3.1.6 \(\int (e x)^m (a+b x^2)^2 (A+B x^2) (c+d x^2)^2 \, dx\)

Optimal. Leaf size=216 \[ \frac {(e x)^{m+5} \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{e^5 (m+5)}+\frac {(e x)^{m+7} \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{e^7 (m+7)}+\frac {a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac {b d (e x)^{m+9} (2 a B d+A b d+2 b B c)}{e^9 (m+9)}+\frac {a c (e x)^{m+3} (2 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac {b^2 B d^2 (e x)^{m+11}}{e^{11} (m+11)} \]

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Rubi [A]  time = 0.25, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {570} \begin {gather*} \frac {(e x)^{m+5} \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{e^5 (m+5)}+\frac {(e x)^{m+7} \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{e^7 (m+7)}+\frac {a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac {a c (e x)^{m+3} (2 A (a d+b c)+a B c)}{e^3 (m+3)}+\frac {b d (e x)^{m+9} (2 a B d+A b d+2 b B c)}{e^9 (m+9)}+\frac {b^2 B d^2 (e x)^{m+11}}{e^{11} (m+11)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*A*c^2*(e*x)^(1 + m))/(e*(1 + m)) + (a*c*(a*B*c + 2*A*(b*c + a*d))*(e*x)^(3 + m))/(e^3*(3 + m)) + ((2*a*B*
c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*(e*x)^(5 + m))/(e^5*(5 + m)) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c
+ A*d) + b^2*c*(B*c + 2*A*d))*(e*x)^(7 + m))/(e^7*(7 + m)) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*(e*x)^(9 + m))/
(e^9*(9 + m)) + (b^2*B*d^2*(e*x)^(11 + m))/(e^11*(11 + m))

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

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

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Mathematica [A]  time = 0.35, size = 178, normalized size = 0.82 \begin {gather*} x (e x)^m \left (\frac {x^4 \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+5}+\frac {x^6 \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+7}+\frac {a^2 A c^2}{m+1}+\frac {b d x^8 (2 a B d+A b d+2 b B c)}{m+9}+\frac {a c x^2 (2 A (a d+b c)+a B c)}{m+3}+\frac {b^2 B d^2 x^{10}}{m+11}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*(e*x)^m*((a^2*A*c^2)/(1 + m) + (a*c*(a*B*c + 2*A*(b*c + a*d))*x^2)/(3 + m) + ((2*a*B*c*(b*c + a*d) + A*(b^2*
c^2 + 4*a*b*c*d + a^2*d^2))*x^4)/(5 + m) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^6)/(7
+ m) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^8)/(9 + m) + (b^2*B*d^2*x^10)/(11 + m))

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IntegrateAlgebraic [F]  time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m \left (a+b x^2\right )^2 \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(e*x)^m*(a + b*x^2)^2*(A + B*x^2)*(c + d*x^2)^2,x]

[Out]

Defer[IntegrateAlgebraic][(e*x)^m*(a + b*x^2)^2*(A + B*x^2)*(c + d*x^2)^2, x]

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fricas [B]  time = 1.03, size = 1043, normalized size = 4.83

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x^2+a)^2*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="fricas")

[Out]

((B*b^2*d^2*m^5 + 25*B*b^2*d^2*m^4 + 230*B*b^2*d^2*m^3 + 950*B*b^2*d^2*m^2 + 1689*B*b^2*d^2*m + 945*B*b^2*d^2)
*x^11 + ((2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^5 + 2310*B*b^2*c*d + 27*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)
*m^4 + 262*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + 1155*(2*B*a*b + A*b^2)*d^2 + 1122*(2*B*b^2*c*d + (2*B*a
*b + A*b^2)*d^2)*m^2 + 2041*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*x^9 + ((B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c
*d + (B*a^2 + 2*A*a*b)*d^2)*m^5 + 1485*B*b^2*c^2 + 29*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)
*d^2)*m^4 + 302*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^3 + 2970*(2*B*a*b + A*b^2)*c*d
+ 1485*(B*a^2 + 2*A*a*b)*d^2 + 1366*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 2577*
(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*x^7 + ((A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2
*(B*a^2 + 2*A*a*b)*c*d)*m^5 + 2079*A*a^2*d^2 + 31*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d
)*m^4 + 350*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + 2079*(2*B*a*b + A*b^2)*c^2 + 4
158*(B*a^2 + 2*A*a*b)*c*d + 1730*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 3489*(A*a
^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m)*x^5 + ((2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^5
+ 6930*A*a^2*c*d + 33*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^4 + 406*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^
3 + 3465*(B*a^2 + 2*A*a*b)*c^2 + 2262*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 5353*(2*A*a^2*c*d + (B*a^2 +
2*A*a*b)*c^2)*m)*x^3 + (A*a^2*c^2*m^5 + 35*A*a^2*c^2*m^4 + 470*A*a^2*c^2*m^3 + 3010*A*a^2*c^2*m^2 + 9129*A*a^
2*c^2*m + 10395*A*a^2*c^2)*x)*(e*x)^m/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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giac [B]  time = 0.68, size = 2010, normalized size = 9.31

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x^2+a)^2*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="giac")

[Out]

(B*b^2*d^2*m^5*x^11*x^m*e^m + 25*B*b^2*d^2*m^4*x^11*x^m*e^m + 2*B*b^2*c*d*m^5*x^9*x^m*e^m + 2*B*a*b*d^2*m^5*x^
9*x^m*e^m + A*b^2*d^2*m^5*x^9*x^m*e^m + 230*B*b^2*d^2*m^3*x^11*x^m*e^m + 54*B*b^2*c*d*m^4*x^9*x^m*e^m + 54*B*a
*b*d^2*m^4*x^9*x^m*e^m + 27*A*b^2*d^2*m^4*x^9*x^m*e^m + 950*B*b^2*d^2*m^2*x^11*x^m*e^m + B*b^2*c^2*m^5*x^7*x^m
*e^m + 4*B*a*b*c*d*m^5*x^7*x^m*e^m + 2*A*b^2*c*d*m^5*x^7*x^m*e^m + B*a^2*d^2*m^5*x^7*x^m*e^m + 2*A*a*b*d^2*m^5
*x^7*x^m*e^m + 524*B*b^2*c*d*m^3*x^9*x^m*e^m + 524*B*a*b*d^2*m^3*x^9*x^m*e^m + 262*A*b^2*d^2*m^3*x^9*x^m*e^m +
1689*B*b^2*d^2*m*x^11*x^m*e^m + 29*B*b^2*c^2*m^4*x^7*x^m*e^m + 116*B*a*b*c*d*m^4*x^7*x^m*e^m + 58*A*b^2*c*d*m
^4*x^7*x^m*e^m + 29*B*a^2*d^2*m^4*x^7*x^m*e^m + 58*A*a*b*d^2*m^4*x^7*x^m*e^m + 2244*B*b^2*c*d*m^2*x^9*x^m*e^m
+ 2244*B*a*b*d^2*m^2*x^9*x^m*e^m + 1122*A*b^2*d^2*m^2*x^9*x^m*e^m + 945*B*b^2*d^2*x^11*x^m*e^m + 2*B*a*b*c^2*m
^5*x^5*x^m*e^m + A*b^2*c^2*m^5*x^5*x^m*e^m + 2*B*a^2*c*d*m^5*x^5*x^m*e^m + 4*A*a*b*c*d*m^5*x^5*x^m*e^m + A*a^2
*d^2*m^5*x^5*x^m*e^m + 302*B*b^2*c^2*m^3*x^7*x^m*e^m + 1208*B*a*b*c*d*m^3*x^7*x^m*e^m + 604*A*b^2*c*d*m^3*x^7*
x^m*e^m + 302*B*a^2*d^2*m^3*x^7*x^m*e^m + 604*A*a*b*d^2*m^3*x^7*x^m*e^m + 4082*B*b^2*c*d*m*x^9*x^m*e^m + 4082*
B*a*b*d^2*m*x^9*x^m*e^m + 2041*A*b^2*d^2*m*x^9*x^m*e^m + 62*B*a*b*c^2*m^4*x^5*x^m*e^m + 31*A*b^2*c^2*m^4*x^5*x
^m*e^m + 62*B*a^2*c*d*m^4*x^5*x^m*e^m + 124*A*a*b*c*d*m^4*x^5*x^m*e^m + 31*A*a^2*d^2*m^4*x^5*x^m*e^m + 1366*B*
b^2*c^2*m^2*x^7*x^m*e^m + 5464*B*a*b*c*d*m^2*x^7*x^m*e^m + 2732*A*b^2*c*d*m^2*x^7*x^m*e^m + 1366*B*a^2*d^2*m^2
*x^7*x^m*e^m + 2732*A*a*b*d^2*m^2*x^7*x^m*e^m + 2310*B*b^2*c*d*x^9*x^m*e^m + 2310*B*a*b*d^2*x^9*x^m*e^m + 1155
*A*b^2*d^2*x^9*x^m*e^m + B*a^2*c^2*m^5*x^3*x^m*e^m + 2*A*a*b*c^2*m^5*x^3*x^m*e^m + 2*A*a^2*c*d*m^5*x^3*x^m*e^m
+ 700*B*a*b*c^2*m^3*x^5*x^m*e^m + 350*A*b^2*c^2*m^3*x^5*x^m*e^m + 700*B*a^2*c*d*m^3*x^5*x^m*e^m + 1400*A*a*b*
c*d*m^3*x^5*x^m*e^m + 350*A*a^2*d^2*m^3*x^5*x^m*e^m + 2577*B*b^2*c^2*m*x^7*x^m*e^m + 10308*B*a*b*c*d*m*x^7*x^m
*e^m + 5154*A*b^2*c*d*m*x^7*x^m*e^m + 2577*B*a^2*d^2*m*x^7*x^m*e^m + 5154*A*a*b*d^2*m*x^7*x^m*e^m + 33*B*a^2*c
^2*m^4*x^3*x^m*e^m + 66*A*a*b*c^2*m^4*x^3*x^m*e^m + 66*A*a^2*c*d*m^4*x^3*x^m*e^m + 3460*B*a*b*c^2*m^2*x^5*x^m*
e^m + 1730*A*b^2*c^2*m^2*x^5*x^m*e^m + 3460*B*a^2*c*d*m^2*x^5*x^m*e^m + 6920*A*a*b*c*d*m^2*x^5*x^m*e^m + 1730*
A*a^2*d^2*m^2*x^5*x^m*e^m + 1485*B*b^2*c^2*x^7*x^m*e^m + 5940*B*a*b*c*d*x^7*x^m*e^m + 2970*A*b^2*c*d*x^7*x^m*e
^m + 1485*B*a^2*d^2*x^7*x^m*e^m + 2970*A*a*b*d^2*x^7*x^m*e^m + A*a^2*c^2*m^5*x*x^m*e^m + 406*B*a^2*c^2*m^3*x^3
*x^m*e^m + 812*A*a*b*c^2*m^3*x^3*x^m*e^m + 812*A*a^2*c*d*m^3*x^3*x^m*e^m + 6978*B*a*b*c^2*m*x^5*x^m*e^m + 3489
*A*b^2*c^2*m*x^5*x^m*e^m + 6978*B*a^2*c*d*m*x^5*x^m*e^m + 13956*A*a*b*c*d*m*x^5*x^m*e^m + 3489*A*a^2*d^2*m*x^5
*x^m*e^m + 35*A*a^2*c^2*m^4*x*x^m*e^m + 2262*B*a^2*c^2*m^2*x^3*x^m*e^m + 4524*A*a*b*c^2*m^2*x^3*x^m*e^m + 4524
*A*a^2*c*d*m^2*x^3*x^m*e^m + 4158*B*a*b*c^2*x^5*x^m*e^m + 2079*A*b^2*c^2*x^5*x^m*e^m + 4158*B*a^2*c*d*x^5*x^m*
e^m + 8316*A*a*b*c*d*x^5*x^m*e^m + 2079*A*a^2*d^2*x^5*x^m*e^m + 470*A*a^2*c^2*m^3*x*x^m*e^m + 5353*B*a^2*c^2*m
*x^3*x^m*e^m + 10706*A*a*b*c^2*m*x^3*x^m*e^m + 10706*A*a^2*c*d*m*x^3*x^m*e^m + 3010*A*a^2*c^2*m^2*x*x^m*e^m +
3465*B*a^2*c^2*x^3*x^m*e^m + 6930*A*a*b*c^2*x^3*x^m*e^m + 6930*A*a^2*c*d*x^3*x^m*e^m + 9129*A*a^2*c^2*m*x*x^m*
e^m + 10395*A*a^2*c^2*x*x^m*e^m)/(m^6 + 36*m^5 + 505*m^4 + 3480*m^3 + 12139*m^2 + 19524*m + 10395)

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maple [B]  time = 0.01, size = 1471, normalized size = 6.81

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x^2+a)^2*(B*x^2+A)*(d*x^2+c)^2,x)

[Out]

x*(B*b^2*d^2*m^5*x^10+25*B*b^2*d^2*m^4*x^10+A*b^2*d^2*m^5*x^8+2*B*a*b*d^2*m^5*x^8+2*B*b^2*c*d*m^5*x^8+230*B*b^
2*d^2*m^3*x^10+27*A*b^2*d^2*m^4*x^8+54*B*a*b*d^2*m^4*x^8+54*B*b^2*c*d*m^4*x^8+950*B*b^2*d^2*m^2*x^10+2*A*a*b*d
^2*m^5*x^6+2*A*b^2*c*d*m^5*x^6+262*A*b^2*d^2*m^3*x^8+B*a^2*d^2*m^5*x^6+4*B*a*b*c*d*m^5*x^6+524*B*a*b*d^2*m^3*x
^8+B*b^2*c^2*m^5*x^6+524*B*b^2*c*d*m^3*x^8+1689*B*b^2*d^2*m*x^10+58*A*a*b*d^2*m^4*x^6+58*A*b^2*c*d*m^4*x^6+112
2*A*b^2*d^2*m^2*x^8+29*B*a^2*d^2*m^4*x^6+116*B*a*b*c*d*m^4*x^6+2244*B*a*b*d^2*m^2*x^8+29*B*b^2*c^2*m^4*x^6+224
4*B*b^2*c*d*m^2*x^8+945*B*b^2*d^2*x^10+A*a^2*d^2*m^5*x^4+4*A*a*b*c*d*m^5*x^4+604*A*a*b*d^2*m^3*x^6+A*b^2*c^2*m
^5*x^4+604*A*b^2*c*d*m^3*x^6+2041*A*b^2*d^2*m*x^8+2*B*a^2*c*d*m^5*x^4+302*B*a^2*d^2*m^3*x^6+2*B*a*b*c^2*m^5*x^
4+1208*B*a*b*c*d*m^3*x^6+4082*B*a*b*d^2*m*x^8+302*B*b^2*c^2*m^3*x^6+4082*B*b^2*c*d*m*x^8+31*A*a^2*d^2*m^4*x^4+
124*A*a*b*c*d*m^4*x^4+2732*A*a*b*d^2*m^2*x^6+31*A*b^2*c^2*m^4*x^4+2732*A*b^2*c*d*m^2*x^6+1155*A*b^2*d^2*x^8+62
*B*a^2*c*d*m^4*x^4+1366*B*a^2*d^2*m^2*x^6+62*B*a*b*c^2*m^4*x^4+5464*B*a*b*c*d*m^2*x^6+2310*B*a*b*d^2*x^8+1366*
B*b^2*c^2*m^2*x^6+2310*B*b^2*c*d*x^8+2*A*a^2*c*d*m^5*x^2+350*A*a^2*d^2*m^3*x^4+2*A*a*b*c^2*m^5*x^2+1400*A*a*b*
c*d*m^3*x^4+5154*A*a*b*d^2*m*x^6+350*A*b^2*c^2*m^3*x^4+5154*A*b^2*c*d*m*x^6+B*a^2*c^2*m^5*x^2+700*B*a^2*c*d*m^
3*x^4+2577*B*a^2*d^2*m*x^6+700*B*a*b*c^2*m^3*x^4+10308*B*a*b*c*d*m*x^6+2577*B*b^2*c^2*m*x^6+66*A*a^2*c*d*m^4*x
^2+1730*A*a^2*d^2*m^2*x^4+66*A*a*b*c^2*m^4*x^2+6920*A*a*b*c*d*m^2*x^4+2970*A*a*b*d^2*x^6+1730*A*b^2*c^2*m^2*x^
4+2970*A*b^2*c*d*x^6+33*B*a^2*c^2*m^4*x^2+3460*B*a^2*c*d*m^2*x^4+1485*B*a^2*d^2*x^6+3460*B*a*b*c^2*m^2*x^4+594
0*B*a*b*c*d*x^6+1485*B*b^2*c^2*x^6+A*a^2*c^2*m^5+812*A*a^2*c*d*m^3*x^2+3489*A*a^2*d^2*m*x^4+812*A*a*b*c^2*m^3*
x^2+13956*A*a*b*c*d*m*x^4+3489*A*b^2*c^2*m*x^4+406*B*a^2*c^2*m^3*x^2+6978*B*a^2*c*d*m*x^4+6978*B*a*b*c^2*m*x^4
+35*A*a^2*c^2*m^4+4524*A*a^2*c*d*m^2*x^2+2079*A*a^2*d^2*x^4+4524*A*a*b*c^2*m^2*x^2+8316*A*a*b*c*d*x^4+2079*A*b
^2*c^2*x^4+2262*B*a^2*c^2*m^2*x^2+4158*B*a^2*c*d*x^4+4158*B*a*b*c^2*x^4+470*A*a^2*c^2*m^3+10706*A*a^2*c*d*m*x^
2+10706*A*a*b*c^2*m*x^2+5353*B*a^2*c^2*m*x^2+3010*A*a^2*c^2*m^2+6930*A*a^2*c*d*x^2+6930*A*a*b*c^2*x^2+3465*B*a
^2*c^2*x^2+9129*A*a^2*c^2*m+10395*A*a^2*c^2)*(e*x)^m/(m+11)/(m+9)/(m+7)/(m+5)/(m+3)/(m+1)

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maxima [A]  time = 2.06, size = 396, normalized size = 1.83 \begin {gather*} \frac {B b^{2} d^{2} e^{m} x^{11} x^{m}}{m + 11} + \frac {2 \, B b^{2} c d e^{m} x^{9} x^{m}}{m + 9} + \frac {2 \, B a b d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {A b^{2} d^{2} e^{m} x^{9} x^{m}}{m + 9} + \frac {B b^{2} c^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {4 \, B a b c d e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, A b^{2} c d e^{m} x^{7} x^{m}}{m + 7} + \frac {B a^{2} d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, A a b d^{2} e^{m} x^{7} x^{m}}{m + 7} + \frac {2 \, B a b c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {A b^{2} c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {2 \, B a^{2} c d e^{m} x^{5} x^{m}}{m + 5} + \frac {4 \, A a b c d e^{m} x^{5} x^{m}}{m + 5} + \frac {A a^{2} d^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {B a^{2} c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a b c^{2} e^{m} x^{3} x^{m}}{m + 3} + \frac {2 \, A a^{2} c d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a^{2} c^{2}}{e {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b^2*d^2*e^m*x^11*x^m/(m + 11) + 2*B*b^2*c*d*e^m*x^9*x^m/(m + 9) + 2*B*a*b*d^2*e^m*x^9*x^m/(m + 9) + A*b^2*d^
2*e^m*x^9*x^m/(m + 9) + B*b^2*c^2*e^m*x^7*x^m/(m + 7) + 4*B*a*b*c*d*e^m*x^7*x^m/(m + 7) + 2*A*b^2*c*d*e^m*x^7*
x^m/(m + 7) + B*a^2*d^2*e^m*x^7*x^m/(m + 7) + 2*A*a*b*d^2*e^m*x^7*x^m/(m + 7) + 2*B*a*b*c^2*e^m*x^5*x^m/(m + 5
) + A*b^2*c^2*e^m*x^5*x^m/(m + 5) + 2*B*a^2*c*d*e^m*x^5*x^m/(m + 5) + 4*A*a*b*c*d*e^m*x^5*x^m/(m + 5) + A*a^2*
d^2*e^m*x^5*x^m/(m + 5) + B*a^2*c^2*e^m*x^3*x^m/(m + 3) + 2*A*a*b*c^2*e^m*x^3*x^m/(m + 3) + 2*A*a^2*c*d*e^m*x^
3*x^m/(m + 3) + (e*x)^(m + 1)*A*a^2*c^2/(e*(m + 1))

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mupad [B]  time = 1.39, size = 499, normalized size = 2.31 \begin {gather*} \frac {x^5\,{\left (e\,x\right )}^m\,\left (2\,B\,a^2\,c\,d+A\,a^2\,d^2+2\,B\,a\,b\,c^2+4\,A\,a\,b\,c\,d+A\,b^2\,c^2\right )\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {x^7\,{\left (e\,x\right )}^m\,\left (B\,a^2\,d^2+4\,B\,a\,b\,c\,d+2\,A\,a\,b\,d^2+B\,b^2\,c^2+2\,A\,b^2\,c\,d\right )\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {a\,c\,x^3\,{\left (e\,x\right )}^m\,\left (2\,A\,a\,d+2\,A\,b\,c+B\,a\,c\right )\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b\,d\,x^9\,{\left (e\,x\right )}^m\,\left (A\,b\,d+2\,B\,a\,d+2\,B\,b\,c\right )\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {A\,a^2\,c^2\,x\,{\left (e\,x\right )}^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {B\,b^2\,d^2\,x^{11}\,{\left (e\,x\right )}^m\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)*(e*x)^m*(a + b*x^2)^2*(c + d*x^2)^2,x)

[Out]

(x^5*(e*x)^m*(A*a^2*d^2 + A*b^2*c^2 + 2*B*a*b*c^2 + 2*B*a^2*c*d + 4*A*a*b*c*d)*(3489*m + 1730*m^2 + 350*m^3 +
31*m^4 + m^5 + 2079))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (x^7*(e*x)^m*(B*a^2*
d^2 + B*b^2*c^2 + 2*A*a*b*d^2 + 2*A*b^2*c*d + 4*B*a*b*c*d)*(2577*m + 1366*m^2 + 302*m^3 + 29*m^4 + m^5 + 1485)
)/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (a*c*x^3*(e*x)^m*(2*A*a*d + 2*A*b*c + B*
a*c)*(5353*m + 2262*m^2 + 406*m^3 + 33*m^4 + m^5 + 3465))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 +
m^6 + 10395) + (b*d*x^9*(e*x)^m*(A*b*d + 2*B*a*d + 2*B*b*c)*(2041*m + 1122*m^2 + 262*m^3 + 27*m^4 + m^5 + 115
5))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (A*a^2*c^2*x*(e*x)^m*(9129*m + 3010*m^
2 + 470*m^3 + 35*m^4 + m^5 + 10395))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^4 + 36*m^5 + m^6 + 10395) + (B*b^
2*d^2*x^11*(e*x)^m*(1689*m + 950*m^2 + 230*m^3 + 25*m^4 + m^5 + 945))/(19524*m + 12139*m^2 + 3480*m^3 + 505*m^
4 + 36*m^5 + m^6 + 10395)

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sympy [A]  time = 10.02, size = 7019, normalized size = 32.50

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(b*x**2+a)**2*(B*x**2+A)*(d*x**2+c)**2,x)

[Out]

Piecewise(((-A*a**2*c**2/(10*x**10) - A*a**2*c*d/(4*x**8) - A*a**2*d**2/(6*x**6) - A*a*b*c**2/(4*x**8) - 2*A*a
*b*c*d/(3*x**6) - A*a*b*d**2/(2*x**4) - A*b**2*c**2/(6*x**6) - A*b**2*c*d/(2*x**4) - A*b**2*d**2/(2*x**2) - B*
a**2*c**2/(8*x**8) - B*a**2*c*d/(3*x**6) - B*a**2*d**2/(4*x**4) - B*a*b*c**2/(3*x**6) - B*a*b*c*d/x**4 - B*a*b
*d**2/x**2 - B*b**2*c**2/(4*x**4) - B*b**2*c*d/x**2 + B*b**2*d**2*log(x))/e**11, Eq(m, -11)), ((-A*a**2*c**2/(
8*x**8) - A*a**2*c*d/(3*x**6) - A*a**2*d**2/(4*x**4) - A*a*b*c**2/(3*x**6) - A*a*b*c*d/x**4 - A*a*b*d**2/x**2
- A*b**2*c**2/(4*x**4) - A*b**2*c*d/x**2 + A*b**2*d**2*log(x) - B*a**2*c**2/(6*x**6) - B*a**2*c*d/(2*x**4) - B
*a**2*d**2/(2*x**2) - B*a*b*c**2/(2*x**4) - 2*B*a*b*c*d/x**2 + 2*B*a*b*d**2*log(x) - B*b**2*c**2/(2*x**2) + 2*
B*b**2*c*d*log(x) + B*b**2*d**2*x**2/2)/e**9, Eq(m, -9)), ((-A*a**2*c**2/(6*x**6) - A*a**2*c*d/(2*x**4) - A*a*
*2*d**2/(2*x**2) - A*a*b*c**2/(2*x**4) - 2*A*a*b*c*d/x**2 + 2*A*a*b*d**2*log(x) - A*b**2*c**2/(2*x**2) + 2*A*b
**2*c*d*log(x) + A*b**2*d**2*x**2/2 - B*a**2*c**2/(4*x**4) - B*a**2*c*d/x**2 + B*a**2*d**2*log(x) - B*a*b*c**2
/x**2 + 4*B*a*b*c*d*log(x) + B*a*b*d**2*x**2 + B*b**2*c**2*log(x) + B*b**2*c*d*x**2 + B*b**2*d**2*x**4/4)/e**7
, Eq(m, -7)), ((-A*a**2*c**2/(4*x**4) - A*a**2*c*d/x**2 + A*a**2*d**2*log(x) - A*a*b*c**2/x**2 + 4*A*a*b*c*d*l
og(x) + A*a*b*d**2*x**2 + A*b**2*c**2*log(x) + A*b**2*c*d*x**2 + A*b**2*d**2*x**4/4 - B*a**2*c**2/(2*x**2) + 2
*B*a**2*c*d*log(x) + B*a**2*d**2*x**2/2 + 2*B*a*b*c**2*log(x) + 2*B*a*b*c*d*x**2 + B*a*b*d**2*x**4/2 + B*b**2*
c**2*x**2/2 + B*b**2*c*d*x**4/2 + B*b**2*d**2*x**6/6)/e**5, Eq(m, -5)), ((-A*a**2*c**2/(2*x**2) + 2*A*a**2*c*d
*log(x) + A*a**2*d**2*x**2/2 + 2*A*a*b*c**2*log(x) + 2*A*a*b*c*d*x**2 + A*a*b*d**2*x**4/2 + A*b**2*c**2*x**2/2
+ A*b**2*c*d*x**4/2 + A*b**2*d**2*x**6/6 + B*a**2*c**2*log(x) + B*a**2*c*d*x**2 + B*a**2*d**2*x**4/4 + B*a*b*
c**2*x**2 + B*a*b*c*d*x**4 + B*a*b*d**2*x**6/3 + B*b**2*c**2*x**4/4 + B*b**2*c*d*x**6/3 + B*b**2*d**2*x**8/8)/
e**3, Eq(m, -3)), ((A*a**2*c**2*log(x) + A*a**2*c*d*x**2 + A*a**2*d**2*x**4/4 + A*a*b*c**2*x**2 + A*a*b*c*d*x*
*4 + A*a*b*d**2*x**6/3 + A*b**2*c**2*x**4/4 + A*b**2*c*d*x**6/3 + A*b**2*d**2*x**8/8 + B*a**2*c**2*x**2/2 + B*
a**2*c*d*x**4/2 + B*a**2*d**2*x**6/6 + B*a*b*c**2*x**4/2 + 2*B*a*b*c*d*x**6/3 + B*a*b*d**2*x**8/4 + B*b**2*c**
2*x**6/6 + B*b**2*c*d*x**8/4 + B*b**2*d**2*x**10/10)/e, Eq(m, -1)), (A*a**2*c**2*e**m*m**5*x*x**m/(m**6 + 36*m
**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 35*A*a**2*c**2*e**m*m**4*x*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 470*A*a**2*c**2*e**m*m**3*x*x**m/(m**6 + 36*m**5 + 505
*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3010*A*a**2*c**2*e**m*m**2*x*x**m/(m**6 + 36*m**5 + 505*m*
*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 9129*A*a**2*c**2*e**m*m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3
480*m**3 + 12139*m**2 + 19524*m + 10395) + 10395*A*a**2*c**2*e**m*x*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**
3 + 12139*m**2 + 19524*m + 10395) + 2*A*a**2*c*d*e**m*m**5*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 +
12139*m**2 + 19524*m + 10395) + 66*A*a**2*c*d*e**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 121
39*m**2 + 19524*m + 10395) + 812*A*a**2*c*d*e**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139
*m**2 + 19524*m + 10395) + 4524*A*a**2*c*d*e**m*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*
m**2 + 19524*m + 10395) + 10706*A*a**2*c*d*e**m*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**
2 + 19524*m + 10395) + 6930*A*a**2*c*d*e**m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19
524*m + 10395) + A*a**2*d**2*e**m*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + 31*A*a**2*d**2*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +
10395) + 350*A*a**2*d**2*e**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +
10395) + 1730*A*a**2*d**2*e**m*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m +
10395) + 3489*A*a**2*d**2*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + 2079*A*a**2*d**2*e**m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) +
2*A*a*b*c**2*e**m*m**5*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 66*
A*a*b*c**2*e**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 812*A*
a*b*c**2*e**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4524*A*a
*b*c**2*e**m*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 10706*A*a
*b*c**2*e**m*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6930*A*a*b*c
**2*e**m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4*A*a*b*c*d*e**m*m
**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 124*A*a*b*c*d*e**m*m**4
*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1400*A*a*b*c*d*e**m*m**3*x
**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6920*A*a*b*c*d*e**m*m**2*x**
5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 13956*A*a*b*c*d*e**m*m*x**5*x*
*m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 8316*A*a*b*c*d*e**m*x**5*x**m/(m**
6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*A*a*b*d**2*e**m*m**5*x**7*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 58*A*a*b*d**2*e**m*m**4*x**7*x**m/(m**6 + 36*
m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 604*A*a*b*d**2*e**m*m**3*x**7*x**m/(m**6 + 36*m*
*5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2732*A*a*b*d**2*e**m*m**2*x**7*x**m/(m**6 + 36*m**
5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5154*A*a*b*d**2*e**m*m*x**7*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2970*A*a*b*d**2*e**m*x**7*x**m/(m**6 + 36*m**5 + 505*m*
*4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + A*b**2*c**2*e**m*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 31*A*b**2*c**2*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 34
80*m**3 + 12139*m**2 + 19524*m + 10395) + 350*A*b**2*c**2*e**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 348
0*m**3 + 12139*m**2 + 19524*m + 10395) + 1730*A*b**2*c**2*e**m*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 348
0*m**3 + 12139*m**2 + 19524*m + 10395) + 3489*A*b**2*c**2*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m
**3 + 12139*m**2 + 19524*m + 10395) + 2079*A*b**2*c**2*e**m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 +
12139*m**2 + 19524*m + 10395) + 2*A*b**2*c*d*e**m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 121
39*m**2 + 19524*m + 10395) + 58*A*b**2*c*d*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*
m**2 + 19524*m + 10395) + 604*A*b**2*c*d*e**m*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m*
*2 + 19524*m + 10395) + 2732*A*b**2*c*d*e**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**
2 + 19524*m + 10395) + 5154*A*b**2*c*d*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +
19524*m + 10395) + 2970*A*b**2*c*d*e**m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*
m + 10395) + A*b**2*d**2*e**m*m**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1
0395) + 27*A*b**2*d**2*e**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 103
95) + 262*A*b**2*d**2*e**m*m**3*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039
5) + 1122*A*b**2*d**2*e**m*m**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039
5) + 2041*A*b**2*d**2*e**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 1155*A*b**2*d**2*e**m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*a
**2*c**2*e**m*m**5*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 33*B*a**
2*c**2*e**m*m**4*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 406*B*a**2
*c**2*e**m*m**3*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2262*B*a**2
*c**2*e**m*m**2*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 5353*B*a**2
*c**2*e**m*m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3465*B*a**2*c*
*2*e**m*x**3*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*B*a**2*c*d*e**m*m
**5*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 62*B*a**2*c*d*e**m*m**4
*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 700*B*a**2*c*d*e**m*m**3*x
**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3460*B*a**2*c*d*e**m*m**2*x*
*5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6978*B*a**2*c*d*e**m*m*x**5*x
**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4158*B*a**2*c*d*e**m*x**5*x**m/(m
**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*a**2*d**2*e**m*m**5*x**7*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 29*B*a**2*d**2*e**m*m**4*x**7*x**m/(m**6 + 3
6*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 302*B*a**2*d**2*e**m*m**3*x**7*x**m/(m**6 + 36
*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1366*B*a**2*d**2*e**m*m**2*x**7*x**m/(m**6 + 36
*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2577*B*a**2*d**2*e**m*m*x**7*x**m/(m**6 + 36*m*
*5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1485*B*a**2*d**2*e**m*x**7*x**m/(m**6 + 36*m**5 +
505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*B*a*b*c**2*e**m*m**5*x**5*x**m/(m**6 + 36*m**5 + 505*
m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 62*B*a*b*c**2*e**m*m**4*x**5*x**m/(m**6 + 36*m**5 + 505*m**
4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 700*B*a*b*c**2*e**m*m**3*x**5*x**m/(m**6 + 36*m**5 + 505*m**4
+ 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 3460*B*a*b*c**2*e**m*m**2*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 +
3480*m**3 + 12139*m**2 + 19524*m + 10395) + 6978*B*a*b*c**2*e**m*m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 348
0*m**3 + 12139*m**2 + 19524*m + 10395) + 4158*B*a*b*c**2*e**m*x**5*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3
+ 12139*m**2 + 19524*m + 10395) + 4*B*a*b*c*d*e**m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12
139*m**2 + 19524*m + 10395) + 116*B*a*b*c*d*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139
*m**2 + 19524*m + 10395) + 1208*B*a*b*c*d*e**m*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m
**2 + 19524*m + 10395) + 5464*B*a*b*c*d*e**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**
2 + 19524*m + 10395) + 10308*B*a*b*c*d*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 +
19524*m + 10395) + 5940*B*a*b*c*d*e**m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m
+ 10395) + 2*B*a*b*d**2*e**m*m**5*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1
0395) + 54*B*a*b*d**2*e**m*m**4*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 1039
5) + 524*B*a*b*d**2*e**m*m**3*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 2244*B*a*b*d**2*e**m*m**2*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395)
+ 4082*B*a*b*d**2*e**m*m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 23
10*B*a*b*d**2*e**m*x**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*b**2*c
**2*e**m*m**5*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 29*B*b**2*c**
2*e**m*m**4*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 302*B*b**2*c**2
*e**m*m**3*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1366*B*b**2*c**2
*e**m*m**2*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2577*B*b**2*c**2
*e**m*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1485*B*b**2*c**2*e*
*m*x**7*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2*B*b**2*c*d*e**m*m**5*x
**9*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 54*B*b**2*c*d*e**m*m**4*x**9
*x**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 524*B*b**2*c*d*e**m*m**3*x**9*x
**m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2244*B*b**2*c*d*e**m*m**2*x**9*x*
*m/(m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 4082*B*b**2*c*d*e**m*m*x**9*x**m/(
m**6 + 36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 2310*B*b**2*c*d*e**m*x**9*x**m/(m**6 +
36*m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + B*b**2*d**2*e**m*m**5*x**11*x**m/(m**6 + 36*
m**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 25*B*b**2*d**2*e**m*m**4*x**11*x**m/(m**6 + 36*m
**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 230*B*b**2*d**2*e**m*m**3*x**11*x**m/(m**6 + 36*m
**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 950*B*b**2*d**2*e**m*m**2*x**11*x**m/(m**6 + 36*m
**5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 1689*B*b**2*d**2*e**m*m*x**11*x**m/(m**6 + 36*m**
5 + 505*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395) + 945*B*b**2*d**2*e**m*x**11*x**m/(m**6 + 36*m**5 + 5
05*m**4 + 3480*m**3 + 12139*m**2 + 19524*m + 10395), True))

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