3.2551 \(\int (d+e x)^m (a+b x+c x^2)^2 \, dx\)
Optimal. Leaf size=178 \[ \frac {(d+e x)^{m+3} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 (m+3)}+\frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^5 (m+1)}-\frac {2 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )}{e^5 (m+2)}-\frac {2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]
[Out]
(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(1+m)/e^5/(1+m)-2*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(e*x+d)^(2+m)/e^5/(2+m)+(6*c^
2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x+d)^(3+m)/e^5/(3+m)-2*c*(-b*e+2*c*d)*(e*x+d)^(4+m)/e^5/(4+m)+c^2*(e*x+d)
^(5+m)/e^5/(5+m)
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Rubi [A] time = 0.11, antiderivative size = 178, 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 =
{698} \[ \frac {(d+e x)^{m+3} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 (m+3)}+\frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^5 (m+1)}-\frac {2 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )}{e^5 (m+2)}-\frac {2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^m*(a + b*x + c*x^2)^2,x]
[Out]
((c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*
x)^(2 + m))/(e^5*(2 + m)) + ((6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(3 + m))/(e^5*(3 + m)) - (2
*c*(2*c*d - b*e)*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (c^2*(d + e*x)^(5 + m))/(e^5*(5 + m))
Rule 698
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))
Rubi steps
\begin {align*} \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^m}{e^4}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{1+m}}{e^4}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{2+m}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{3+m}}{e^4}+\frac {c^2 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {2 c (2 c d-b e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 178, normalized size = 1.00 \[ \frac {(d+e x)^{m+1} \left (\frac {2 (d+e x) \left (\frac {6 (2 c d-b e) \left (e (a e-b d)+c d^2\right )}{m+2}-\frac {(d+e x) \left (4 c e (a e (m+4)-3 b d)-b^2 e^2 (m+1)+12 c^2 d^2\right )}{m+3}\right )}{e^4 (m+4) (m+5)}-\frac {2 (d+e x) (a+x (b+c x)) (b e (m+7)-6 c d+2 c e (m+4) x)}{e^2 (m+4) (m+5)}+(a+x (b+c x))^2\right )}{e (m+1)} \]
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^m*(a + b*x + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*((-2*(d + e*x)*(-6*c*d + b*e*(7 + m) + 2*c*e*(4 + m)*x)*(a + x*(b + c*x)))/(e^2*(4 + m)*(5
+ m)) + (a + x*(b + c*x))^2 + (2*(d + e*x)*((6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(2 + m) - ((12*c^2*d^
2 - b^2*e^2*(1 + m) + 4*c*e*(-3*b*d + a*e*(4 + m)))*(d + e*x))/(3 + m)))/(e^4*(4 + m)*(5 + m))))/(e*(1 + m))
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fricas [B] time = 1.13, size = 869, normalized size = 4.88 \[ \frac {{\left (a^{2} d e^{4} m^{4} + 24 \, c^{2} d^{5} - 60 \, b c d^{4} e - 120 \, a b d^{2} e^{3} + 120 \, a^{2} d e^{4} + 40 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + {\left (c^{2} e^{5} m^{4} + 10 \, c^{2} e^{5} m^{3} + 35 \, c^{2} e^{5} m^{2} + 50 \, c^{2} e^{5} m + 24 \, c^{2} e^{5}\right )} x^{5} + {\left (60 \, b c e^{5} + {\left (c^{2} d e^{4} + 2 \, b c e^{5}\right )} m^{4} + 2 \, {\left (3 \, c^{2} d e^{4} + 11 \, b c e^{5}\right )} m^{3} + {\left (11 \, c^{2} d e^{4} + 82 \, b c e^{5}\right )} m^{2} + 2 \, {\left (3 \, c^{2} d e^{4} + 61 \, b c e^{5}\right )} m\right )} x^{4} - 2 \, {\left (a b d^{2} e^{3} - 7 \, a^{2} d e^{4}\right )} m^{3} + {\left (40 \, {\left (b^{2} + 2 \, a c\right )} e^{5} + {\left (2 \, b c d e^{4} + {\left (b^{2} + 2 \, a c\right )} e^{5}\right )} m^{4} - 4 \, {\left (c^{2} d^{2} e^{3} - 4 \, b c d e^{4} - 3 \, {\left (b^{2} + 2 \, a c\right )} e^{5}\right )} m^{3} - {\left (12 \, c^{2} d^{2} e^{3} - 34 \, b c d e^{4} - 49 \, {\left (b^{2} + 2 \, a c\right )} e^{5}\right )} m^{2} - 2 \, {\left (4 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} - 39 \, {\left (b^{2} + 2 \, a c\right )} e^{5}\right )} m\right )} x^{3} - {\left (24 \, a b d^{2} e^{3} - 71 \, a^{2} d e^{4} - 2 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2}\right )} m^{2} + {\left (120 \, a b e^{5} + {\left (2 \, a b e^{5} + {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} m^{4} - 2 \, {\left (3 \, b c d^{2} e^{3} - 13 \, a b e^{5} - 5 \, {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} m^{3} + {\left (12 \, c^{2} d^{3} e^{2} - 36 \, b c d^{2} e^{3} + 118 \, a b e^{5} + 29 \, {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} m^{2} + 2 \, {\left (6 \, c^{2} d^{3} e^{2} - 15 \, b c d^{2} e^{3} + 107 \, a b e^{5} + 10 \, {\left (b^{2} + 2 \, a c\right )} d e^{4}\right )} m\right )} x^{2} - 2 \, {\left (6 \, b c d^{4} e + 47 \, a b d^{2} e^{3} - 77 \, a^{2} d e^{4} - 9 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2}\right )} m + {\left (120 \, a^{2} e^{5} + {\left (2 \, a b d e^{4} + a^{2} e^{5}\right )} m^{4} + 2 \, {\left (12 \, a b d e^{4} + 7 \, a^{2} e^{5} - {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} m^{3} + {\left (12 \, b c d^{3} e^{2} + 94 \, a b d e^{4} + 71 \, a^{2} e^{5} - 18 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} m^{2} - 2 \, {\left (12 \, c^{2} d^{4} e - 30 \, b c d^{3} e^{2} - 60 \, a b d e^{4} - 77 \, a^{2} e^{5} + 20 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(c*x^2+b*x+a)^2,x, algorithm="fricas")
[Out]
(a^2*d*e^4*m^4 + 24*c^2*d^5 - 60*b*c*d^4*e - 120*a*b*d^2*e^3 + 120*a^2*d*e^4 + 40*(b^2 + 2*a*c)*d^3*e^2 + (c^2
*e^5*m^4 + 10*c^2*e^5*m^3 + 35*c^2*e^5*m^2 + 50*c^2*e^5*m + 24*c^2*e^5)*x^5 + (60*b*c*e^5 + (c^2*d*e^4 + 2*b*c
*e^5)*m^4 + 2*(3*c^2*d*e^4 + 11*b*c*e^5)*m^3 + (11*c^2*d*e^4 + 82*b*c*e^5)*m^2 + 2*(3*c^2*d*e^4 + 61*b*c*e^5)*
m)*x^4 - 2*(a*b*d^2*e^3 - 7*a^2*d*e^4)*m^3 + (40*(b^2 + 2*a*c)*e^5 + (2*b*c*d*e^4 + (b^2 + 2*a*c)*e^5)*m^4 - 4
*(c^2*d^2*e^3 - 4*b*c*d*e^4 - 3*(b^2 + 2*a*c)*e^5)*m^3 - (12*c^2*d^2*e^3 - 34*b*c*d*e^4 - 49*(b^2 + 2*a*c)*e^5
)*m^2 - 2*(4*c^2*d^2*e^3 - 10*b*c*d*e^4 - 39*(b^2 + 2*a*c)*e^5)*m)*x^3 - (24*a*b*d^2*e^3 - 71*a^2*d*e^4 - 2*(b
^2 + 2*a*c)*d^3*e^2)*m^2 + (120*a*b*e^5 + (2*a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*m^4 - 2*(3*b*c*d^2*e^3 - 13*a*b*e^
5 - 5*(b^2 + 2*a*c)*d*e^4)*m^3 + (12*c^2*d^3*e^2 - 36*b*c*d^2*e^3 + 118*a*b*e^5 + 29*(b^2 + 2*a*c)*d*e^4)*m^2
+ 2*(6*c^2*d^3*e^2 - 15*b*c*d^2*e^3 + 107*a*b*e^5 + 10*(b^2 + 2*a*c)*d*e^4)*m)*x^2 - 2*(6*b*c*d^4*e + 47*a*b*d
^2*e^3 - 77*a^2*d*e^4 - 9*(b^2 + 2*a*c)*d^3*e^2)*m + (120*a^2*e^5 + (2*a*b*d*e^4 + a^2*e^5)*m^4 + 2*(12*a*b*d*
e^4 + 7*a^2*e^5 - (b^2 + 2*a*c)*d^2*e^3)*m^3 + (12*b*c*d^3*e^2 + 94*a*b*d*e^4 + 71*a^2*e^5 - 18*(b^2 + 2*a*c)*
d^2*e^3)*m^2 - 2*(12*c^2*d^4*e - 30*b*c*d^3*e^2 - 60*a*b*d*e^4 - 77*a^2*e^5 + 20*(b^2 + 2*a*c)*d^2*e^3)*m)*x)*
(e*x + d)^m/(e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e^5*m^2 + 274*e^5*m + 120*e^5)
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giac [B] time = 0.23, size = 1717, normalized size = 9.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(c*x^2+b*x+a)^2,x, algorithm="giac")
[Out]
((x*e + d)^m*c^2*m^4*x^5*e^5 + (x*e + d)^m*c^2*d*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c*m^4*x^4*e^5 + 10*(x*e + d)^m*
c^2*m^3*x^5*e^5 + 2*(x*e + d)^m*b*c*d*m^4*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m^3*x^4*e^4 - 4*(x*e + d)^m*c^2*d^2*m^
3*x^3*e^3 + (x*e + d)^m*b^2*m^4*x^3*e^5 + 2*(x*e + d)^m*a*c*m^4*x^3*e^5 + 22*(x*e + d)^m*b*c*m^3*x^4*e^5 + 35*
(x*e + d)^m*c^2*m^2*x^5*e^5 + (x*e + d)^m*b^2*d*m^4*x^2*e^4 + 2*(x*e + d)^m*a*c*d*m^4*x^2*e^4 + 16*(x*e + d)^m
*b*c*d*m^3*x^3*e^4 + 11*(x*e + d)^m*c^2*d*m^2*x^4*e^4 - 6*(x*e + d)^m*b*c*d^2*m^3*x^2*e^3 - 12*(x*e + d)^m*c^2
*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*c^2*d^3*m^2*x^2*e^2 + 2*(x*e + d)^m*a*b*m^4*x^2*e^5 + 12*(x*e + d)^m*b^2*m^3
*x^3*e^5 + 24*(x*e + d)^m*a*c*m^3*x^3*e^5 + 82*(x*e + d)^m*b*c*m^2*x^4*e^5 + 50*(x*e + d)^m*c^2*m*x^5*e^5 + 2*
(x*e + d)^m*a*b*d*m^4*x*e^4 + 10*(x*e + d)^m*b^2*d*m^3*x^2*e^4 + 20*(x*e + d)^m*a*c*d*m^3*x^2*e^4 + 34*(x*e +
d)^m*b*c*d*m^2*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m*x^4*e^4 - 2*(x*e + d)^m*b^2*d^2*m^3*x*e^3 - 4*(x*e + d)^m*a*c*d
^2*m^3*x*e^3 - 36*(x*e + d)^m*b*c*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*c^2*d^2*m*x^3*e^3 + 12*(x*e + d)^m*b*c*d^3*m
^2*x*e^2 + 12*(x*e + d)^m*c^2*d^3*m*x^2*e^2 - 24*(x*e + d)^m*c^2*d^4*m*x*e + (x*e + d)^m*a^2*m^4*x*e^5 + 26*(x
*e + d)^m*a*b*m^3*x^2*e^5 + 49*(x*e + d)^m*b^2*m^2*x^3*e^5 + 98*(x*e + d)^m*a*c*m^2*x^3*e^5 + 122*(x*e + d)^m*
b*c*m*x^4*e^5 + 24*(x*e + d)^m*c^2*x^5*e^5 + (x*e + d)^m*a^2*d*m^4*e^4 + 24*(x*e + d)^m*a*b*d*m^3*x*e^4 + 29*(
x*e + d)^m*b^2*d*m^2*x^2*e^4 + 58*(x*e + d)^m*a*c*d*m^2*x^2*e^4 + 20*(x*e + d)^m*b*c*d*m*x^3*e^4 - 2*(x*e + d)
^m*a*b*d^2*m^3*e^3 - 18*(x*e + d)^m*b^2*d^2*m^2*x*e^3 - 36*(x*e + d)^m*a*c*d^2*m^2*x*e^3 - 30*(x*e + d)^m*b*c*
d^2*m*x^2*e^3 + 2*(x*e + d)^m*b^2*d^3*m^2*e^2 + 4*(x*e + d)^m*a*c*d^3*m^2*e^2 + 60*(x*e + d)^m*b*c*d^3*m*x*e^2
- 12*(x*e + d)^m*b*c*d^4*m*e + 24*(x*e + d)^m*c^2*d^5 + 14*(x*e + d)^m*a^2*m^3*x*e^5 + 118*(x*e + d)^m*a*b*m^
2*x^2*e^5 + 78*(x*e + d)^m*b^2*m*x^3*e^5 + 156*(x*e + d)^m*a*c*m*x^3*e^5 + 60*(x*e + d)^m*b*c*x^4*e^5 + 14*(x*
e + d)^m*a^2*d*m^3*e^4 + 94*(x*e + d)^m*a*b*d*m^2*x*e^4 + 20*(x*e + d)^m*b^2*d*m*x^2*e^4 + 40*(x*e + d)^m*a*c*
d*m*x^2*e^4 - 24*(x*e + d)^m*a*b*d^2*m^2*e^3 - 40*(x*e + d)^m*b^2*d^2*m*x*e^3 - 80*(x*e + d)^m*a*c*d^2*m*x*e^3
+ 18*(x*e + d)^m*b^2*d^3*m*e^2 + 36*(x*e + d)^m*a*c*d^3*m*e^2 - 60*(x*e + d)^m*b*c*d^4*e + 71*(x*e + d)^m*a^2
*m^2*x*e^5 + 214*(x*e + d)^m*a*b*m*x^2*e^5 + 40*(x*e + d)^m*b^2*x^3*e^5 + 80*(x*e + d)^m*a*c*x^3*e^5 + 71*(x*e
+ d)^m*a^2*d*m^2*e^4 + 120*(x*e + d)^m*a*b*d*m*x*e^4 - 94*(x*e + d)^m*a*b*d^2*m*e^3 + 40*(x*e + d)^m*b^2*d^3*
e^2 + 80*(x*e + d)^m*a*c*d^3*e^2 + 154*(x*e + d)^m*a^2*m*x*e^5 + 120*(x*e + d)^m*a*b*x^2*e^5 + 154*(x*e + d)^m
*a^2*d*m*e^4 - 120*(x*e + d)^m*a*b*d^2*e^3 + 120*(x*e + d)^m*a^2*x*e^5 + 120*(x*e + d)^m*a^2*d*e^4)/(m^5*e^5 +
15*m^4*e^5 + 85*m^3*e^5 + 225*m^2*e^5 + 274*m*e^5 + 120*e^5)
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maple [B] time = 0.06, size = 822, normalized size = 4.62 \[ \frac {\left (c^{2} e^{4} m^{4} x^{4}+2 b c \,e^{4} m^{4} x^{3}+10 c^{2} e^{4} m^{3} x^{4}+2 a c \,e^{4} m^{4} x^{2}+b^{2} e^{4} m^{4} x^{2}+22 b c \,e^{4} m^{3} x^{3}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+2 a b \,e^{4} m^{4} x +24 a c \,e^{4} m^{3} x^{2}+12 b^{2} e^{4} m^{3} x^{2}-6 b c d \,e^{3} m^{3} x^{2}+82 b c \,e^{4} m^{2} x^{3}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}+a^{2} e^{4} m^{4}+26 a b \,e^{4} m^{3} x -4 a c d \,e^{3} m^{3} x +98 a c \,e^{4} m^{2} x^{2}-2 b^{2} d \,e^{3} m^{3} x +49 b^{2} e^{4} m^{2} x^{2}-48 b c d \,e^{3} m^{2} x^{2}+122 b c \,e^{4} m \,x^{3}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} x^{4} e^{4}+14 a^{2} e^{4} m^{3}-2 a b d \,e^{3} m^{3}+118 a b \,e^{4} m^{2} x -40 a c d \,e^{3} m^{2} x +156 a c \,e^{4} m \,x^{2}-20 b^{2} d \,e^{3} m^{2} x +78 b^{2} e^{4} m \,x^{2}+12 b c \,d^{2} e^{2} m^{2} x -102 b c d \,e^{3} m \,x^{2}+60 b c \,e^{4} x^{3}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 c^{2} d \,e^{3} x^{3}+71 a^{2} e^{4} m^{2}-24 a b d \,e^{3} m^{2}+214 a b \,e^{4} m x +4 a c \,d^{2} e^{2} m^{2}-116 a c d \,e^{3} m x +80 a c \,e^{4} x^{2}+2 b^{2} d^{2} e^{2} m^{2}-58 b^{2} d \,e^{3} m x +40 b^{2} e^{4} x^{2}+72 b c \,d^{2} e^{2} m x -60 b c d \,e^{3} x^{2}-24 c^{2} d^{3} e m x +24 c^{2} d^{2} e^{2} x^{2}+154 a^{2} e^{4} m -94 a b d \,e^{3} m +120 a b x \,e^{4}+36 a c \,d^{2} e^{2} m -80 a c d \,e^{3} x +18 b^{2} d^{2} e^{2} m -40 b^{2} d \,e^{3} x -12 b c \,d^{3} e m +60 b c \,d^{2} e^{2} x -24 c^{2} d^{3} x e +120 a^{2} e^{4}-120 a b d \,e^{3}+80 a c \,d^{2} e^{2}+40 b^{2} d^{2} e^{2}-60 b c \,d^{3} e +24 c^{2} d^{4}\right ) \left (e x +d \right )^{m +1}}{\left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^m*(c*x^2+b*x+a)^2,x)
[Out]
(e*x+d)^(m+1)*(c^2*e^4*m^4*x^4+2*b*c*e^4*m^4*x^3+10*c^2*e^4*m^3*x^4+2*a*c*e^4*m^4*x^2+b^2*e^4*m^4*x^2+22*b*c*e
^4*m^3*x^3-4*c^2*d*e^3*m^3*x^3+35*c^2*e^4*m^2*x^4+2*a*b*e^4*m^4*x+24*a*c*e^4*m^3*x^2+12*b^2*e^4*m^3*x^2-6*b*c*
d*e^3*m^3*x^2+82*b*c*e^4*m^2*x^3-24*c^2*d*e^3*m^2*x^3+50*c^2*e^4*m*x^4+a^2*e^4*m^4+26*a*b*e^4*m^3*x-4*a*c*d*e^
3*m^3*x+98*a*c*e^4*m^2*x^2-2*b^2*d*e^3*m^3*x+49*b^2*e^4*m^2*x^2-48*b*c*d*e^3*m^2*x^2+122*b*c*e^4*m*x^3+12*c^2*
d^2*e^2*m^2*x^2-44*c^2*d*e^3*m*x^3+24*c^2*e^4*x^4+14*a^2*e^4*m^3-2*a*b*d*e^3*m^3+118*a*b*e^4*m^2*x-40*a*c*d*e^
3*m^2*x+156*a*c*e^4*m*x^2-20*b^2*d*e^3*m^2*x+78*b^2*e^4*m*x^2+12*b*c*d^2*e^2*m^2*x-102*b*c*d*e^3*m*x^2+60*b*c*
e^4*x^3+36*c^2*d^2*e^2*m*x^2-24*c^2*d*e^3*x^3+71*a^2*e^4*m^2-24*a*b*d*e^3*m^2+214*a*b*e^4*m*x+4*a*c*d^2*e^2*m^
2-116*a*c*d*e^3*m*x+80*a*c*e^4*x^2+2*b^2*d^2*e^2*m^2-58*b^2*d*e^3*m*x+40*b^2*e^4*x^2+72*b*c*d^2*e^2*m*x-60*b*c
*d*e^3*x^2-24*c^2*d^3*e*m*x+24*c^2*d^2*e^2*x^2+154*a^2*e^4*m-94*a*b*d*e^3*m+120*a*b*e^4*x+36*a*c*d^2*e^2*m-80*
a*c*d*e^3*x+18*b^2*d^2*e^2*m-40*b^2*d*e^3*x-12*b*c*d^3*e*m+60*b*c*d^2*e^2*x-24*c^2*d^3*e*x+120*a^2*e^4-120*a*b
*d*e^3+80*a*c*d^2*e^2+40*b^2*d^2*e^2-60*b*c*d^3*e+24*c^2*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)
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maxima [B] time = 1.02, size = 455, normalized size = 2.56 \[ \frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{2}}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} b c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(c*x^2+b*x+a)^2,x, algorithm="maxima")
[Out]
2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a*b/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a^2/(e*(m + 1))
+ ((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*b^2/((m^3 + 6*m^2 + 11*m +
6)*e^3) + 2*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a*c/((m^3 + 6*m
^2 + 11*m + 6)*e^3) + 2*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^
2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*b*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m
^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2
+ m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^
5)
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mupad [B] time = 1.55, size = 895, normalized size = 5.03 \[ \frac {{\left (d+e\,x\right )}^m\,\left (a^2\,d\,e^4\,m^4+14\,a^2\,d\,e^4\,m^3+71\,a^2\,d\,e^4\,m^2+154\,a^2\,d\,e^4\,m+120\,a^2\,d\,e^4-2\,a\,b\,d^2\,e^3\,m^3-24\,a\,b\,d^2\,e^3\,m^2-94\,a\,b\,d^2\,e^3\,m-120\,a\,b\,d^2\,e^3+4\,a\,c\,d^3\,e^2\,m^2+36\,a\,c\,d^3\,e^2\,m+80\,a\,c\,d^3\,e^2+2\,b^2\,d^3\,e^2\,m^2+18\,b^2\,d^3\,e^2\,m+40\,b^2\,d^3\,e^2-12\,b\,c\,d^4\,e\,m-60\,b\,c\,d^4\,e+24\,c^2\,d^5\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^5\,m^4+14\,a^2\,e^5\,m^3+71\,a^2\,e^5\,m^2+154\,a^2\,e^5\,m+120\,a^2\,e^5+2\,a\,b\,d\,e^4\,m^4+24\,a\,b\,d\,e^4\,m^3+94\,a\,b\,d\,e^4\,m^2+120\,a\,b\,d\,e^4\,m-4\,a\,c\,d^2\,e^3\,m^3-36\,a\,c\,d^2\,e^3\,m^2-80\,a\,c\,d^2\,e^3\,m-2\,b^2\,d^2\,e^3\,m^3-18\,b^2\,d^2\,e^3\,m^2-40\,b^2\,d^2\,e^3\,m+12\,b\,c\,d^3\,e^2\,m^2+60\,b\,c\,d^3\,e^2\,m-24\,c^2\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c^2\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2+2\,b\,c\,d\,e\,m^2+10\,b\,c\,d\,e\,m-4\,c^2\,d^2\,m+2\,a\,c\,e^2\,m^2+18\,a\,c\,e^2\,m+40\,a\,c\,e^2\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (b^2\,d\,e^2\,m^3+9\,b^2\,d\,e^2\,m^2+20\,b^2\,d\,e^2\,m-6\,b\,c\,d^2\,e\,m^2-30\,b\,c\,d^2\,e\,m+2\,a\,b\,e^3\,m^3+24\,a\,b\,e^3\,m^2+94\,a\,b\,e^3\,m+120\,a\,b\,e^3+12\,c^2\,d^3\,m+2\,a\,c\,d\,e^2\,m^3+18\,a\,c\,d\,e^2\,m^2+40\,a\,c\,d\,e^2\,m\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c\,x^4\,{\left (d+e\,x\right )}^m\,\left (10\,b\,e+2\,b\,e\,m+c\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d + e*x)^m*(a + b*x + c*x^2)^2,x)
[Out]
((d + e*x)^m*(24*c^2*d^5 + 120*a^2*d*e^4 + 40*b^2*d^3*e^2 + 71*a^2*d*e^4*m^2 + 14*a^2*d*e^4*m^3 + a^2*d*e^4*m^
4 + 18*b^2*d^3*e^2*m - 60*b*c*d^4*e + 2*b^2*d^3*e^2*m^2 - 120*a*b*d^2*e^3 + 80*a*c*d^3*e^2 + 154*a^2*d*e^4*m -
94*a*b*d^2*e^3*m + 36*a*c*d^3*e^2*m - 24*a*b*d^2*e^3*m^2 - 2*a*b*d^2*e^3*m^3 + 4*a*c*d^3*e^2*m^2 - 12*b*c*d^4
*e*m))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*(d + e*x)^m*(120*a^2*e^5 + 154*a^2*e^5*m + 7
1*a^2*e^5*m^2 + 14*a^2*e^5*m^3 + a^2*e^5*m^4 - 40*b^2*d^2*e^3*m - 18*b^2*d^2*e^3*m^2 - 2*b^2*d^2*e^3*m^3 - 24*
c^2*d^4*e*m + 94*a*b*d*e^4*m^2 + 24*a*b*d*e^4*m^3 + 2*a*b*d*e^4*m^4 - 80*a*c*d^2*e^3*m + 60*b*c*d^3*e^2*m - 36
*a*c*d^2*e^3*m^2 - 4*a*c*d^2*e^3*m^3 + 12*b*c*d^3*e^2*m^2 + 120*a*b*d*e^4*m))/(e^5*(274*m + 225*m^2 + 85*m^3 +
15*m^4 + m^5 + 120)) + (c^2*x^5*(d + e*x)^m*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 +
15*m^4 + m^5 + 120) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(20*b^2*e^2 + 9*b^2*e^2*m - 4*c^2*d^2*m + b^2*e^2*m^2 +
40*a*c*e^2 + 18*a*c*e^2*m + 2*a*c*e^2*m^2 + 10*b*c*d*e*m + 2*b*c*d*e*m^2))/(e^2*(274*m + 225*m^2 + 85*m^3 + 1
5*m^4 + m^5 + 120)) + (x^2*(m + 1)*(d + e*x)^m*(12*c^2*d^3*m + 120*a*b*e^3 + 9*b^2*d*e^2*m^2 + b^2*d*e^2*m^3 +
94*a*b*e^3*m + 24*a*b*e^3*m^2 + 2*a*b*e^3*m^3 + 20*b^2*d*e^2*m + 18*a*c*d*e^2*m^2 + 2*a*c*d*e^2*m^3 - 6*b*c*d
^2*e*m^2 + 40*a*c*d*e^2*m - 30*b*c*d^2*e*m))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (c*x^4*(d
+ e*x)^m*(10*b*e + 2*b*e*m + c*d*m)*(11*m + 6*m^2 + m^3 + 6))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 1
20))
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sympy [A] time = 10.31, size = 10171, normalized size = 57.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**m*(c*x**2+b*x+a)**2,x)
[Out]
Piecewise((d**m*(a**2*x + a*b*x**2 + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), Eq(e, 0)), (-3*a*
*2*e**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 2*a*b*d*e**3/(12
*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 8*a*b*e**4*x/(12*d**4*e**5
+ 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 2*a*c*d**2*e**2/(12*d**4*e**5 + 48*d**
3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 8*a*c*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x
+ 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 12*a*c*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d*
*2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - b**2*d**2*e**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x
**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*b**2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48
*d*e**8*x**3 + 12*e**9*x**4) - 6*b**2*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8
*x**3 + 12*e**9*x**4) - 6*b*c*d**3*e/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*
e**9*x**4) - 24*b*c*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*
x**4) - 36*b*c*d*e**3*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4)
- 24*b*c*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 12*c
**2*d**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 25
*c**2*d**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48*c**2*d**3*
e*x*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 88*c**2
*d**3*e*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 72*c**2*d**2*e
**2*x**2*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 10
8*c**2*d**2*e**2*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) + 48
*c**2*d*e**3*x**3*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x
**4) + 48*c**2*d*e**3*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4)
+ 12*c**2*e**4*x**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**
9*x**4), Eq(m, -5)), (-a**2*e**4/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - a*b*d*e**3/(3*d
**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 3*a*b*e**4*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7
*x**2 + 3*e**8*x**3) - 2*a*c*d**2*e**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 6*a*c*d*e
**3*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 6*a*c*e**4*x**2/(3*d**3*e**5 + 9*d**2*e**6
*x + 9*d*e**7*x**2 + 3*e**8*x**3) - b**2*d**2*e**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3)
- 3*b**2*d*e**3*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 3*b**2*e**4*x**2/(3*d**3*e**5
+ 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 6*b*c*d**3*e*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d
*e**7*x**2 + 3*e**8*x**3) + 11*b*c*d**3*e/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 18*b*c
*d**2*e**2*x*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 27*b*c*d**2*e**2*x/(3*
d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 18*b*c*d*e**3*x**2*log(d/e + x)/(3*d**3*e**5 + 9*d*
*2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 18*b*c*d*e**3*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3
*e**8*x**3) + 6*b*c*e**4*x**3*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 12*c*
*2*d**4*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 22*c**2*d**4/(3*d**3*e**5 +
9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*c**2*d**3*e*x*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9
*d*e**7*x**2 + 3*e**8*x**3) - 54*c**2*d**3*e*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 3
6*c**2*d**2*e**2*x**2*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*c**2*d**2*
e**2*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 12*c**2*d*e**3*x**3*log(d/e + x)/(3*d*
*3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 3*c**2*e**4*x**4/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e
**7*x**2 + 3*e**8*x**3), Eq(m, -4)), (-a**2*e**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 2*a*b*d*e**3/(2*d*
*2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*a*b*e**4*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*a*c*d**2*e**2*
log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 6*a*c*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**
2) + 8*a*c*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 8*a*c*d*e**3*x/(2*d**2*e**5 + 4*d*
e**6*x + 2*e**7*x**2) + 4*a*c*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*d**2*e*
*2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 3*b**2*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7
*x**2) + 4*b**2*d*e**3*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**3*x/(2*d**2*e**5
+ 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*e**4*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*
d**3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 18*b*c*d**3*e/(2*d**2*e**5 + 4*d*e**6*x + 2*e**
7*x**2) - 24*b*c*d**2*e**2*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*b*c*d**2*e**2*x/(2*d**
2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*d*e**3*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2)
+ 4*b*c*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d**4*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6
*x + 2*e**7*x**2) + 18*c**2*d**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x*log(d/e + x)/(2*d
**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d**
2*e**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 4*c**2*d*e**3*x**3/(2*d**2*e**5 + 4*d*e**6
*x + 2*e**7*x**2) + c**2*e**4*x**4/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2), Eq(m, -3)), (-3*a**2*e**4/(3*d*e*
*5 + 3*e**6*x) + 6*a*b*d*e**3*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 6*a*b*d*e**3/(3*d*e**5 + 3*e**6*x) + 6*a*b*
e**4*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*a*c*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*a*c*d**2*
e**2/(3*d*e**5 + 3*e**6*x) - 12*a*c*d*e**3*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 6*a*c*e**4*x**2/(3*d*e**5 +
3*e**6*x) - 6*b**2*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 6*b**2*d**2*e**2/(3*d*e**5 + 3*e**6*x) - 6*b
**2*d*e**3*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 3*b**2*e**4*x**2/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**3*e*log(d
/e + x)/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**3*e/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**2*e**2*x*log(d/e + x)/(3*d*e**
5 + 3*e**6*x) - 9*b*c*d*e**3*x**2/(3*d*e**5 + 3*e**6*x) + 3*b*c*e**4*x**3/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**4
*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**4/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**3*e*x*log(d/e + x)/(3*d*
e**5 + 3*e**6*x) + 6*c**2*d**2*e**2*x**2/(3*d*e**5 + 3*e**6*x) - 2*c**2*d*e**3*x**3/(3*d*e**5 + 3*e**6*x) + c*
*2*e**4*x**4/(3*d*e**5 + 3*e**6*x), Eq(m, -2)), (a**2*log(d/e + x)/e - 2*a*b*d*log(d/e + x)/e**2 + 2*a*b*x/e +
2*a*c*d**2*log(d/e + x)/e**3 - 2*a*c*d*x/e**2 + a*c*x**2/e + b**2*d**2*log(d/e + x)/e**3 - b**2*d*x/e**2 + b*
*2*x**2/(2*e) - 2*b*c*d**3*log(d/e + x)/e**4 + 2*b*c*d**2*x/e**3 - b*c*d*x**2/e**2 + 2*b*c*x**3/(3*e) + c**2*d
**4*log(d/e + x)/e**5 - c**2*d**3*x/e**4 + c**2*d**2*x**2/(2*e**3) - c**2*d*x**3/(3*e**2) + c**2*x**4/(4*e), E
q(m, -1)), (a**2*d*e**4*m**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*
m + 120*e**5) + 14*a**2*d*e**4*m**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 27
4*e**5*m + 120*e**5) + 71*a**2*d*e**4*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m*
*2 + 274*e**5*m + 120*e**5) + 154*a**2*d*e**4*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e*
*5*m**2 + 274*e**5*m + 120*e**5) + 120*a**2*d*e**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225
*e**5*m**2 + 274*e**5*m + 120*e**5) + a**2*e**5*m**4*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5) + 14*a**2*e**5*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 71*a**2*e**5*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 154*a**2*e**5*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m*
*4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*a**2*e**5*x*(d + e*x)**m/(e**5*m**5 + 15*e**5
*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 2*a*b*d**2*e**3*m**3*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 24*a*b*d**2*e**3*m**2*(d + e*x)**m/(e*
*5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 94*a*b*d**2*e**3*m*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 120*a*b*d**2*e**3*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*a*b*d*e**4*m**4*
x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*a*b*d*e*
*4*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 94*
a*b*d*e**4*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**
5) + 120*a*b*d*e**4*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 1
20*e**5) + 2*a*b*e**5*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e*
*5*m + 120*e**5) + 26*a*b*e**5*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + 118*a*b*e**5*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225
*e**5*m**2 + 274*e**5*m + 120*e**5) + 214*a*b*e**5*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**
3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 120*a*b*e**5*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**
5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 4*a*c*d**3*e**2*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 36*a*c*d**3*e**2*m*(d + e*x)**m/(e**5*m**5 + 15*e**5
*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 80*a*c*d**3*e**2*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*a*c*d**2*e**3*m**3*x*(d + e*x)**m/(e**5*
m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 36*a*c*d**2*e**3*m**2*x*(d + e*x
)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 80*a*c*d**2*e**3*m*x*
(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*a*c*d*e**4*
m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 20*
a*c*d*e**4*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*
e**5) + 58*a*c*d*e**4*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e*
*5*m + 120*e**5) + 40*a*c*d*e**4*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + 2*a*c*e**5*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e*
*5*m**2 + 274*e**5*m + 120*e**5) + 24*a*c*e**5*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 98*a*c*e**5*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*
e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 156*a*c*e**5*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**
4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 80*a*c*e**5*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5
*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*b**2*d**3*e**2*m**2*(d + e*x)**m/(e**5*m**5
+ 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 18*b**2*d**3*e**2*m*(d + e*x)**m/(e**
5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 40*b**2*d**3*e**2*(d + e*x)**m
/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 2*b**2*d**2*e**3*m**3*x*(
d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 18*b**2*d**2*e
**3*m**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 40
*b**2*d**2*e**3*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e
**5) + b**2*d*e**4*m**4*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*
m + 120*e**5) + 10*b**2*d*e**4*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + 29*b**2*d*e**4*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 2
25*e**5*m**2 + 274*e**5*m + 120*e**5) + 20*b**2*d*e**4*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + b**2*e**5*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*b**2*e**5*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e*
*5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 49*b**2*e**5*m**2*x**3*(d + e*x)**m/(e**5*m*
*5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 78*b**2*e**5*m*x**3*(d + e*x)**m/(
e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 40*b**2*e**5*x**3*(d + e*x)
**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 12*b*c*d**4*e*m*(d + e
*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 60*b*c*d**4*e*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*b*c*d**3*e**2*m
**2*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 60*b*c*
d**3*e**2*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) -
6*b*c*d**2*e**3*m**3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m
+ 120*e**5) - 36*b*c*d**2*e**3*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) - 30*b*c*d**2*e**3*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 22
5*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*b*c*d*e**4*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 16*b*c*d*e**4*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**
4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 34*b*c*d*e**4*m**2*x**3*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 20*b*c*d*e**4*m*x**3*(d + e*x)**m/(e**5
*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*b*c*e**5*m**4*x**4*(d + e*x)*
*m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 22*b*c*e**5*m**3*x**4*(
d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 82*b*c*e**5*m*
*2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 122*b
*c*e**5*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
+ 60*b*c*e**5*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e*
*5) + 24*c**2*d**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e*
*5) - 24*c**2*d**4*e*m*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m +
120*e**5) + 12*c**2*d**3*e**2*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2
+ 274*e**5*m + 120*e**5) + 12*c**2*d**3*e**2*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 22
5*e**5*m**2 + 274*e**5*m + 120*e**5) - 4*c**2*d**2*e**3*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*
e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 12*c**2*d**2*e**3*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*
e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 8*c**2*d**2*e**3*m*x**3*(d + e*x)**m/(e**5
*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + c**2*d*e**4*m**4*x**4*(d + e*x)
**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*c**2*d*e**4*m**3*x**
4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 11*c**2*d*e
**4*m**2*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) +
6*c**2*d*e**4*m*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120
*e**5) + c**2*e**5*m**4*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*
m + 120*e**5) + 10*c**2*e**5*m**3*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) + 35*c**2*e**5*m**2*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e
**5*m**2 + 274*e**5*m + 120*e**5) + 50*c**2*e**5*m*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3
+ 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*c**2*e**5*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*
m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5), True))
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