3.326 \(\int x^m \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx\)

Optimal. Leaf size=109 \[ \frac{x^{m+5} \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{m+5}+\frac{a^2 c^2 x^{m+1}}{m+1}+\frac{2 a c x^{m+3} (a d+b c)}{m+3}+\frac{2 b d x^{m+7} (a d+b c)}{m+7}+\frac{b^2 d^2 x^{m+9}}{m+9} \]

[Out]

(a^2*c^2*x^(1 + m))/(1 + m) + (2*a*c*(b*c + a*d)*x^(3 + m))/(3 + m) + ((b^2*c^2
+ 4*a*b*c*d + a^2*d^2)*x^(5 + m))/(5 + m) + (2*b*d*(b*c + a*d)*x^(7 + m))/(7 + m
) + (b^2*d^2*x^(9 + m))/(9 + m)

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Rubi [A]  time = 0.168043, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{x^{m+5} \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{m+5}+\frac{a^2 c^2 x^{m+1}}{m+1}+\frac{2 a c x^{m+3} (a d+b c)}{m+3}+\frac{2 b d x^{m+7} (a d+b c)}{m+7}+\frac{b^2 d^2 x^{m+9}}{m+9} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

(a^2*c^2*x^(1 + m))/(1 + m) + (2*a*c*(b*c + a*d)*x^(3 + m))/(3 + m) + ((b^2*c^2
+ 4*a*b*c*d + a^2*d^2)*x^(5 + m))/(5 + m) + (2*b*d*(b*c + a*d)*x^(7 + m))/(7 + m
) + (b^2*d^2*x^(9 + m))/(9 + m)

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Rubi in Sympy [A]  time = 28.2352, size = 100, normalized size = 0.92 \[ \frac{a^{2} c^{2} x^{m + 1}}{m + 1} + \frac{2 a c x^{m + 3} \left (a d + b c\right )}{m + 3} + \frac{b^{2} d^{2} x^{m + 9}}{m + 9} + \frac{2 b d x^{m + 7} \left (a d + b c\right )}{m + 7} + \frac{x^{m + 5} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{m + 5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(b*x**2+a)**2*(d*x**2+c)**2,x)

[Out]

a**2*c**2*x**(m + 1)/(m + 1) + 2*a*c*x**(m + 3)*(a*d + b*c)/(m + 3) + b**2*d**2*
x**(m + 9)/(m + 9) + 2*b*d*x**(m + 7)*(a*d + b*c)/(m + 7) + x**(m + 5)*(a**2*d**
2 + 4*a*b*c*d + b**2*c**2)/(m + 5)

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Mathematica [A]  time = 0.110726, size = 101, normalized size = 0.93 \[ x^m \left (\frac{x^5 \left (a^2 d^2+4 a b c d+b^2 c^2\right )}{m+5}+\frac{a^2 c^2 x}{m+1}+\frac{2 b d x^7 (a d+b c)}{m+7}+\frac{2 a c x^3 (a d+b c)}{m+3}+\frac{b^2 d^2 x^9}{m+9}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

x^m*((a^2*c^2*x)/(1 + m) + (2*a*c*(b*c + a*d)*x^3)/(3 + m) + ((b^2*c^2 + 4*a*b*c
*d + a^2*d^2)*x^5)/(5 + m) + (2*b*d*(b*c + a*d)*x^7)/(7 + m) + (b^2*d^2*x^9)/(9
+ m))

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Maple [B]  time = 0.009, size = 569, normalized size = 5.2 \[{\frac{{x}^{1+m} \left ({b}^{2}{d}^{2}{m}^{4}{x}^{8}+16\,{b}^{2}{d}^{2}{m}^{3}{x}^{8}+2\,ab{d}^{2}{m}^{4}{x}^{6}+2\,{b}^{2}cd{m}^{4}{x}^{6}+86\,{b}^{2}{d}^{2}{m}^{2}{x}^{8}+36\,ab{d}^{2}{m}^{3}{x}^{6}+36\,{b}^{2}cd{m}^{3}{x}^{6}+176\,{b}^{2}{d}^{2}m{x}^{8}+{a}^{2}{d}^{2}{m}^{4}{x}^{4}+4\,abcd{m}^{4}{x}^{4}+208\,ab{d}^{2}{m}^{2}{x}^{6}+{b}^{2}{c}^{2}{m}^{4}{x}^{4}+208\,{b}^{2}cd{m}^{2}{x}^{6}+105\,{b}^{2}{d}^{2}{x}^{8}+20\,{a}^{2}{d}^{2}{m}^{3}{x}^{4}+80\,abcd{m}^{3}{x}^{4}+444\,ab{d}^{2}m{x}^{6}+20\,{b}^{2}{c}^{2}{m}^{3}{x}^{4}+444\,{b}^{2}cdm{x}^{6}+2\,{a}^{2}cd{m}^{4}{x}^{2}+130\,{a}^{2}{d}^{2}{m}^{2}{x}^{4}+2\,ab{c}^{2}{m}^{4}{x}^{2}+520\,abcd{m}^{2}{x}^{4}+270\,{x}^{6}ab{d}^{2}+130\,{b}^{2}{c}^{2}{m}^{2}{x}^{4}+270\,{x}^{6}{b}^{2}cd+44\,{a}^{2}cd{m}^{3}{x}^{2}+300\,{a}^{2}{d}^{2}m{x}^{4}+44\,ab{c}^{2}{m}^{3}{x}^{2}+1200\,abcdm{x}^{4}+300\,{b}^{2}{c}^{2}m{x}^{4}+{a}^{2}{c}^{2}{m}^{4}+328\,{a}^{2}cd{m}^{2}{x}^{2}+189\,{x}^{4}{a}^{2}{d}^{2}+328\,ab{c}^{2}{m}^{2}{x}^{2}+756\,{x}^{4}abcd+189\,{x}^{4}{b}^{2}{c}^{2}+24\,{a}^{2}{c}^{2}{m}^{3}+916\,{a}^{2}cdm{x}^{2}+916\,ab{c}^{2}m{x}^{2}+206\,{a}^{2}{c}^{2}{m}^{2}+630\,{x}^{2}{a}^{2}cd+630\,a{c}^{2}b{x}^{2}+744\,{a}^{2}{c}^{2}m+945\,{a}^{2}{c}^{2} \right ) }{ \left ( 9+m \right ) \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(b*x^2+a)^2*(d*x^2+c)^2,x)

[Out]

x^(1+m)*(b^2*d^2*m^4*x^8+16*b^2*d^2*m^3*x^8+2*a*b*d^2*m^4*x^6+2*b^2*c*d*m^4*x^6+
86*b^2*d^2*m^2*x^8+36*a*b*d^2*m^3*x^6+36*b^2*c*d*m^3*x^6+176*b^2*d^2*m*x^8+a^2*d
^2*m^4*x^4+4*a*b*c*d*m^4*x^4+208*a*b*d^2*m^2*x^6+b^2*c^2*m^4*x^4+208*b^2*c*d*m^2
*x^6+105*b^2*d^2*x^8+20*a^2*d^2*m^3*x^4+80*a*b*c*d*m^3*x^4+444*a*b*d^2*m*x^6+20*
b^2*c^2*m^3*x^4+444*b^2*c*d*m*x^6+2*a^2*c*d*m^4*x^2+130*a^2*d^2*m^2*x^4+2*a*b*c^
2*m^4*x^2+520*a*b*c*d*m^2*x^4+270*a*b*d^2*x^6+130*b^2*c^2*m^2*x^4+270*b^2*c*d*x^
6+44*a^2*c*d*m^3*x^2+300*a^2*d^2*m*x^4+44*a*b*c^2*m^3*x^2+1200*a*b*c*d*m*x^4+300
*b^2*c^2*m*x^4+a^2*c^2*m^4+328*a^2*c*d*m^2*x^2+189*a^2*d^2*x^4+328*a*b*c^2*m^2*x
^2+756*a*b*c*d*x^4+189*b^2*c^2*x^4+24*a^2*c^2*m^3+916*a^2*c*d*m*x^2+916*a*b*c^2*
m*x^2+206*a^2*c^2*m^2+630*a^2*c*d*x^2+630*a*b*c^2*x^2+744*a^2*c^2*m+945*a^2*c^2)
/(9+m)/(7+m)/(5+m)/(3+m)/(1+m)

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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((b*x^2 + a)^2*(d*x^2 + c)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237938, size = 597, normalized size = 5.48 \[ \frac{{\left ({\left (b^{2} d^{2} m^{4} + 16 \, b^{2} d^{2} m^{3} + 86 \, b^{2} d^{2} m^{2} + 176 \, b^{2} d^{2} m + 105 \, b^{2} d^{2}\right )} x^{9} + 2 \,{\left ({\left (b^{2} c d + a b d^{2}\right )} m^{4} + 135 \, b^{2} c d + 135 \, a b d^{2} + 18 \,{\left (b^{2} c d + a b d^{2}\right )} m^{3} + 104 \,{\left (b^{2} c d + a b d^{2}\right )} m^{2} + 222 \,{\left (b^{2} c d + a b d^{2}\right )} m\right )} x^{7} +{\left ({\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{4} + 189 \, b^{2} c^{2} + 756 \, a b c d + 189 \, a^{2} d^{2} + 20 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{3} + 130 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m^{2} + 300 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} m\right )} x^{5} + 2 \,{\left ({\left (a b c^{2} + a^{2} c d\right )} m^{4} + 315 \, a b c^{2} + 315 \, a^{2} c d + 22 \,{\left (a b c^{2} + a^{2} c d\right )} m^{3} + 164 \,{\left (a b c^{2} + a^{2} c d\right )} m^{2} + 458 \,{\left (a b c^{2} + a^{2} c d\right )} m\right )} x^{3} +{\left (a^{2} c^{2} m^{4} + 24 \, a^{2} c^{2} m^{3} + 206 \, a^{2} c^{2} m^{2} + 744 \, a^{2} c^{2} m + 945 \, a^{2} c^{2}\right )} x\right )} x^{m}}{m^{5} + 25 \, m^{4} + 230 \, m^{3} + 950 \, m^{2} + 1689 \, m + 945} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^2*x^m,x, algorithm="fricas")

[Out]

((b^2*d^2*m^4 + 16*b^2*d^2*m^3 + 86*b^2*d^2*m^2 + 176*b^2*d^2*m + 105*b^2*d^2)*x
^9 + 2*((b^2*c*d + a*b*d^2)*m^4 + 135*b^2*c*d + 135*a*b*d^2 + 18*(b^2*c*d + a*b*
d^2)*m^3 + 104*(b^2*c*d + a*b*d^2)*m^2 + 222*(b^2*c*d + a*b*d^2)*m)*x^7 + ((b^2*
c^2 + 4*a*b*c*d + a^2*d^2)*m^4 + 189*b^2*c^2 + 756*a*b*c*d + 189*a^2*d^2 + 20*(b
^2*c^2 + 4*a*b*c*d + a^2*d^2)*m^3 + 130*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m^2 + 30
0*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*m)*x^5 + 2*((a*b*c^2 + a^2*c*d)*m^4 + 315*a*b*
c^2 + 315*a^2*c*d + 22*(a*b*c^2 + a^2*c*d)*m^3 + 164*(a*b*c^2 + a^2*c*d)*m^2 + 4
58*(a*b*c^2 + a^2*c*d)*m)*x^3 + (a^2*c^2*m^4 + 24*a^2*c^2*m^3 + 206*a^2*c^2*m^2
+ 744*a^2*c^2*m + 945*a^2*c^2)*x)*x^m/(m^5 + 25*m^4 + 230*m^3 + 950*m^2 + 1689*m
+ 945)

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Sympy [A]  time = 9.85582, size = 2363, normalized size = 21.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(b*x**2+a)**2*(d*x**2+c)**2,x)

[Out]

Piecewise((-a**2*c**2/(8*x**8) - a**2*c*d/(3*x**6) - a**2*d**2/(4*x**4) - a*b*c*
*2/(3*x**6) - a*b*c*d/x**4 - a*b*d**2/x**2 - b**2*c**2/(4*x**4) - b**2*c*d/x**2
+ b**2*d**2*log(x), Eq(m, -9)), (-a**2*c**2/(6*x**6) - a**2*c*d/(2*x**4) - a**2*
d**2/(2*x**2) - a*b*c**2/(2*x**4) - 2*a*b*c*d/x**2 + 2*a*b*d**2*log(x) - b**2*c*
*2/(2*x**2) + 2*b**2*c*d*log(x) + b**2*d**2*x**2/2, Eq(m, -7)), (-a**2*c**2/(4*x
**4) - a**2*c*d/x**2 + a**2*d**2*log(x) - a*b*c**2/x**2 + 4*a*b*c*d*log(x) + a*b
*d**2*x**2 + b**2*c**2*log(x) + b**2*c*d*x**2 + b**2*d**2*x**4/4, Eq(m, -5)), (-
a**2*c**2/(2*x**2) + 2*a**2*c*d*log(x) + a**2*d**2*x**2/2 + 2*a*b*c**2*log(x) +
2*a*b*c*d*x**2 + a*b*d**2*x**4/2 + b**2*c**2*x**2/2 + b**2*c*d*x**4/2 + b**2*d**
2*x**6/6, Eq(m, -3)), (a**2*c**2*log(x) + a**2*c*d*x**2 + a**2*d**2*x**4/4 + a*b
*c**2*x**2 + a*b*c*d*x**4 + a*b*d**2*x**6/3 + b**2*c**2*x**4/4 + b**2*c*d*x**6/3
+ b**2*d**2*x**8/8, Eq(m, -1)), (a**2*c**2*m**4*x*x**m/(m**5 + 25*m**4 + 230*m*
*3 + 950*m**2 + 1689*m + 945) + 24*a**2*c**2*m**3*x*x**m/(m**5 + 25*m**4 + 230*m
**3 + 950*m**2 + 1689*m + 945) + 206*a**2*c**2*m**2*x*x**m/(m**5 + 25*m**4 + 230
*m**3 + 950*m**2 + 1689*m + 945) + 744*a**2*c**2*m*x*x**m/(m**5 + 25*m**4 + 230*
m**3 + 950*m**2 + 1689*m + 945) + 945*a**2*c**2*x*x**m/(m**5 + 25*m**4 + 230*m**
3 + 950*m**2 + 1689*m + 945) + 2*a**2*c*d*m**4*x**3*x**m/(m**5 + 25*m**4 + 230*m
**3 + 950*m**2 + 1689*m + 945) + 44*a**2*c*d*m**3*x**3*x**m/(m**5 + 25*m**4 + 23
0*m**3 + 950*m**2 + 1689*m + 945) + 328*a**2*c*d*m**2*x**3*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 916*a**2*c*d*m*x**3*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 630*a**2*c*d*x**3*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + a**2*d**2*m**4*x**5*x**m/(m**5 + 25*m**4
+ 230*m**3 + 950*m**2 + 1689*m + 945) + 20*a**2*d**2*m**3*x**5*x**m/(m**5 + 25*
m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 130*a**2*d**2*m**2*x**5*x**m/(m**5
+ 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 300*a**2*d**2*m*x**5*x**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 189*a**2*d**2*x**5*x**m/(m**
5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a*b*c**2*m**4*x**3*x**m/(m
**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 44*a*b*c**2*m**3*x**3*x**m
/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 328*a*b*c**2*m**2*x**3*
x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 916*a*b*c**2*m*x**3
*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 630*a*b*c**2*x**3*
x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 4*a*b*c*d*m**4*x**5
*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 80*a*b*c*d*m**3*x*
*5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 520*a*b*c*d*m**2
*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 1200*a*b*c*d*
m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 756*a*b*c*d*
x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 2*a*b*d**2*m**
4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 36*a*b*d**2*
m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 208*a*b*d
**2*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 444*a
*b*d**2*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 270*
a*b*d**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + b**2*
c**2*m**4*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) + 20*b
**2*c**2*m**3*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 945) +
130*b**2*c**2*m**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m + 94
5) + 300*b**2*c**2*m*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 189*b**2*c**2*x**5*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m +
945) + 2*b**2*c*d*m**4*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689*m
+ 945) + 36*b**2*c*d*m**3*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 + 1689
*m + 945) + 208*b**2*c*d*m**2*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 444*b**2*c*d*m*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 270*b**2*c*d*x**7*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + b**2*d**2*m**4*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m**2 +
1689*m + 945) + 16*b**2*d**2*m**3*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 950*m*
*2 + 1689*m + 945) + 86*b**2*d**2*m**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 + 95
0*m**2 + 1689*m + 945) + 176*b**2*d**2*m*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945) + 105*b**2*d**2*x**9*x**m/(m**5 + 25*m**4 + 230*m**3 +
950*m**2 + 1689*m + 945), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228424, size = 1071, normalized size = 9.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^2*(d*x^2 + c)^2*x^m,x, algorithm="giac")

[Out]

(b^2*d^2*m^4*x^9*e^(m*ln(x)) + 16*b^2*d^2*m^3*x^9*e^(m*ln(x)) + 2*b^2*c*d*m^4*x^
7*e^(m*ln(x)) + 2*a*b*d^2*m^4*x^7*e^(m*ln(x)) + 86*b^2*d^2*m^2*x^9*e^(m*ln(x)) +
36*b^2*c*d*m^3*x^7*e^(m*ln(x)) + 36*a*b*d^2*m^3*x^7*e^(m*ln(x)) + 176*b^2*d^2*m
*x^9*e^(m*ln(x)) + b^2*c^2*m^4*x^5*e^(m*ln(x)) + 4*a*b*c*d*m^4*x^5*e^(m*ln(x)) +
a^2*d^2*m^4*x^5*e^(m*ln(x)) + 208*b^2*c*d*m^2*x^7*e^(m*ln(x)) + 208*a*b*d^2*m^2
*x^7*e^(m*ln(x)) + 105*b^2*d^2*x^9*e^(m*ln(x)) + 20*b^2*c^2*m^3*x^5*e^(m*ln(x))
+ 80*a*b*c*d*m^3*x^5*e^(m*ln(x)) + 20*a^2*d^2*m^3*x^5*e^(m*ln(x)) + 444*b^2*c*d*
m*x^7*e^(m*ln(x)) + 444*a*b*d^2*m*x^7*e^(m*ln(x)) + 2*a*b*c^2*m^4*x^3*e^(m*ln(x)
) + 2*a^2*c*d*m^4*x^3*e^(m*ln(x)) + 130*b^2*c^2*m^2*x^5*e^(m*ln(x)) + 520*a*b*c*
d*m^2*x^5*e^(m*ln(x)) + 130*a^2*d^2*m^2*x^5*e^(m*ln(x)) + 270*b^2*c*d*x^7*e^(m*l
n(x)) + 270*a*b*d^2*x^7*e^(m*ln(x)) + 44*a*b*c^2*m^3*x^3*e^(m*ln(x)) + 44*a^2*c*
d*m^3*x^3*e^(m*ln(x)) + 300*b^2*c^2*m*x^5*e^(m*ln(x)) + 1200*a*b*c*d*m*x^5*e^(m*
ln(x)) + 300*a^2*d^2*m*x^5*e^(m*ln(x)) + a^2*c^2*m^4*x*e^(m*ln(x)) + 328*a*b*c^2
*m^2*x^3*e^(m*ln(x)) + 328*a^2*c*d*m^2*x^3*e^(m*ln(x)) + 189*b^2*c^2*x^5*e^(m*ln
(x)) + 756*a*b*c*d*x^5*e^(m*ln(x)) + 189*a^2*d^2*x^5*e^(m*ln(x)) + 24*a^2*c^2*m^
3*x*e^(m*ln(x)) + 916*a*b*c^2*m*x^3*e^(m*ln(x)) + 916*a^2*c*d*m*x^3*e^(m*ln(x))
+ 206*a^2*c^2*m^2*x*e^(m*ln(x)) + 630*a*b*c^2*x^3*e^(m*ln(x)) + 630*a^2*c*d*x^3*
e^(m*ln(x)) + 744*a^2*c^2*m*x*e^(m*ln(x)) + 945*a^2*c^2*x*e^(m*ln(x)))/(m^5 + 25
*m^4 + 230*m^3 + 950*m^2 + 1689*m + 945)