3.5.34 \(\int (e x)^m (A+B x) (a+c x^2)^3 \, dx\)

Optimal. Leaf size=169 \[ \frac {a^3 A (e x)^{m+1}}{e (m+1)}+\frac {a^3 B (e x)^{m+2}}{e^2 (m+2)}+\frac {3 a^2 A c (e x)^{m+3}}{e^3 (m+3)}+\frac {3 a^2 B c (e x)^{m+4}}{e^4 (m+4)}+\frac {3 a A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac {3 a B c^2 (e x)^{m+6}}{e^6 (m+6)}+\frac {A c^3 (e x)^{m+7}}{e^7 (m+7)}+\frac {B c^3 (e x)^{m+8}}{e^8 (m+8)} \]

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Rubi [A]  time = 0.10, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {766} \begin {gather*} \frac {3 a^2 A c (e x)^{m+3}}{e^3 (m+3)}+\frac {a^3 A (e x)^{m+1}}{e (m+1)}+\frac {3 a^2 B c (e x)^{m+4}}{e^4 (m+4)}+\frac {a^3 B (e x)^{m+2}}{e^2 (m+2)}+\frac {3 a A c^2 (e x)^{m+5}}{e^5 (m+5)}+\frac {3 a B c^2 (e x)^{m+6}}{e^6 (m+6)}+\frac {A c^3 (e x)^{m+7}}{e^7 (m+7)}+\frac {B c^3 (e x)^{m+8}}{e^8 (m+8)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(A + B*x)*(a + c*x^2)^3,x]

[Out]

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

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 101, normalized size = 0.60 \begin {gather*} x (e x)^m \left (a^3 \left (\frac {A}{m+1}+\frac {B x}{m+2}\right )+3 a^2 c x^2 \left (\frac {A}{m+3}+\frac {B x}{m+4}\right )+3 a c^2 x^4 \left (\frac {A}{m+5}+\frac {B x}{m+6}\right )+c^3 x^6 \left (\frac {A}{m+7}+\frac {B x}{m+8}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(A + B*x)*(a + c*x^2)^3,x]

[Out]

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

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IntegrateAlgebraic [F]  time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m (A+B x) \left (a+c x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(e*x)^m*(A + B*x)*(a + c*x^2)^3,x]

[Out]

Defer[IntegrateAlgebraic][(e*x)^m*(A + B*x)*(a + c*x^2)^3, x]

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fricas [B]  time = 0.45, size = 649, normalized size = 3.84 \begin {gather*} \frac {{\left ({\left (B c^{3} m^{7} + 28 \, B c^{3} m^{6} + 322 \, B c^{3} m^{5} + 1960 \, B c^{3} m^{4} + 6769 \, B c^{3} m^{3} + 13132 \, B c^{3} m^{2} + 13068 \, B c^{3} m + 5040 \, B c^{3}\right )} x^{8} + {\left (A c^{3} m^{7} + 29 \, A c^{3} m^{6} + 343 \, A c^{3} m^{5} + 2135 \, A c^{3} m^{4} + 7504 \, A c^{3} m^{3} + 14756 \, A c^{3} m^{2} + 14832 \, A c^{3} m + 5760 \, A c^{3}\right )} x^{7} + 3 \, {\left (B a c^{2} m^{7} + 30 \, B a c^{2} m^{6} + 366 \, B a c^{2} m^{5} + 2340 \, B a c^{2} m^{4} + 8409 \, B a c^{2} m^{3} + 16830 \, B a c^{2} m^{2} + 17144 \, B a c^{2} m + 6720 \, B a c^{2}\right )} x^{6} + 3 \, {\left (A a c^{2} m^{7} + 31 \, A a c^{2} m^{6} + 391 \, A a c^{2} m^{5} + 2581 \, A a c^{2} m^{4} + 9544 \, A a c^{2} m^{3} + 19564 \, A a c^{2} m^{2} + 20304 \, A a c^{2} m + 8064 \, A a c^{2}\right )} x^{5} + 3 \, {\left (B a^{2} c m^{7} + 32 \, B a^{2} c m^{6} + 418 \, B a^{2} c m^{5} + 2864 \, B a^{2} c m^{4} + 10993 \, B a^{2} c m^{3} + 23312 \, B a^{2} c m^{2} + 24876 \, B a^{2} c m + 10080 \, B a^{2} c\right )} x^{4} + 3 \, {\left (A a^{2} c m^{7} + 33 \, A a^{2} c m^{6} + 447 \, A a^{2} c m^{5} + 3195 \, A a^{2} c m^{4} + 12864 \, A a^{2} c m^{3} + 28692 \, A a^{2} c m^{2} + 32048 \, A a^{2} c m + 13440 \, A a^{2} c\right )} x^{3} + {\left (B a^{3} m^{7} + 34 \, B a^{3} m^{6} + 478 \, B a^{3} m^{5} + 3580 \, B a^{3} m^{4} + 15289 \, B a^{3} m^{3} + 36706 \, B a^{3} m^{2} + 44712 \, B a^{3} m + 20160 \, B a^{3}\right )} x^{2} + {\left (A a^{3} m^{7} + 35 \, A a^{3} m^{6} + 511 \, A a^{3} m^{5} + 4025 \, A a^{3} m^{4} + 18424 \, A a^{3} m^{3} + 48860 \, A a^{3} m^{2} + 69264 \, A a^{3} m + 40320 \, A a^{3}\right )} x\right )} \left (e x\right )^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

((B*c^3*m^7 + 28*B*c^3*m^6 + 322*B*c^3*m^5 + 1960*B*c^3*m^4 + 6769*B*c^3*m^3 + 13132*B*c^3*m^2 + 13068*B*c^3*m
+ 5040*B*c^3)*x^8 + (A*c^3*m^7 + 29*A*c^3*m^6 + 343*A*c^3*m^5 + 2135*A*c^3*m^4 + 7504*A*c^3*m^3 + 14756*A*c^3
*m^2 + 14832*A*c^3*m + 5760*A*c^3)*x^7 + 3*(B*a*c^2*m^7 + 30*B*a*c^2*m^6 + 366*B*a*c^2*m^5 + 2340*B*a*c^2*m^4
+ 8409*B*a*c^2*m^3 + 16830*B*a*c^2*m^2 + 17144*B*a*c^2*m + 6720*B*a*c^2)*x^6 + 3*(A*a*c^2*m^7 + 31*A*a*c^2*m^6
+ 391*A*a*c^2*m^5 + 2581*A*a*c^2*m^4 + 9544*A*a*c^2*m^3 + 19564*A*a*c^2*m^2 + 20304*A*a*c^2*m + 8064*A*a*c^2)
*x^5 + 3*(B*a^2*c*m^7 + 32*B*a^2*c*m^6 + 418*B*a^2*c*m^5 + 2864*B*a^2*c*m^4 + 10993*B*a^2*c*m^3 + 23312*B*a^2*
c*m^2 + 24876*B*a^2*c*m + 10080*B*a^2*c)*x^4 + 3*(A*a^2*c*m^7 + 33*A*a^2*c*m^6 + 447*A*a^2*c*m^5 + 3195*A*a^2*
c*m^4 + 12864*A*a^2*c*m^3 + 28692*A*a^2*c*m^2 + 32048*A*a^2*c*m + 13440*A*a^2*c)*x^3 + (B*a^3*m^7 + 34*B*a^3*m
^6 + 478*B*a^3*m^5 + 3580*B*a^3*m^4 + 15289*B*a^3*m^3 + 36706*B*a^3*m^2 + 44712*B*a^3*m + 20160*B*a^3)*x^2 + (
A*a^3*m^7 + 35*A*a^3*m^6 + 511*A*a^3*m^5 + 4025*A*a^3*m^4 + 18424*A*a^3*m^3 + 48860*A*a^3*m^2 + 69264*A*a^3*m
+ 40320*A*a^3)*x)*(e*x)^m/(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 22449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m +
40320)

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giac [B]  time = 0.20, size = 1102, normalized size = 6.52 \begin {gather*} \frac {B c^{3} m^{7} x^{8} x^{m} e^{m} + A c^{3} m^{7} x^{7} x^{m} e^{m} + 28 \, B c^{3} m^{6} x^{8} x^{m} e^{m} + 3 \, B a c^{2} m^{7} x^{6} x^{m} e^{m} + 29 \, A c^{3} m^{6} x^{7} x^{m} e^{m} + 322 \, B c^{3} m^{5} x^{8} x^{m} e^{m} + 3 \, A a c^{2} m^{7} x^{5} x^{m} e^{m} + 90 \, B a c^{2} m^{6} x^{6} x^{m} e^{m} + 343 \, A c^{3} m^{5} x^{7} x^{m} e^{m} + 1960 \, B c^{3} m^{4} x^{8} x^{m} e^{m} + 3 \, B a^{2} c m^{7} x^{4} x^{m} e^{m} + 93 \, A a c^{2} m^{6} x^{5} x^{m} e^{m} + 1098 \, B a c^{2} m^{5} x^{6} x^{m} e^{m} + 2135 \, A c^{3} m^{4} x^{7} x^{m} e^{m} + 6769 \, B c^{3} m^{3} x^{8} x^{m} e^{m} + 3 \, A a^{2} c m^{7} x^{3} x^{m} e^{m} + 96 \, B a^{2} c m^{6} x^{4} x^{m} e^{m} + 1173 \, A a c^{2} m^{5} x^{5} x^{m} e^{m} + 7020 \, B a c^{2} m^{4} x^{6} x^{m} e^{m} + 7504 \, A c^{3} m^{3} x^{7} x^{m} e^{m} + 13132 \, B c^{3} m^{2} x^{8} x^{m} e^{m} + B a^{3} m^{7} x^{2} x^{m} e^{m} + 99 \, A a^{2} c m^{6} x^{3} x^{m} e^{m} + 1254 \, B a^{2} c m^{5} x^{4} x^{m} e^{m} + 7743 \, A a c^{2} m^{4} x^{5} x^{m} e^{m} + 25227 \, B a c^{2} m^{3} x^{6} x^{m} e^{m} + 14756 \, A c^{3} m^{2} x^{7} x^{m} e^{m} + 13068 \, B c^{3} m x^{8} x^{m} e^{m} + A a^{3} m^{7} x x^{m} e^{m} + 34 \, B a^{3} m^{6} x^{2} x^{m} e^{m} + 1341 \, A a^{2} c m^{5} x^{3} x^{m} e^{m} + 8592 \, B a^{2} c m^{4} x^{4} x^{m} e^{m} + 28632 \, A a c^{2} m^{3} x^{5} x^{m} e^{m} + 50490 \, B a c^{2} m^{2} x^{6} x^{m} e^{m} + 14832 \, A c^{3} m x^{7} x^{m} e^{m} + 5040 \, B c^{3} x^{8} x^{m} e^{m} + 35 \, A a^{3} m^{6} x x^{m} e^{m} + 478 \, B a^{3} m^{5} x^{2} x^{m} e^{m} + 9585 \, A a^{2} c m^{4} x^{3} x^{m} e^{m} + 32979 \, B a^{2} c m^{3} x^{4} x^{m} e^{m} + 58692 \, A a c^{2} m^{2} x^{5} x^{m} e^{m} + 51432 \, B a c^{2} m x^{6} x^{m} e^{m} + 5760 \, A c^{3} x^{7} x^{m} e^{m} + 511 \, A a^{3} m^{5} x x^{m} e^{m} + 3580 \, B a^{3} m^{4} x^{2} x^{m} e^{m} + 38592 \, A a^{2} c m^{3} x^{3} x^{m} e^{m} + 69936 \, B a^{2} c m^{2} x^{4} x^{m} e^{m} + 60912 \, A a c^{2} m x^{5} x^{m} e^{m} + 20160 \, B a c^{2} x^{6} x^{m} e^{m} + 4025 \, A a^{3} m^{4} x x^{m} e^{m} + 15289 \, B a^{3} m^{3} x^{2} x^{m} e^{m} + 86076 \, A a^{2} c m^{2} x^{3} x^{m} e^{m} + 74628 \, B a^{2} c m x^{4} x^{m} e^{m} + 24192 \, A a c^{2} x^{5} x^{m} e^{m} + 18424 \, A a^{3} m^{3} x x^{m} e^{m} + 36706 \, B a^{3} m^{2} x^{2} x^{m} e^{m} + 96144 \, A a^{2} c m x^{3} x^{m} e^{m} + 30240 \, B a^{2} c x^{4} x^{m} e^{m} + 48860 \, A a^{3} m^{2} x x^{m} e^{m} + 44712 \, B a^{3} m x^{2} x^{m} e^{m} + 40320 \, A a^{2} c x^{3} x^{m} e^{m} + 69264 \, A a^{3} m x x^{m} e^{m} + 20160 \, B a^{3} x^{2} x^{m} e^{m} + 40320 \, A a^{3} x x^{m} e^{m}}{m^{8} + 36 \, m^{7} + 546 \, m^{6} + 4536 \, m^{5} + 22449 \, m^{4} + 67284 \, m^{3} + 118124 \, m^{2} + 109584 \, m + 40320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+a)^3,x, algorithm="giac")

[Out]

(B*c^3*m^7*x^8*x^m*e^m + A*c^3*m^7*x^7*x^m*e^m + 28*B*c^3*m^6*x^8*x^m*e^m + 3*B*a*c^2*m^7*x^6*x^m*e^m + 29*A*c
^3*m^6*x^7*x^m*e^m + 322*B*c^3*m^5*x^8*x^m*e^m + 3*A*a*c^2*m^7*x^5*x^m*e^m + 90*B*a*c^2*m^6*x^6*x^m*e^m + 343*
A*c^3*m^5*x^7*x^m*e^m + 1960*B*c^3*m^4*x^8*x^m*e^m + 3*B*a^2*c*m^7*x^4*x^m*e^m + 93*A*a*c^2*m^6*x^5*x^m*e^m +
1098*B*a*c^2*m^5*x^6*x^m*e^m + 2135*A*c^3*m^4*x^7*x^m*e^m + 6769*B*c^3*m^3*x^8*x^m*e^m + 3*A*a^2*c*m^7*x^3*x^m
*e^m + 96*B*a^2*c*m^6*x^4*x^m*e^m + 1173*A*a*c^2*m^5*x^5*x^m*e^m + 7020*B*a*c^2*m^4*x^6*x^m*e^m + 7504*A*c^3*m
^3*x^7*x^m*e^m + 13132*B*c^3*m^2*x^8*x^m*e^m + B*a^3*m^7*x^2*x^m*e^m + 99*A*a^2*c*m^6*x^3*x^m*e^m + 1254*B*a^2
*c*m^5*x^4*x^m*e^m + 7743*A*a*c^2*m^4*x^5*x^m*e^m + 25227*B*a*c^2*m^3*x^6*x^m*e^m + 14756*A*c^3*m^2*x^7*x^m*e^
m + 13068*B*c^3*m*x^8*x^m*e^m + A*a^3*m^7*x*x^m*e^m + 34*B*a^3*m^6*x^2*x^m*e^m + 1341*A*a^2*c*m^5*x^3*x^m*e^m
+ 8592*B*a^2*c*m^4*x^4*x^m*e^m + 28632*A*a*c^2*m^3*x^5*x^m*e^m + 50490*B*a*c^2*m^2*x^6*x^m*e^m + 14832*A*c^3*m
*x^7*x^m*e^m + 5040*B*c^3*x^8*x^m*e^m + 35*A*a^3*m^6*x*x^m*e^m + 478*B*a^3*m^5*x^2*x^m*e^m + 9585*A*a^2*c*m^4*
x^3*x^m*e^m + 32979*B*a^2*c*m^3*x^4*x^m*e^m + 58692*A*a*c^2*m^2*x^5*x^m*e^m + 51432*B*a*c^2*m*x^6*x^m*e^m + 57
60*A*c^3*x^7*x^m*e^m + 511*A*a^3*m^5*x*x^m*e^m + 3580*B*a^3*m^4*x^2*x^m*e^m + 38592*A*a^2*c*m^3*x^3*x^m*e^m +
69936*B*a^2*c*m^2*x^4*x^m*e^m + 60912*A*a*c^2*m*x^5*x^m*e^m + 20160*B*a*c^2*x^6*x^m*e^m + 4025*A*a^3*m^4*x*x^m
*e^m + 15289*B*a^3*m^3*x^2*x^m*e^m + 86076*A*a^2*c*m^2*x^3*x^m*e^m + 74628*B*a^2*c*m*x^4*x^m*e^m + 24192*A*a*c
^2*x^5*x^m*e^m + 18424*A*a^3*m^3*x*x^m*e^m + 36706*B*a^3*m^2*x^2*x^m*e^m + 96144*A*a^2*c*m*x^3*x^m*e^m + 30240
*B*a^2*c*x^4*x^m*e^m + 48860*A*a^3*m^2*x*x^m*e^m + 44712*B*a^3*m*x^2*x^m*e^m + 40320*A*a^2*c*x^3*x^m*e^m + 692
64*A*a^3*m*x*x^m*e^m + 20160*B*a^3*x^2*x^m*e^m + 40320*A*a^3*x*x^m*e^m)/(m^8 + 36*m^7 + 546*m^6 + 4536*m^5 + 2
2449*m^4 + 67284*m^3 + 118124*m^2 + 109584*m + 40320)

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maple [B]  time = 0.06, size = 765, normalized size = 4.53 \begin {gather*} \frac {\left (B \,c^{3} m^{7} x^{7}+A \,c^{3} m^{7} x^{6}+28 B \,c^{3} m^{6} x^{7}+29 A \,c^{3} m^{6} x^{6}+3 B a \,c^{2} m^{7} x^{5}+322 B \,c^{3} m^{5} x^{7}+3 A a \,c^{2} m^{7} x^{4}+343 A \,c^{3} m^{5} x^{6}+90 B a \,c^{2} m^{6} x^{5}+1960 B \,c^{3} m^{4} x^{7}+93 A a \,c^{2} m^{6} x^{4}+2135 A \,c^{3} m^{4} x^{6}+3 B \,a^{2} c \,m^{7} x^{3}+1098 B a \,c^{2} m^{5} x^{5}+6769 B \,c^{3} m^{3} x^{7}+3 A \,a^{2} c \,m^{7} x^{2}+1173 A a \,c^{2} m^{5} x^{4}+7504 A \,c^{3} m^{3} x^{6}+96 B \,a^{2} c \,m^{6} x^{3}+7020 B a \,c^{2} m^{4} x^{5}+13132 B \,c^{3} m^{2} x^{7}+99 A \,a^{2} c \,m^{6} x^{2}+7743 A a \,c^{2} m^{4} x^{4}+14756 A \,c^{3} m^{2} x^{6}+B \,a^{3} m^{7} x +1254 B \,a^{2} c \,m^{5} x^{3}+25227 B a \,c^{2} m^{3} x^{5}+13068 B \,c^{3} m \,x^{7}+A \,a^{3} m^{7}+1341 A \,a^{2} c \,m^{5} x^{2}+28632 A a \,c^{2} m^{3} x^{4}+14832 A \,c^{3} m \,x^{6}+34 B \,a^{3} m^{6} x +8592 B \,a^{2} c \,m^{4} x^{3}+50490 B a \,c^{2} m^{2} x^{5}+5040 B \,c^{3} x^{7}+35 A \,a^{3} m^{6}+9585 A \,a^{2} c \,m^{4} x^{2}+58692 A a \,c^{2} m^{2} x^{4}+5760 A \,c^{3} x^{6}+478 B \,a^{3} m^{5} x +32979 B \,a^{2} c \,m^{3} x^{3}+51432 B a \,c^{2} m \,x^{5}+511 A \,a^{3} m^{5}+38592 A \,a^{2} c \,m^{3} x^{2}+60912 A a \,c^{2} m \,x^{4}+3580 B \,a^{3} m^{4} x +69936 B \,a^{2} c \,m^{2} x^{3}+20160 B a \,c^{2} x^{5}+4025 A \,a^{3} m^{4}+86076 A \,a^{2} c \,m^{2} x^{2}+24192 A a \,c^{2} x^{4}+15289 B \,a^{3} m^{3} x +74628 B \,a^{2} c m \,x^{3}+18424 A \,a^{3} m^{3}+96144 A \,a^{2} c m \,x^{2}+36706 B \,a^{3} m^{2} x +30240 B \,a^{2} c \,x^{3}+48860 A \,a^{3} m^{2}+40320 A \,a^{2} c \,x^{2}+44712 B \,a^{3} m x +69264 A \,a^{3} m +20160 B \,a^{3} x +40320 A \,a^{3}\right ) x \left (e x \right )^{m}}{\left (m +8\right ) \left (m +7\right ) \left (m +6\right ) \left (m +5\right ) \left (m +4\right ) \left (m +3\right ) \left (m +2\right ) \left (m +1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(B*x+A)*(c*x^2+a)^3,x)

[Out]

x*(B*c^3*m^7*x^7+A*c^3*m^7*x^6+28*B*c^3*m^6*x^7+29*A*c^3*m^6*x^6+3*B*a*c^2*m^7*x^5+322*B*c^3*m^5*x^7+3*A*a*c^2
*m^7*x^4+343*A*c^3*m^5*x^6+90*B*a*c^2*m^6*x^5+1960*B*c^3*m^4*x^7+93*A*a*c^2*m^6*x^4+2135*A*c^3*m^4*x^6+3*B*a^2
*c*m^7*x^3+1098*B*a*c^2*m^5*x^5+6769*B*c^3*m^3*x^7+3*A*a^2*c*m^7*x^2+1173*A*a*c^2*m^5*x^4+7504*A*c^3*m^3*x^6+9
6*B*a^2*c*m^6*x^3+7020*B*a*c^2*m^4*x^5+13132*B*c^3*m^2*x^7+99*A*a^2*c*m^6*x^2+7743*A*a*c^2*m^4*x^4+14756*A*c^3
*m^2*x^6+B*a^3*m^7*x+1254*B*a^2*c*m^5*x^3+25227*B*a*c^2*m^3*x^5+13068*B*c^3*m*x^7+A*a^3*m^7+1341*A*a^2*c*m^5*x
^2+28632*A*a*c^2*m^3*x^4+14832*A*c^3*m*x^6+34*B*a^3*m^6*x+8592*B*a^2*c*m^4*x^3+50490*B*a*c^2*m^2*x^5+5040*B*c^
3*x^7+35*A*a^3*m^6+9585*A*a^2*c*m^4*x^2+58692*A*a*c^2*m^2*x^4+5760*A*c^3*x^6+478*B*a^3*m^5*x+32979*B*a^2*c*m^3
*x^3+51432*B*a*c^2*m*x^5+511*A*a^3*m^5+38592*A*a^2*c*m^3*x^2+60912*A*a*c^2*m*x^4+3580*B*a^3*m^4*x+69936*B*a^2*
c*m^2*x^3+20160*B*a*c^2*x^5+4025*A*a^3*m^4+86076*A*a^2*c*m^2*x^2+24192*A*a*c^2*x^4+15289*B*a^3*m^3*x+74628*B*a
^2*c*m*x^3+18424*A*a^3*m^3+96144*A*a^2*c*m*x^2+36706*B*a^3*m^2*x+30240*B*a^2*c*x^3+48860*A*a^3*m^2+40320*A*a^2
*c*x^2+44712*B*a^3*m*x+69264*A*a^3*m+20160*B*a^3*x+40320*A*a^3)*(e*x)^m/(m+8)/(m+7)/(m+6)/(m+5)/(m+4)/(m+3)/(m
+2)/(m+1)

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maxima [A]  time = 0.62, size = 162, normalized size = 0.96 \begin {gather*} \frac {B c^{3} e^{m} x^{8} x^{m}}{m + 8} + \frac {A c^{3} e^{m} x^{7} x^{m}}{m + 7} + \frac {3 \, B a c^{2} e^{m} x^{6} x^{m}}{m + 6} + \frac {3 \, A a c^{2} e^{m} x^{5} x^{m}}{m + 5} + \frac {3 \, B a^{2} c e^{m} x^{4} x^{m}}{m + 4} + \frac {3 \, A a^{2} c e^{m} x^{3} x^{m}}{m + 3} + \frac {B a^{3} e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} A a^{3}}{e {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(B*x+A)*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.48, size = 695, normalized size = 4.11 \begin {gather*} \frac {A\,a^3\,x\,{\left (e\,x\right )}^m\,\left (m^7+35\,m^6+511\,m^5+4025\,m^4+18424\,m^3+48860\,m^2+69264\,m+40320\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {B\,a^3\,x^2\,{\left (e\,x\right )}^m\,\left (m^7+34\,m^6+478\,m^5+3580\,m^4+15289\,m^3+36706\,m^2+44712\,m+20160\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {A\,c^3\,x^7\,{\left (e\,x\right )}^m\,\left (m^7+29\,m^6+343\,m^5+2135\,m^4+7504\,m^3+14756\,m^2+14832\,m+5760\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {B\,c^3\,x^8\,{\left (e\,x\right )}^m\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,A\,a\,c^2\,x^5\,{\left (e\,x\right )}^m\,\left (m^7+31\,m^6+391\,m^5+2581\,m^4+9544\,m^3+19564\,m^2+20304\,m+8064\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,A\,a^2\,c\,x^3\,{\left (e\,x\right )}^m\,\left (m^7+33\,m^6+447\,m^5+3195\,m^4+12864\,m^3+28692\,m^2+32048\,m+13440\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,B\,a\,c^2\,x^6\,{\left (e\,x\right )}^m\,\left (m^7+30\,m^6+366\,m^5+2340\,m^4+8409\,m^3+16830\,m^2+17144\,m+6720\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320}+\frac {3\,B\,a^2\,c\,x^4\,{\left (e\,x\right )}^m\,\left (m^7+32\,m^6+418\,m^5+2864\,m^4+10993\,m^3+23312\,m^2+24876\,m+10080\right )}{m^8+36\,m^7+546\,m^6+4536\,m^5+22449\,m^4+67284\,m^3+118124\,m^2+109584\,m+40320} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(a + c*x^2)^3*(A + B*x),x)

[Out]

(A*a^3*x*(e*x)^m*(69264*m + 48860*m^2 + 18424*m^3 + 4025*m^4 + 511*m^5 + 35*m^6 + m^7 + 40320))/(109584*m + 11
8124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (B*a^3*x^2*(e*x)^m*(44712*m +
36706*m^2 + 15289*m^3 + 3580*m^4 + 478*m^5 + 34*m^6 + m^7 + 20160))/(109584*m + 118124*m^2 + 67284*m^3 + 22449
*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (A*c^3*x^7*(e*x)^m*(14832*m + 14756*m^2 + 7504*m^3 + 2135*
m^4 + 343*m^5 + 29*m^6 + m^7 + 5760))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36
*m^7 + m^8 + 40320) + (B*c^3*x^8*(e*x)^m*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 +
5040))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (3*A*a*c
^2*x^5*(e*x)^m*(20304*m + 19564*m^2 + 9544*m^3 + 2581*m^4 + 391*m^5 + 31*m^6 + m^7 + 8064))/(109584*m + 118124
*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (3*A*a^2*c*x^3*(e*x)^m*(32048*m +
28692*m^2 + 12864*m^3 + 3195*m^4 + 447*m^5 + 33*m^6 + m^7 + 13440))/(109584*m + 118124*m^2 + 67284*m^3 + 22449
*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320) + (3*B*a*c^2*x^6*(e*x)^m*(17144*m + 16830*m^2 + 8409*m^3 + 2
340*m^4 + 366*m^5 + 30*m^6 + m^7 + 6720))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6
+ 36*m^7 + m^8 + 40320) + (3*B*a^2*c*x^4*(e*x)^m*(24876*m + 23312*m^2 + 10993*m^3 + 2864*m^4 + 418*m^5 + 32*m^
6 + m^7 + 10080))/(109584*m + 118124*m^2 + 67284*m^3 + 22449*m^4 + 4536*m^5 + 546*m^6 + 36*m^7 + m^8 + 40320)

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sympy [A]  time = 3.18, size = 4507, normalized size = 26.67

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(B*x+A)*(c*x**2+a)**3,x)

[Out]

Piecewise(((-A*a**3/(7*x**7) - 3*A*a**2*c/(5*x**5) - A*a*c**2/x**3 - A*c**3/x - B*a**3/(6*x**6) - 3*B*a**2*c/(
4*x**4) - 3*B*a*c**2/(2*x**2) + B*c**3*log(x))/e**8, Eq(m, -8)), ((-A*a**3/(6*x**6) - 3*A*a**2*c/(4*x**4) - 3*
A*a*c**2/(2*x**2) + A*c**3*log(x) - B*a**3/(5*x**5) - B*a**2*c/x**3 - 3*B*a*c**2/x + B*c**3*x)/e**7, Eq(m, -7)
), ((-A*a**3/(5*x**5) - A*a**2*c/x**3 - 3*A*a*c**2/x + A*c**3*x - B*a**3/(4*x**4) - 3*B*a**2*c/(2*x**2) + 3*B*
a*c**2*log(x) + B*c**3*x**2/2)/e**6, Eq(m, -6)), ((-A*a**3/(4*x**4) - 3*A*a**2*c/(2*x**2) + 3*A*a*c**2*log(x)
+ A*c**3*x**2/2 - B*a**3/(3*x**3) - 3*B*a**2*c/x + 3*B*a*c**2*x + B*c**3*x**3/3)/e**5, Eq(m, -5)), ((-A*a**3/(
3*x**3) - 3*A*a**2*c/x + 3*A*a*c**2*x + A*c**3*x**3/3 - B*a**3/(2*x**2) + 3*B*a**2*c*log(x) + 3*B*a*c**2*x**2/
2 + B*c**3*x**4/4)/e**4, Eq(m, -4)), ((-A*a**3/(2*x**2) + 3*A*a**2*c*log(x) + 3*A*a*c**2*x**2/2 + A*c**3*x**4/
4 - B*a**3/x + 3*B*a**2*c*x + B*a*c**2*x**3 + B*c**3*x**5/5)/e**3, Eq(m, -3)), ((-A*a**3/x + 3*A*a**2*c*x + A*
a*c**2*x**3 + A*c**3*x**5/5 + B*a**3*log(x) + 3*B*a**2*c*x**2/2 + 3*B*a*c**2*x**4/4 + B*c**3*x**6/6)/e**2, Eq(
m, -2)), ((A*a**3*log(x) + 3*A*a**2*c*x**2/2 + 3*A*a*c**2*x**4/4 + A*c**3*x**6/6 + B*a**3*x + B*a**2*c*x**3 +
3*B*a*c**2*x**5/5 + B*c**3*x**7/7)/e, Eq(m, -1)), (A*a**3*e**m*m**7*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m
**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 35*A*a**3*e**m*m**6*x*x**m/(m**8 + 36*m**7 +
546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 511*A*a**3*e**m*m**5*x*x**
m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 4025*A*
a**3*e**m*m**4*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*
m + 40320) + 18424*A*a**3*e**m*m**3*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 +
118124*m**2 + 109584*m + 40320) + 48860*A*a**3*e**m*m**2*x*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449
*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 69264*A*a**3*e**m*m*x*x**m/(m**8 + 36*m**7 + 546*m**6 +
4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 40320*A*a**3*e**m*x*x**m/(m**8 + 36*m
**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 3*A*a**2*c*e**m*m**7*
x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) +
99*A*a**2*c*e**m*m**6*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**
2 + 109584*m + 40320) + 1341*A*a**2*c*e**m*m**5*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4
+ 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 9585*A*a**2*c*e**m*m**4*x**3*x**m/(m**8 + 36*m**7 + 546*m**6
+ 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 38592*A*a**2*c*e**m*m**3*x**3*x**m/(
m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 86076*A*a*
*2*c*e**m*m**2*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 1095
84*m + 40320) + 96144*A*a**2*c*e**m*m*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m*
*3 + 118124*m**2 + 109584*m + 40320) + 40320*A*a**2*c*e**m*x**3*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 +
22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 3*A*a*c**2*e**m*m**7*x**5*x**m/(m**8 + 36*m**7 + 5
46*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 93*A*a*c**2*e**m*m**6*x**5*x
**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 1173*
A*a*c**2*e**m*m**5*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 +
109584*m + 40320) + 7743*A*a*c**2*e**m*m**4*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67
284*m**3 + 118124*m**2 + 109584*m + 40320) + 28632*A*a*c**2*e**m*m**3*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4
536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 58692*A*a*c**2*e**m*m**2*x**5*x**m/(m**
8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 60912*A*a*c**
2*e**m*m*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m +
40320) + 24192*A*a*c**2*e**m*x**5*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118
124*m**2 + 109584*m + 40320) + A*c**3*e**m*m**7*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4
+ 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 29*A*c**3*e**m*m**6*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 45
36*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 343*A*c**3*e**m*m**5*x**7*x**m/(m**8 + 3
6*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 2135*A*c**3*e**m*m
**4*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 4032
0) + 7504*A*c**3*e**m*m**3*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124
*m**2 + 109584*m + 40320) + 14756*A*c**3*e**m*m**2*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m*
*4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 14832*A*c**3*e**m*m*x**7*x**m/(m**8 + 36*m**7 + 546*m**6 +
4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 5760*A*c**3*e**m*x**7*x**m/(m**8 + 36
*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + B*a**3*e**m*m**7*x*
*2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 3
4*B*a**3*e**m*m**6*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 +
109584*m + 40320) + 478*B*a**3*e**m*m**5*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284
*m**3 + 118124*m**2 + 109584*m + 40320) + 3580*B*a**3*e**m*m**4*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m*
*5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 15289*B*a**3*e**m*m**3*x**2*x**m/(m**8 + 36*m
**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 36706*B*a**3*e**m*m**
2*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320)
+ 44712*B*a**3*e**m*m*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**
2 + 109584*m + 40320) + 20160*B*a**3*e**m*x**2*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 6728
4*m**3 + 118124*m**2 + 109584*m + 40320) + 3*B*a**2*c*e**m*m**7*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m*
*5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 96*B*a**2*c*e**m*m**6*x**4*x**m/(m**8 + 36*m*
*7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 1254*B*a**2*c*e**m*m**
5*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320)
+ 8592*B*a**2*c*e**m*m**4*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124
*m**2 + 109584*m + 40320) + 32979*B*a**2*c*e**m*m**3*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*
m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 69936*B*a**2*c*e**m*m**2*x**4*x**m/(m**8 + 36*m**7 + 546
*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 74628*B*a**2*c*e**m*m*x**4*x**
m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 30240*B
*a**2*c*e**m*x**4*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584
*m + 40320) + 3*B*a*c**2*e**m*m**7*x**6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3
+ 118124*m**2 + 109584*m + 40320) + 90*B*a*c**2*e**m*m**6*x**6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 2
2449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 1098*B*a*c**2*e**m*m**5*x**6*x**m/(m**8 + 36*m**7 +
546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 7020*B*a*c**2*e**m*m**4*x*
*6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 2
5227*B*a*c**2*e**m*m**3*x**6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m*
*2 + 109584*m + 40320) + 50490*B*a*c**2*e**m*m**2*x**6*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**
4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 51432*B*a*c**2*e**m*m*x**6*x**m/(m**8 + 36*m**7 + 546*m**6
+ 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 20160*B*a*c**2*e**m*x**6*x**m/(m**8
+ 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + B*c**3*e**m*m**
7*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320)
+ 28*B*c**3*e**m*m**6*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**
2 + 109584*m + 40320) + 322*B*c**3*e**m*m**5*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 6
7284*m**3 + 118124*m**2 + 109584*m + 40320) + 1960*B*c**3*e**m*m**4*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 453
6*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 6769*B*c**3*e**m*m**3*x**8*x**m/(m**8 + 3
6*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 40320) + 13132*B*c**3*e**m*
m**2*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*m**2 + 109584*m + 403
20) + 13068*B*c**3*e**m*m*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67284*m**3 + 118124*
m**2 + 109584*m + 40320) + 5040*B*c**3*e**m*x**8*x**m/(m**8 + 36*m**7 + 546*m**6 + 4536*m**5 + 22449*m**4 + 67
284*m**3 + 118124*m**2 + 109584*m + 40320), True))

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