3.319 \(\int x^m (a+b x^2)^3 (A+B x^2) \, dx\)
Optimal. Leaf size=96 \[ \frac{a^2 x^{m+3} (a B+3 A b)}{m+3}+\frac{a^3 A x^{m+1}}{m+1}+\frac{b^2 x^{m+7} (3 a B+A b)}{m+7}+\frac{3 a b x^{m+5} (a B+A b)}{m+5}+\frac{b^3 B x^{m+9}}{m+9} \]
[Out]
(a^3*A*x^(1 + m))/(1 + m) + (a^2*(3*A*b + a*B)*x^(3 + m))/(3 + m) + (3*a*b*(A*b + a*B)*x^(5 + m))/(5 + m) + (b
^2*(A*b + 3*a*B)*x^(7 + m))/(7 + m) + (b^3*B*x^(9 + m))/(9 + m)
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Rubi [A] time = 0.0651068, antiderivative size = 96, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used =
{448} \[ \frac{a^2 x^{m+3} (a B+3 A b)}{m+3}+\frac{a^3 A x^{m+1}}{m+1}+\frac{b^2 x^{m+7} (3 a B+A b)}{m+7}+\frac{3 a b x^{m+5} (a B+A b)}{m+5}+\frac{b^3 B x^{m+9}}{m+9} \]
Antiderivative was successfully verified.
[In]
Int[x^m*(a + b*x^2)^3*(A + B*x^2),x]
[Out]
(a^3*A*x^(1 + m))/(1 + m) + (a^2*(3*A*b + a*B)*x^(3 + m))/(3 + m) + (3*a*b*(A*b + a*B)*x^(5 + m))/(5 + m) + (b
^2*(A*b + 3*a*B)*x^(7 + m))/(7 + m) + (b^3*B*x^(9 + m))/(9 + m)
Rule 448
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]
Rubi steps
\begin{align*} \int x^m \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx &=\int \left (a^3 A x^m+a^2 (3 A b+a B) x^{2+m}+3 a b (A b+a B) x^{4+m}+b^2 (A b+3 a B) x^{6+m}+b^3 B x^{8+m}\right ) \, dx\\ &=\frac{a^3 A x^{1+m}}{1+m}+\frac{a^2 (3 A b+a B) x^{3+m}}{3+m}+\frac{3 a b (A b+a B) x^{5+m}}{5+m}+\frac{b^2 (A b+3 a B) x^{7+m}}{7+m}+\frac{b^3 B x^{9+m}}{9+m}\\ \end{align*}
Mathematica [A] time = 0.102434, size = 89, normalized size = 0.93 \[ x^{m+1} \left (\frac{a^2 x^2 (a B+3 A b)}{m+3}+\frac{a^3 A}{m+1}+\frac{b^2 x^6 (3 a B+A b)}{m+7}+\frac{3 a b x^4 (a B+A b)}{m+5}+\frac{b^3 B x^8}{m+9}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^m*(a + b*x^2)^3*(A + B*x^2),x]
[Out]
x^(1 + m)*((a^3*A)/(1 + m) + (a^2*(3*A*b + a*B)*x^2)/(3 + m) + (3*a*b*(A*b + a*B)*x^4)/(5 + m) + (b^2*(A*b + 3
*a*B)*x^6)/(7 + m) + (b^3*B*x^8)/(9 + m))
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Maple [B] time = 0.007, size = 474, normalized size = 4.9 \begin{align*}{\frac{{x}^{1+m} \left ( B{b}^{3}{m}^{4}{x}^{8}+16\,B{b}^{3}{m}^{3}{x}^{8}+A{b}^{3}{m}^{4}{x}^{6}+3\,Ba{b}^{2}{m}^{4}{x}^{6}+86\,B{b}^{3}{m}^{2}{x}^{8}+18\,A{b}^{3}{m}^{3}{x}^{6}+54\,Ba{b}^{2}{m}^{3}{x}^{6}+176\,B{b}^{3}m{x}^{8}+3\,Aa{b}^{2}{m}^{4}{x}^{4}+104\,A{b}^{3}{m}^{2}{x}^{6}+3\,B{a}^{2}b{m}^{4}{x}^{4}+312\,Ba{b}^{2}{m}^{2}{x}^{6}+105\,B{b}^{3}{x}^{8}+60\,Aa{b}^{2}{m}^{3}{x}^{4}+222\,A{b}^{3}m{x}^{6}+60\,B{a}^{2}b{m}^{3}{x}^{4}+666\,Ba{b}^{2}m{x}^{6}+3\,A{a}^{2}b{m}^{4}{x}^{2}+390\,Aa{b}^{2}{m}^{2}{x}^{4}+135\,A{b}^{3}{x}^{6}+B{a}^{3}{m}^{4}{x}^{2}+390\,B{a}^{2}b{m}^{2}{x}^{4}+405\,Ba{b}^{2}{x}^{6}+66\,A{a}^{2}b{m}^{3}{x}^{2}+900\,Aa{b}^{2}m{x}^{4}+22\,B{a}^{3}{m}^{3}{x}^{2}+900\,B{a}^{2}bm{x}^{4}+A{a}^{3}{m}^{4}+492\,A{a}^{2}b{m}^{2}{x}^{2}+567\,Aa{b}^{2}{x}^{4}+164\,B{a}^{3}{m}^{2}{x}^{2}+567\,B{a}^{2}b{x}^{4}+24\,A{a}^{3}{m}^{3}+1374\,A{a}^{2}bm{x}^{2}+458\,B{a}^{3}m{x}^{2}+206\,A{a}^{3}{m}^{2}+945\,A{a}^{2}b{x}^{2}+315\,B{a}^{3}{x}^{2}+744\,A{a}^{3}m+945\,A{a}^{3} \right ) }{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^m*(b*x^2+a)^3*(B*x^2+A),x)
[Out]
x^(1+m)*(B*b^3*m^4*x^8+16*B*b^3*m^3*x^8+A*b^3*m^4*x^6+3*B*a*b^2*m^4*x^6+86*B*b^3*m^2*x^8+18*A*b^3*m^3*x^6+54*B
*a*b^2*m^3*x^6+176*B*b^3*m*x^8+3*A*a*b^2*m^4*x^4+104*A*b^3*m^2*x^6+3*B*a^2*b*m^4*x^4+312*B*a*b^2*m^2*x^6+105*B
*b^3*x^8+60*A*a*b^2*m^3*x^4+222*A*b^3*m*x^6+60*B*a^2*b*m^3*x^4+666*B*a*b^2*m*x^6+3*A*a^2*b*m^4*x^2+390*A*a*b^2
*m^2*x^4+135*A*b^3*x^6+B*a^3*m^4*x^2+390*B*a^2*b*m^2*x^4+405*B*a*b^2*x^6+66*A*a^2*b*m^3*x^2+900*A*a*b^2*m*x^4+
22*B*a^3*m^3*x^2+900*B*a^2*b*m*x^4+A*a^3*m^4+492*A*a^2*b*m^2*x^2+567*A*a*b^2*x^4+164*B*a^3*m^2*x^2+567*B*a^2*b
*x^4+24*A*a^3*m^3+1374*A*a^2*b*m*x^2+458*B*a^3*m*x^2+206*A*a^3*m^2+945*A*a^2*b*x^2+315*B*a^3*x^2+744*A*a^3*m+9
45*A*a^3)/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)
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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(x^m*(b*x^2+a)^3*(B*x^2+A),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 1.07209, size = 865, normalized size = 9.01 \begin{align*} \frac{{\left ({\left (B b^{3} m^{4} + 16 \, B b^{3} m^{3} + 86 \, B b^{3} m^{2} + 176 \, B b^{3} m + 105 \, B b^{3}\right )} x^{9} +{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} m^{4} + 405 \, B a b^{2} + 135 \, A b^{3} + 18 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m^{3} + 104 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m^{2} + 222 \,{\left (3 \, B a b^{2} + A b^{3}\right )} m\right )} x^{7} + 3 \,{\left ({\left (B a^{2} b + A a b^{2}\right )} m^{4} + 189 \, B a^{2} b + 189 \, A a b^{2} + 20 \,{\left (B a^{2} b + A a b^{2}\right )} m^{3} + 130 \,{\left (B a^{2} b + A a b^{2}\right )} m^{2} + 300 \,{\left (B a^{2} b + A a b^{2}\right )} m\right )} x^{5} +{\left ({\left (B a^{3} + 3 \, A a^{2} b\right )} m^{4} + 315 \, B a^{3} + 945 \, A a^{2} b + 22 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m^{3} + 164 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m^{2} + 458 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} m\right )} x^{3} +{\left (A a^{3} m^{4} + 24 \, A a^{3} m^{3} + 206 \, A a^{3} m^{2} + 744 \, A a^{3} m + 945 \, A a^{3}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^2+a)^3*(B*x^2+A),x, algorithm="fricas")
[Out]
((B*b^3*m^4 + 16*B*b^3*m^3 + 86*B*b^3*m^2 + 176*B*b^3*m + 105*B*b^3)*x^9 + ((3*B*a*b^2 + A*b^3)*m^4 + 405*B*a*
b^2 + 135*A*b^3 + 18*(3*B*a*b^2 + A*b^3)*m^3 + 104*(3*B*a*b^2 + A*b^3)*m^2 + 222*(3*B*a*b^2 + A*b^3)*m)*x^7 +
3*((B*a^2*b + A*a*b^2)*m^4 + 189*B*a^2*b + 189*A*a*b^2 + 20*(B*a^2*b + A*a*b^2)*m^3 + 130*(B*a^2*b + A*a*b^2)*
m^2 + 300*(B*a^2*b + A*a*b^2)*m)*x^5 + ((B*a^3 + 3*A*a^2*b)*m^4 + 315*B*a^3 + 945*A*a^2*b + 22*(B*a^3 + 3*A*a^
2*b)*m^3 + 164*(B*a^3 + 3*A*a^2*b)*m^2 + 458*(B*a^3 + 3*A*a^2*b)*m)*x^3 + (A*a^3*m^4 + 24*A*a^3*m^3 + 206*A*a^
3*m^2 + 744*A*a^3*m + 945*A*a^3)*x)*x^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)
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Sympy [A] time = 2.95055, size = 2069, normalized size = 21.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**m*(b*x**2+a)**3*(B*x**2+A),x)
[Out]
Piecewise((-A*a**3/(8*x**8) - A*a**2*b/(2*x**6) - 3*A*a*b**2/(4*x**4) - A*b**3/(2*x**2) - B*a**3/(6*x**6) - 3*
B*a**2*b/(4*x**4) - 3*B*a*b**2/(2*x**2) + B*b**3*log(x), Eq(m, -9)), (-A*a**3/(6*x**6) - 3*A*a**2*b/(4*x**4) -
3*A*a*b**2/(2*x**2) + A*b**3*log(x) - B*a**3/(4*x**4) - 3*B*a**2*b/(2*x**2) + 3*B*a*b**2*log(x) + B*b**3*x**2
/2, Eq(m, -7)), (-A*a**3/(4*x**4) - 3*A*a**2*b/(2*x**2) + 3*A*a*b**2*log(x) + A*b**3*x**2/2 - B*a**3/(2*x**2)
+ 3*B*a**2*b*log(x) + 3*B*a*b**2*x**2/2 + B*b**3*x**4/4, Eq(m, -5)), (-A*a**3/(2*x**2) + 3*A*a**2*b*log(x) + 3
*A*a*b**2*x**2/2 + A*b**3*x**4/4 + B*a**3*log(x) + 3*B*a**2*b*x**2/2 + 3*B*a*b**2*x**4/4 + B*b**3*x**6/6, Eq(m
, -3)), (A*a**3*log(x) + 3*A*a**2*b*x**2/2 + 3*A*a*b**2*x**4/4 + A*b**3*x**6/6 + B*a**3*x**2/2 + 3*B*a**2*b*x*
*4/4 + B*a*b**2*x**6/2 + B*b**3*x**8/8, Eq(m, -1)), (A*a**3*m**4*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2
+ 1689*m + 945) + 24*A*a**3*m**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 206*A*a**3*m**
2*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 744*A*a**3*m*x*x**m/(m**5 + 25*m**4 + 230*m**
3 + 950*m**2 + 1689*m + 945) + 945*A*a**3*x*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3*A*a
**2*b*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 66*A*a**2*b*m**3*x**3*x**m/(m**5
+ 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 492*A*a**2*b*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950
*m**2 + 1689*m + 945) + 1374*A*a**2*b*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 945*
A*a**2*b*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3*A*a*b**2*m**4*x**5*x**m/(m**5 + 2
5*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 60*A*a*b**2*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**
2 + 1689*m + 945) + 390*A*a*b**2*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 900*A*
a*b**2*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 567*A*a*b**2*x**5*x**m/(m**5 + 25*m
**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + A*b**3*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689
*m + 945) + 18*A*b**3*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 104*A*b**3*m**2*x
**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 222*A*b**3*m*x**7*x**m/(m**5 + 25*m**4 + 230*
m**3 + 950*m**2 + 1689*m + 945) + 135*A*b**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
B*a**3*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 22*B*a**3*m**3*x**3*x**m/(m**5
+ 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 164*B*a**3*m**2*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m
**2 + 1689*m + 945) + 458*B*a**3*m*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 315*B*a**
3*x**3*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 3*B*a**2*b*m**4*x**5*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 60*B*a**2*b*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 168
9*m + 945) + 390*B*a**2*b*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 900*B*a**2*b*
m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 567*B*a**2*b*x**5*x**m/(m**5 + 25*m**4 + 2
30*m**3 + 950*m**2 + 1689*m + 945) + 3*B*a*b**2*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 54*B*a*b**2*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 312*B*a*b**2*m**2*
x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 666*B*a*b**2*m*x**7*x**m/(m**5 + 25*m**4 + 2
30*m**3 + 950*m**2 + 1689*m + 945) + 405*B*a*b**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 9
45) + B*b**3*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 16*B*b**3*m**3*x**9*x**m/(
m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 86*B*b**3*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 9
50*m**2 + 1689*m + 945) + 176*B*b**3*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 105*B
*b**3*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945), True))
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Giac [B] time = 1.19561, size = 801, normalized size = 8.34 \begin{align*} \frac{B b^{3} m^{4} x^{9} x^{m} + 16 \, B b^{3} m^{3} x^{9} x^{m} + 3 \, B a b^{2} m^{4} x^{7} x^{m} + A b^{3} m^{4} x^{7} x^{m} + 86 \, B b^{3} m^{2} x^{9} x^{m} + 54 \, B a b^{2} m^{3} x^{7} x^{m} + 18 \, A b^{3} m^{3} x^{7} x^{m} + 176 \, B b^{3} m x^{9} x^{m} + 3 \, B a^{2} b m^{4} x^{5} x^{m} + 3 \, A a b^{2} m^{4} x^{5} x^{m} + 312 \, B a b^{2} m^{2} x^{7} x^{m} + 104 \, A b^{3} m^{2} x^{7} x^{m} + 105 \, B b^{3} x^{9} x^{m} + 60 \, B a^{2} b m^{3} x^{5} x^{m} + 60 \, A a b^{2} m^{3} x^{5} x^{m} + 666 \, B a b^{2} m x^{7} x^{m} + 222 \, A b^{3} m x^{7} x^{m} + B a^{3} m^{4} x^{3} x^{m} + 3 \, A a^{2} b m^{4} x^{3} x^{m} + 390 \, B a^{2} b m^{2} x^{5} x^{m} + 390 \, A a b^{2} m^{2} x^{5} x^{m} + 405 \, B a b^{2} x^{7} x^{m} + 135 \, A b^{3} x^{7} x^{m} + 22 \, B a^{3} m^{3} x^{3} x^{m} + 66 \, A a^{2} b m^{3} x^{3} x^{m} + 900 \, B a^{2} b m x^{5} x^{m} + 900 \, A a b^{2} m x^{5} x^{m} + A a^{3} m^{4} x x^{m} + 164 \, B a^{3} m^{2} x^{3} x^{m} + 492 \, A a^{2} b m^{2} x^{3} x^{m} + 567 \, B a^{2} b x^{5} x^{m} + 567 \, A a b^{2} x^{5} x^{m} + 24 \, A a^{3} m^{3} x x^{m} + 458 \, B a^{3} m x^{3} x^{m} + 1374 \, A a^{2} b m x^{3} x^{m} + 206 \, A a^{3} m^{2} x x^{m} + 315 \, B a^{3} x^{3} x^{m} + 945 \, A a^{2} b x^{3} x^{m} + 744 \, A a^{3} m x x^{m} + 945 \, A a^{3} x x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^m*(b*x^2+a)^3*(B*x^2+A),x, algorithm="giac")
[Out]
(B*b^3*m^4*x^9*x^m + 16*B*b^3*m^3*x^9*x^m + 3*B*a*b^2*m^4*x^7*x^m + A*b^3*m^4*x^7*x^m + 86*B*b^3*m^2*x^9*x^m +
54*B*a*b^2*m^3*x^7*x^m + 18*A*b^3*m^3*x^7*x^m + 176*B*b^3*m*x^9*x^m + 3*B*a^2*b*m^4*x^5*x^m + 3*A*a*b^2*m^4*x
^5*x^m + 312*B*a*b^2*m^2*x^7*x^m + 104*A*b^3*m^2*x^7*x^m + 105*B*b^3*x^9*x^m + 60*B*a^2*b*m^3*x^5*x^m + 60*A*a
*b^2*m^3*x^5*x^m + 666*B*a*b^2*m*x^7*x^m + 222*A*b^3*m*x^7*x^m + B*a^3*m^4*x^3*x^m + 3*A*a^2*b*m^4*x^3*x^m + 3
90*B*a^2*b*m^2*x^5*x^m + 390*A*a*b^2*m^2*x^5*x^m + 405*B*a*b^2*x^7*x^m + 135*A*b^3*x^7*x^m + 22*B*a^3*m^3*x^3*
x^m + 66*A*a^2*b*m^3*x^3*x^m + 900*B*a^2*b*m*x^5*x^m + 900*A*a*b^2*m*x^5*x^m + A*a^3*m^4*x*x^m + 164*B*a^3*m^2
*x^3*x^m + 492*A*a^2*b*m^2*x^3*x^m + 567*B*a^2*b*x^5*x^m + 567*A*a*b^2*x^5*x^m + 24*A*a^3*m^3*x*x^m + 458*B*a^
3*m*x^3*x^m + 1374*A*a^2*b*m*x^3*x^m + 206*A*a^3*m^2*x*x^m + 315*B*a^3*x^3*x^m + 945*A*a^2*b*x^3*x^m + 744*A*a
^3*m*x*x^m + 945*A*a^3*x*x^m)/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)