3\right )}{2\,a^5}}{b\,x^9+a\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^(1/2)*atan((b^(1/2)*x)/a^(1/2))*(9*b^3*c - 3*a^3*f - 7*a*b^2*d + 5*a^2*b*e))/(2*a^(11/2)) - (c/(7*a) - (x^6
*(9*b^3*c - 3*a^3*f - 7*a*b^2*d + 5*a^2*b*e))/(3*a^4) + (x^2*(7*a*d - 9*b*c))/(35*a^2) + (x^4*(9*b^2*c + 5*a^2
*e - 7*a*b*d))/(15*a^3) - (b*x^8*(9*b^3*c - 3*a^3*f - 7*a*b^2*d + 5*a^2*b*e))/(2*a^5))/(a*x^7 + b*x^9)

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