3.12.43 \(\int (b d+2 c d x)^m (a+b x+c x^2)^3 \, dx\)
Optimal. Leaf size=141 \[ -\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac {(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)} \]
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Rubi [A] time = 0.08, antiderivative size = 141, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used =
{683} \begin {gather*} \frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac {(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3,x]
[Out]
-((b^2 - 4*a*c)^3*(b*d + 2*c*d*x)^(1 + m))/(128*c^4*d*(1 + m)) + (3*(b^2 - 4*a*c)^2*(b*d + 2*c*d*x)^(3 + m))/(
128*c^4*d^3*(3 + m)) - (3*(b^2 - 4*a*c)*(b*d + 2*c*d*x)^(5 + m))/(128*c^4*d^5*(5 + m)) + (b*d + 2*c*d*x)^(7 +
m)/(128*c^4*d^7*(7 + m))
Rule 683
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] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] && !(EqQ[m, 3] && NeQ[p, 1])
Rubi steps
\begin {align*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^3 \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3 (b d+2 c d x)^m}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{2+m}}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{4+m}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{6+m}}{64 c^3 d^6}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{1+m}}{128 c^4 d (1+m)}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{3+m}}{128 c^4 d^3 (3+m)}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{5+m}}{128 c^4 d^5 (5+m)}+\frac {(b d+2 c d x)^{7+m}}{128 c^4 d^7 (7+m)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 103, normalized size = 0.73 \begin {gather*} \frac {(b+2 c x) \left (-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)^4}{m+5}+\frac {3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{m+3}-\frac {\left (b^2-4 a c\right )^3}{m+1}+\frac {(b+2 c x)^6}{m+7}\right ) (d (b+2 c x))^m}{128 c^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3,x]
[Out]
((b + 2*c*x)*(d*(b + 2*c*x))^m*(-((b^2 - 4*a*c)^3/(1 + m)) + (3*(b^2 - 4*a*c)^2*(b + 2*c*x)^2)/(3 + m) - (3*(b
^2 - 4*a*c)*(b + 2*c*x)^4)/(5 + m) + (b + 2*c*x)^6/(7 + m)))/(128*c^4)
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IntegrateAlgebraic [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int (b d+2 c d x)^m \left (a+b x+c x^2\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3,x]
[Out]
Defer[IntegrateAlgebraic][(b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3, x]
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fricas [B] time = 0.42, size = 691, normalized size = 4.90 \begin {gather*} \frac {{\left (4 \, a^{3} b c^{3} m^{3} + 8 \, {\left (c^{7} m^{3} + 9 \, c^{7} m^{2} + 23 \, c^{7} m + 15 \, c^{7}\right )} x^{7} - 3 \, b^{7} + 42 \, a b^{5} c - 210 \, a^{2} b^{3} c^{2} + 420 \, a^{3} b c^{3} + 28 \, {\left (b c^{6} m^{3} + 9 \, b c^{6} m^{2} + 23 \, b c^{6} m + 15 \, b c^{6}\right )} x^{6} + 12 \, {\left (42 \, b^{2} c^{5} + 42 \, a c^{6} + {\left (3 \, b^{2} c^{5} + 2 \, a c^{6}\right )} m^{3} + 2 \, {\left (13 \, b^{2} c^{5} + 11 \, a c^{6}\right )} m^{2} + {\left (65 \, b^{2} c^{5} + 62 \, a c^{6}\right )} m\right )} x^{5} + 10 \, {\left (21 \, b^{3} c^{4} + 126 \, a b c^{5} + 2 \, {\left (b^{3} c^{4} + 3 \, a b c^{5}\right )} m^{3} + 3 \, {\left (5 \, b^{3} c^{4} + 22 \, a b c^{5}\right )} m^{2} + 2 \, {\left (17 \, b^{3} c^{4} + 93 \, a b c^{5}\right )} m\right )} x^{4} + 4 \, {\left (210 \, a b^{2} c^{4} + 210 \, a^{2} c^{5} + {\left (b^{4} c^{3} + 12 \, a b^{2} c^{4} + 6 \, a^{2} c^{5}\right )} m^{3} + 3 \, {\left (b^{4} c^{3} + 42 \, a b^{2} c^{4} + 26 \, a^{2} c^{5}\right )} m^{2} + 2 \, {\left (b^{4} c^{3} + 162 \, a b^{2} c^{4} + 141 \, a^{2} c^{5}\right )} m\right )} x^{3} - 6 \, {\left (a^{2} b^{3} c^{2} - 10 \, a^{3} b c^{3}\right )} m^{2} + 6 \, {\left (210 \, a^{2} b c^{4} + 2 \, {\left (a b^{3} c^{3} + 3 \, a^{2} b c^{4}\right )} m^{3} - {\left (b^{5} c^{2} - 16 \, a b^{3} c^{3} - 78 \, a^{2} b c^{4}\right )} m^{2} - {\left (b^{5} c^{2} - 14 \, a b^{3} c^{3} - 282 \, a^{2} b c^{4}\right )} m\right )} x^{2} + 2 \, {\left (3 \, a b^{5} c - 36 \, a^{2} b^{3} c^{2} + 142 \, a^{3} b c^{3}\right )} m + 2 \, {\left (420 \, a^{3} c^{4} + 2 \, {\left (3 \, a^{2} b^{2} c^{3} + 2 \, a^{3} c^{4}\right )} m^{3} - 6 \, {\left (a b^{4} c^{2} - 12 \, a^{2} b^{2} c^{3} - 10 \, a^{3} c^{4}\right )} m^{2} + {\left (3 \, b^{6} c - 42 \, a b^{4} c^{2} + 210 \, a^{2} b^{2} c^{3} + 284 \, a^{3} c^{4}\right )} m\right )} x\right )} {\left (2 \, c d x + b d\right )}^{m}}{8 \, {\left (c^{4} m^{4} + 16 \, c^{4} m^{3} + 86 \, c^{4} m^{2} + 176 \, c^{4} m + 105 \, c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
1/8*(4*a^3*b*c^3*m^3 + 8*(c^7*m^3 + 9*c^7*m^2 + 23*c^7*m + 15*c^7)*x^7 - 3*b^7 + 42*a*b^5*c - 210*a^2*b^3*c^2
+ 420*a^3*b*c^3 + 28*(b*c^6*m^3 + 9*b*c^6*m^2 + 23*b*c^6*m + 15*b*c^6)*x^6 + 12*(42*b^2*c^5 + 42*a*c^6 + (3*b^
2*c^5 + 2*a*c^6)*m^3 + 2*(13*b^2*c^5 + 11*a*c^6)*m^2 + (65*b^2*c^5 + 62*a*c^6)*m)*x^5 + 10*(21*b^3*c^4 + 126*a
*b*c^5 + 2*(b^3*c^4 + 3*a*b*c^5)*m^3 + 3*(5*b^3*c^4 + 22*a*b*c^5)*m^2 + 2*(17*b^3*c^4 + 93*a*b*c^5)*m)*x^4 + 4
*(210*a*b^2*c^4 + 210*a^2*c^5 + (b^4*c^3 + 12*a*b^2*c^4 + 6*a^2*c^5)*m^3 + 3*(b^4*c^3 + 42*a*b^2*c^4 + 26*a^2*
c^5)*m^2 + 2*(b^4*c^3 + 162*a*b^2*c^4 + 141*a^2*c^5)*m)*x^3 - 6*(a^2*b^3*c^2 - 10*a^3*b*c^3)*m^2 + 6*(210*a^2*
b*c^4 + 2*(a*b^3*c^3 + 3*a^2*b*c^4)*m^3 - (b^5*c^2 - 16*a*b^3*c^3 - 78*a^2*b*c^4)*m^2 - (b^5*c^2 - 14*a*b^3*c^
3 - 282*a^2*b*c^4)*m)*x^2 + 2*(3*a*b^5*c - 36*a^2*b^3*c^2 + 142*a^3*b*c^3)*m + 2*(420*a^3*c^4 + 2*(3*a^2*b^2*c
^3 + 2*a^3*c^4)*m^3 - 6*(a*b^4*c^2 - 12*a^2*b^2*c^3 - 10*a^3*c^4)*m^2 + (3*b^6*c - 42*a*b^4*c^2 + 210*a^2*b^2*
c^3 + 284*a^3*c^4)*m)*x)*(2*c*d*x + b*d)^m/(c^4*m^4 + 16*c^4*m^3 + 86*c^4*m^2 + 176*c^4*m + 105*c^4)
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giac [B] time = 0.25, size = 1506, normalized size = 10.68
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
1/8*(8*(2*c*d*x + b*d)^m*c^7*m^3*x^7 + 28*(2*c*d*x + b*d)^m*b*c^6*m^3*x^6 + 72*(2*c*d*x + b*d)^m*c^7*m^2*x^7 +
36*(2*c*d*x + b*d)^m*b^2*c^5*m^3*x^5 + 24*(2*c*d*x + b*d)^m*a*c^6*m^3*x^5 + 252*(2*c*d*x + b*d)^m*b*c^6*m^2*x
^6 + 184*(2*c*d*x + b*d)^m*c^7*m*x^7 + 20*(2*c*d*x + b*d)^m*b^3*c^4*m^3*x^4 + 60*(2*c*d*x + b*d)^m*a*b*c^5*m^3
*x^4 + 312*(2*c*d*x + b*d)^m*b^2*c^5*m^2*x^5 + 264*(2*c*d*x + b*d)^m*a*c^6*m^2*x^5 + 644*(2*c*d*x + b*d)^m*b*c
^6*m*x^6 + 120*(2*c*d*x + b*d)^m*c^7*x^7 + 4*(2*c*d*x + b*d)^m*b^4*c^3*m^3*x^3 + 48*(2*c*d*x + b*d)^m*a*b^2*c^
4*m^3*x^3 + 24*(2*c*d*x + b*d)^m*a^2*c^5*m^3*x^3 + 150*(2*c*d*x + b*d)^m*b^3*c^4*m^2*x^4 + 660*(2*c*d*x + b*d)
^m*a*b*c^5*m^2*x^4 + 780*(2*c*d*x + b*d)^m*b^2*c^5*m*x^5 + 744*(2*c*d*x + b*d)^m*a*c^6*m*x^5 + 420*(2*c*d*x +
b*d)^m*b*c^6*x^6 + 12*(2*c*d*x + b*d)^m*a*b^3*c^3*m^3*x^2 + 36*(2*c*d*x + b*d)^m*a^2*b*c^4*m^3*x^2 + 12*(2*c*d
*x + b*d)^m*b^4*c^3*m^2*x^3 + 504*(2*c*d*x + b*d)^m*a*b^2*c^4*m^2*x^3 + 312*(2*c*d*x + b*d)^m*a^2*c^5*m^2*x^3
+ 340*(2*c*d*x + b*d)^m*b^3*c^4*m*x^4 + 1860*(2*c*d*x + b*d)^m*a*b*c^5*m*x^4 + 504*(2*c*d*x + b*d)^m*b^2*c^5*x
^5 + 504*(2*c*d*x + b*d)^m*a*c^6*x^5 + 12*(2*c*d*x + b*d)^m*a^2*b^2*c^3*m^3*x + 8*(2*c*d*x + b*d)^m*a^3*c^4*m^
3*x - 6*(2*c*d*x + b*d)^m*b^5*c^2*m^2*x^2 + 96*(2*c*d*x + b*d)^m*a*b^3*c^3*m^2*x^2 + 468*(2*c*d*x + b*d)^m*a^2
*b*c^4*m^2*x^2 + 8*(2*c*d*x + b*d)^m*b^4*c^3*m*x^3 + 1296*(2*c*d*x + b*d)^m*a*b^2*c^4*m*x^3 + 1128*(2*c*d*x +
b*d)^m*a^2*c^5*m*x^3 + 210*(2*c*d*x + b*d)^m*b^3*c^4*x^4 + 1260*(2*c*d*x + b*d)^m*a*b*c^5*x^4 + 4*(2*c*d*x + b
*d)^m*a^3*b*c^3*m^3 - 12*(2*c*d*x + b*d)^m*a*b^4*c^2*m^2*x + 144*(2*c*d*x + b*d)^m*a^2*b^2*c^3*m^2*x + 120*(2*
c*d*x + b*d)^m*a^3*c^4*m^2*x - 6*(2*c*d*x + b*d)^m*b^5*c^2*m*x^2 + 84*(2*c*d*x + b*d)^m*a*b^3*c^3*m*x^2 + 1692
*(2*c*d*x + b*d)^m*a^2*b*c^4*m*x^2 + 840*(2*c*d*x + b*d)^m*a*b^2*c^4*x^3 + 840*(2*c*d*x + b*d)^m*a^2*c^5*x^3 -
6*(2*c*d*x + b*d)^m*a^2*b^3*c^2*m^2 + 60*(2*c*d*x + b*d)^m*a^3*b*c^3*m^2 + 6*(2*c*d*x + b*d)^m*b^6*c*m*x - 84
*(2*c*d*x + b*d)^m*a*b^4*c^2*m*x + 420*(2*c*d*x + b*d)^m*a^2*b^2*c^3*m*x + 568*(2*c*d*x + b*d)^m*a^3*c^4*m*x +
1260*(2*c*d*x + b*d)^m*a^2*b*c^4*x^2 + 6*(2*c*d*x + b*d)^m*a*b^5*c*m - 72*(2*c*d*x + b*d)^m*a^2*b^3*c^2*m + 2
84*(2*c*d*x + b*d)^m*a^3*b*c^3*m + 840*(2*c*d*x + b*d)^m*a^3*c^4*x - 3*(2*c*d*x + b*d)^m*b^7 + 42*(2*c*d*x + b
*d)^m*a*b^5*c - 210*(2*c*d*x + b*d)^m*a^2*b^3*c^2 + 420*(2*c*d*x + b*d)^m*a^3*b*c^3)/(c^4*m^4 + 16*c^4*m^3 + 8
6*c^4*m^2 + 176*c^4*m + 105*c^4)
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maple [B] time = 0.05, size = 653, normalized size = 4.63 \begin {gather*} \frac {\left (4 c^{6} m^{3} x^{6}+12 b \,c^{5} m^{3} x^{5}+36 c^{6} m^{2} x^{6}+12 a \,c^{5} m^{3} x^{4}+12 b^{2} c^{4} m^{3} x^{4}+108 b \,c^{5} m^{2} x^{5}+92 c^{6} m \,x^{6}+24 a b \,c^{4} m^{3} x^{3}+132 a \,c^{5} m^{2} x^{4}+4 b^{3} c^{3} m^{3} x^{3}+102 b^{2} c^{4} m^{2} x^{4}+276 b \,c^{5} m \,x^{5}+60 c^{6} x^{6}+12 a^{2} c^{4} m^{3} x^{2}+12 a \,b^{2} c^{3} m^{3} x^{2}+264 a b \,c^{4} m^{2} x^{3}+372 a \,c^{5} m \,x^{4}+24 b^{3} c^{3} m^{2} x^{3}+252 b^{2} c^{4} m \,x^{4}+180 b \,c^{5} x^{5}+12 a^{2} b \,c^{3} m^{3} x +156 a^{2} c^{4} m^{2} x^{2}+120 a \,b^{2} c^{3} m^{2} x^{2}+744 a b \,c^{4} m \,x^{3}+252 a \,c^{5} x^{4}-6 b^{4} c^{2} m^{2} x^{2}+44 b^{3} c^{3} m \,x^{3}+162 b^{2} c^{4} x^{4}+4 a^{3} c^{3} m^{3}+156 a^{2} b \,c^{3} m^{2} x +564 a^{2} c^{4} m \,x^{2}-12 a \,b^{3} c^{2} m^{2} x +276 a \,b^{2} c^{3} m \,x^{2}+504 a b \,c^{4} x^{3}-18 b^{4} c^{2} m \,x^{2}+24 b^{3} c^{3} x^{3}+60 a^{3} c^{3} m^{2}-6 a^{2} b^{2} c^{2} m^{2}+564 a^{2} b \,c^{3} m x +420 a^{2} c^{4} x^{2}-96 a \,b^{3} c^{2} m x +168 a \,b^{2} c^{3} x^{2}+6 b^{5} c m x -12 b^{4} c^{2} x^{2}+284 a^{3} c^{3} m -72 a^{2} b^{2} c^{2} m +420 a^{2} b \,c^{3} x +6 a \,b^{4} c m -84 a \,b^{3} c^{2} x +6 b^{5} c x +420 a^{3} c^{3}-210 a^{2} b^{2} c^{2}+42 a \,b^{4} c -3 b^{6}\right ) \left (2 c x +b \right ) \left (2 c d x +b d \right )^{m}}{8 \left (m^{4}+16 m^{3}+86 m^{2}+176 m +105\right ) c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^3,x)
[Out]
1/8*(2*c*d*x+b*d)^m*(4*c^6*m^3*x^6+12*b*c^5*m^3*x^5+36*c^6*m^2*x^6+12*a*c^5*m^3*x^4+12*b^2*c^4*m^3*x^4+108*b*c
^5*m^2*x^5+92*c^6*m*x^6+24*a*b*c^4*m^3*x^3+132*a*c^5*m^2*x^4+4*b^3*c^3*m^3*x^3+102*b^2*c^4*m^2*x^4+276*b*c^5*m
*x^5+60*c^6*x^6+12*a^2*c^4*m^3*x^2+12*a*b^2*c^3*m^3*x^2+264*a*b*c^4*m^2*x^3+372*a*c^5*m*x^4+24*b^3*c^3*m^2*x^3
+252*b^2*c^4*m*x^4+180*b*c^5*x^5+12*a^2*b*c^3*m^3*x+156*a^2*c^4*m^2*x^2+120*a*b^2*c^3*m^2*x^2+744*a*b*c^4*m*x^
3+252*a*c^5*x^4-6*b^4*c^2*m^2*x^2+44*b^3*c^3*m*x^3+162*b^2*c^4*x^4+4*a^3*c^3*m^3+156*a^2*b*c^3*m^2*x+564*a^2*c
^4*m*x^2-12*a*b^3*c^2*m^2*x+276*a*b^2*c^3*m*x^2+504*a*b*c^4*x^3-18*b^4*c^2*m*x^2+24*b^3*c^3*x^3+60*a^3*c^3*m^2
-6*a^2*b^2*c^2*m^2+564*a^2*b*c^3*m*x+420*a^2*c^4*x^2-96*a*b^3*c^2*m*x+168*a*b^2*c^3*x^2+6*b^5*c*m*x-12*b^4*c^2
*x^2+284*a^3*c^3*m-72*a^2*b^2*c^2*m+420*a^2*b*c^3*x+6*a*b^4*c*m-84*a*b^3*c^2*x+6*b^5*c*x+420*a^3*c^3-210*a^2*b
^2*c^2+42*a*b^4*c-3*b^6)*(2*c*x+b)/c^4/(m^4+16*m^3+86*m^2+176*m+105)
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maxima [B] time = 2.29, size = 1299, normalized size = 9.21
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*d*x+b*d)^m*(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
3/4*(4*c^2*d^m*(m + 1)*x^2 + 2*b*c*d^m*m*x - b^2*d^m)*(2*c*x + b)^m*a^2*b/((m^2 + 3*m + 2)*c^2) + 3/4*(4*(m^2
+ 3*m + 2)*c^3*d^m*x^3 + 2*(m^2 + m)*b*c^2*d^m*x^2 - 2*b^2*c*d^m*m*x + b^3*d^m)*(2*c*x + b)^m*a*b^2/((m^3 + 6*
m^2 + 11*m + 6)*c^3) + 3/4*(4*(m^2 + 3*m + 2)*c^3*d^m*x^3 + 2*(m^2 + m)*b*c^2*d^m*x^2 - 2*b^2*c*d^m*m*x + b^3*
d^m)*(2*c*x + b)^m*a^2/((m^3 + 6*m^2 + 11*m + 6)*c^2) + 1/2*(2*c*d*x + b*d)^(m + 1)*a^3/(c*d*(m + 1)) + 1/8*(8
*(m^3 + 6*m^2 + 11*m + 6)*c^4*d^m*x^4 + 4*(m^3 + 3*m^2 + 2*m)*b*c^3*d^m*x^3 - 6*(m^2 + m)*b^2*c^2*d^m*x^2 + 6*
b^3*c*d^m*m*x - 3*b^4*d^m)*(2*c*x + b)^m*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*c^4) + 3/4*(8*(m^3 + 6*m^2 +
11*m + 6)*c^4*d^m*x^4 + 4*(m^3 + 3*m^2 + 2*m)*b*c^3*d^m*x^3 - 6*(m^2 + m)*b^2*c^2*d^m*x^2 + 6*b^3*c*d^m*m*x -
3*b^4*d^m)*(2*c*x + b)^m*a*b/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*c^3) + 3/4*(4*(m^4 + 10*m^3 + 35*m^2 + 50*m
+ 24)*c^5*d^m*x^5 + 2*(m^4 + 6*m^3 + 11*m^2 + 6*m)*b*c^4*d^m*x^4 - 4*(m^3 + 3*m^2 + 2*m)*b^2*c^3*d^m*x^3 + 6*
(m^2 + m)*b^3*c^2*d^m*x^2 - 6*b^4*c*d^m*m*x + 3*b^5*d^m)*(2*c*x + b)^m*b^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 +
274*m + 120)*c^4) + 3/4*(4*(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*c^5*d^m*x^5 + 2*(m^4 + 6*m^3 + 11*m^2 + 6*m)*b
*c^4*d^m*x^4 - 4*(m^3 + 3*m^2 + 2*m)*b^2*c^3*d^m*x^3 + 6*(m^2 + m)*b^3*c^2*d^m*x^2 - 6*b^4*c*d^m*m*x + 3*b^5*d
^m)*(2*c*x + b)^m*a/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*c^3) + 3/8*(8*(m^5 + 15*m^4 + 85*m^3 + 22
5*m^2 + 274*m + 120)*c^6*d^m*x^6 + 4*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*b*c^5*d^m*x^5 - 10*(m^4 + 6*m^3 +
11*m^2 + 6*m)*b^2*c^4*d^m*x^4 + 20*(m^3 + 3*m^2 + 2*m)*b^3*c^3*d^m*x^3 - 30*(m^2 + m)*b^4*c^2*d^m*x^2 + 30*b^
5*c*d^m*m*x - 15*b^6*d^m)*(2*c*x + b)^m*b/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*c^4) +
1/8*(8*(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*c^7*d^m*x^7 + 4*(m^6 + 15*m^5 + 85*m^4 +
225*m^3 + 274*m^2 + 120*m)*b*c^6*d^m*x^6 - 12*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*b^2*c^5*d^m*x^5 + 30*(m^
4 + 6*m^3 + 11*m^2 + 6*m)*b^3*c^4*d^m*x^4 - 60*(m^3 + 3*m^2 + 2*m)*b^4*c^3*d^m*x^3 + 90*(m^2 + m)*b^5*c^2*d^m*
x^2 - 90*b^6*c*d^m*m*x + 45*b^7*d^m)*(2*c*x + b)^m/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2
+ 13068*m + 5040)*c^4)
________________________________________________________________________________________
mupad [B] time = 1.11, size = 655, normalized size = 4.65 \begin {gather*} {\left (b\,d+2\,c\,d\,x\right )}^m\,\left (\frac {4\,a^3\,b\,c^3\,m^3+60\,a^3\,b\,c^3\,m^2+284\,a^3\,b\,c^3\,m+420\,a^3\,b\,c^3-6\,a^2\,b^3\,c^2\,m^2-72\,a^2\,b^3\,c^2\,m-210\,a^2\,b^3\,c^2+6\,a\,b^5\,c\,m+42\,a\,b^5\,c-3\,b^7}{8\,c^4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {c^3\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {x\,\left (8\,a^3\,c^4\,m^3+120\,a^3\,c^4\,m^2+568\,a^3\,c^4\,m+840\,a^3\,c^4+12\,a^2\,b^2\,c^3\,m^3+144\,a^2\,b^2\,c^3\,m^2+420\,a^2\,b^2\,c^3\,m-12\,a\,b^4\,c^2\,m^2-84\,a\,b^4\,c^2\,m+6\,b^6\,c\,m\right )}{8\,c^4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {5\,b\,x^4\,\left (m^2+4\,m+3\right )\,\left (42\,a\,c+2\,b^2\,m+7\,b^2+6\,a\,c\,m\right )}{4\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {3\,c\,x^5\,\left (m^2+4\,m+3\right )\,\left (14\,a\,c+3\,b^2\,m+14\,b^2+2\,a\,c\,m\right )}{2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {x^3\,\left (m+1\right )\,\left (6\,a^2\,c^2\,m^2+72\,a^2\,c^2\,m+210\,a^2\,c^2+12\,a\,b^2\,c\,m^2+114\,a\,b^2\,c\,m+210\,a\,b^2\,c+b^4\,m^2+2\,b^4\,m\right )}{2\,c\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {7\,b\,c^2\,x^6\,\left (m^3+9\,m^2+23\,m+15\right )}{2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}+\frac {3\,b\,x^2\,\left (m+1\right )\,\left (6\,a^2\,c^2\,m^2+72\,a^2\,c^2\,m+210\,a^2\,c^2+2\,a\,b^2\,c\,m^2+14\,a\,b^2\,c\,m-b^4\,m\right )}{4\,c^2\,\left (m^4+16\,m^3+86\,m^2+176\,m+105\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*d + 2*c*d*x)^m*(a + b*x + c*x^2)^3,x)
[Out]
(b*d + 2*c*d*x)^m*((420*a^3*b*c^3 - 3*b^7 - 210*a^2*b^3*c^2 + 42*a*b^5*c - 72*a^2*b^3*c^2*m + 60*a^3*b*c^3*m^2
+ 4*a^3*b*c^3*m^3 + 6*a*b^5*c*m - 6*a^2*b^3*c^2*m^2 + 284*a^3*b*c^3*m)/(8*c^4*(176*m + 86*m^2 + 16*m^3 + m^4
+ 105)) + (c^3*x^7*(23*m + 9*m^2 + m^3 + 15))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (x*(840*a^3*c^4 + 568*a^
3*c^4*m + 120*a^3*c^4*m^2 + 8*a^3*c^4*m^3 + 6*b^6*c*m + 420*a^2*b^2*c^3*m - 12*a*b^4*c^2*m^2 + 144*a^2*b^2*c^3
*m^2 + 12*a^2*b^2*c^3*m^3 - 84*a*b^4*c^2*m))/(8*c^4*(176*m + 86*m^2 + 16*m^3 + m^4 + 105)) + (5*b*x^4*(4*m + m
^2 + 3)*(42*a*c + 2*b^2*m + 7*b^2 + 6*a*c*m))/(4*(176*m + 86*m^2 + 16*m^3 + m^4 + 105)) + (3*c*x^5*(4*m + m^2
+ 3)*(14*a*c + 3*b^2*m + 14*b^2 + 2*a*c*m))/(2*(176*m + 86*m^2 + 16*m^3 + m^4 + 105)) + (x^3*(m + 1)*(2*b^4*m
+ 210*a^2*c^2 + b^4*m^2 + 72*a^2*c^2*m + 6*a^2*c^2*m^2 + 210*a*b^2*c + 114*a*b^2*c*m + 12*a*b^2*c*m^2))/(2*c*(
176*m + 86*m^2 + 16*m^3 + m^4 + 105)) + (7*b*c^2*x^6*(23*m + 9*m^2 + m^3 + 15))/(2*(176*m + 86*m^2 + 16*m^3 +
m^4 + 105)) + (3*b*x^2*(m + 1)*(210*a^2*c^2 - b^4*m + 72*a^2*c^2*m + 6*a^2*c^2*m^2 + 14*a*b^2*c*m + 2*a*b^2*c*
m^2))/(4*c^2*(176*m + 86*m^2 + 16*m^3 + m^4 + 105)))
________________________________________________________________________________________
sympy [A] time = 10.15, size = 9491, normalized size = 67.31
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*d*x+b*d)**m*(c*x**2+b*x+a)**3,x)
[Out]
Piecewise(((b*d)**m*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b**3*x**4/4), Eq(c, 0)), (-128*a**3*c**3/(1536*b
**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*
c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 48*a**2*b**2*c**2/(1536*b**6*c**4*d**7 + 1
8432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 +
294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 576*a**2*b*c**3*x/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d*
*7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**
7*x**5 + 98304*c**10*d**7*x**6) - 576*a**2*c**4*x**2/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**
4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c
**10*d**7*x**6) - 24*a*b**4*c/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 2457
60*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 288*a
*b**3*c**2*x/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7
*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 1440*a*b**2*c**3*x**2/
(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 36864
0*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 2304*a*b*c**4*x**3/(1536*b**6*c**4*
d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7
*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) - 1152*a*c**5*x**4/(1536*b**6*c**4*d**7 + 18432*b**5*
c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c
**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 12*b**6*log(b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*
x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x
**5 + 98304*c**10*d**7*x**6) + 11*b**6/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x*
*2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6
) + 144*b**5*c*x*log(b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 +
245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 1
08*b**5*c*x/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*
x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 720*b**4*c**2*x**2*log(
b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7
*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 396*b**4*c**2*x**2/(15
36*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b
**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 1920*b**3*c**3*x**3*log(b/(2*c) + x)/(
1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640
*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 576*b**3*c**3*x**3/(1536*b**6*c**4*d
**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*
x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 2880*b**2*c**4*x**4*log(b/(2*c) + x)/(1536*b**6*c**4
*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**
7*x**4 + 294912*b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 288*b**2*c**4*x**4/(1536*b**6*c**4*d**7 + 18432*b*
*5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*
b*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 2304*b*c**5*x**5*log(b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**
5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b
*c**9*d**7*x**5 + 98304*c**10*d**7*x**6) + 768*c**6*x**6*log(b/(2*c) + x)/(1536*b**6*c**4*d**7 + 18432*b**5*c*
*5*d**7*x + 92160*b**4*c**6*d**7*x**2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**
9*d**7*x**5 + 98304*c**10*d**7*x**6), Eq(m, -7)), (-64*a**3*c**3/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x +
12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 48*a**2*b**2*c**2/(512*b**4*c**4*
d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 384
*a**2*b*c**3*x/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**
3 + 8192*c**8*d**5*x**4) - 384*a**2*c**4*x**2/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d*
*5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) + 48*a*b**4*c*log(b/(2*c) + x)/(512*b**4*c**4*d**5 + 4
096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) + 36*a*b**4*c
/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*
d**5*x**4) + 384*a*b**3*c**2*x*log(b/(2*c) + x)/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*
d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) + 192*a*b**3*c**2*x/(512*b**4*c**4*d**5 + 4096*b**3*
c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) + 1152*a*b**2*c**3*x**
2*log(b/(2*c) + x)/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5
*x**3 + 8192*c**8*d**5*x**4) + 192*a*b**2*c**3*x**2/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c
**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) + 1536*a*b*c**4*x**3*log(b/(2*c) + x)/(512*b**4*
c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4)
+ 768*a*c**5*x**4*log(b/(2*c) + x)/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 1
6384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 12*b**6*log(b/(2*c) + x)/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d
**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 33*b**6/(512*b**4*c**4*d**
5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 96*b**
5*c*x*log(b/(2*c) + x)/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*
d**5*x**3 + 8192*c**8*d**5*x**4) - 240*b**5*c*x/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*
d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 288*b**4*c**2*x**2*log(b/(2*c) + x)/(512*b**4*c**4
*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 62
4*b**4*c**2*x**2/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x
**3 + 8192*c**8*d**5*x**4) - 384*b**3*c**3*x**3*log(b/(2*c) + x)/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x +
12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 640*b**3*c**3*x**3/(512*b**4*c**4
*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) - 19
2*b**2*c**4*x**4*log(b/(2*c) + x)/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16
384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) + 384*b*c**5*x**5/(512*b**4*c**4*d**5 + 4096*b**3*c**5*d**5*x + 12
288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) + 128*c**6*x**6/(512*b**4*c**4*d**5 +
4096*b**3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4), Eq(m, -5)),
(-64*a**3*c**3/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 96*a**2*b**2*c**2*log(b/(2*c
) + x)/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 48*a**2*b**2*c**2/(256*b**2*c**4*d**3
+ 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 384*a**2*b*c**3*x*log(b/(2*c) + x)/(256*b**2*c**4*d**3 + 1024*b
*c**5*d**3*x + 1024*c**6*d**3*x**2) + 384*a**2*c**4*x**2*log(b/(2*c) + x)/(256*b**2*c**4*d**3 + 1024*b*c**5*d*
*3*x + 1024*c**6*d**3*x**2) - 48*a*b**4*c*log(b/(2*c) + x)/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**
6*d**3*x**2) - 72*a*b**4*c/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) - 192*a*b**3*c**2*x
*log(b/(2*c) + x)/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) - 192*a*b**3*c**2*x/(256*b**
2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) - 192*a*b**2*c**3*x**2*log(b/(2*c) + x)/(256*b**2*c**4
*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 384*a*b*c**4*x**3/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x
+ 1024*c**6*d**3*x**2) + 192*a*c**5*x**4/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 6*
b**6*log(b/(2*c) + x)/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 9*b**6/(256*b**2*c**4*
d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 24*b**5*c*x*log(b/(2*c) + x)/(256*b**2*c**4*d**3 + 1024*b*c
**5*d**3*x + 1024*c**6*d**3*x**2) + 24*b**5*c*x/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2
) + 24*b**4*c**2*x**2*log(b/(2*c) + x)/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) - 16*b*
*3*c**3*x**3/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 72*b**2*c**4*x**4/(256*b**2*c**
4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2) + 96*b*c**5*x**5/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x +
1024*c**6*d**3*x**2) + 32*c**6*x**6/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x**2), Eq(m, -3
)), (a**3*log(b/(2*c) + x)/(2*c*d) - 3*a**2*b**2*log(b/(2*c) + x)/(8*c**2*d) + 3*a**2*b*x/(4*c*d) + 3*a**2*x**
2/(4*d) + 3*a*b**4*log(b/(2*c) + x)/(32*c**3*d) - 3*a*b**3*x/(16*c**2*d) + 3*a*b**2*x**2/(16*c*d) + 3*a*b*x**3
/(4*d) + 3*a*c*x**4/(8*d) - b**6*log(b/(2*c) + x)/(128*c**4*d) + b**5*x/(64*c**3*d) - b**4*x**2/(64*c**2*d) +
b**3*x**3/(48*c*d) + 7*b**2*x**4/(32*d) + b*c*x**5/(4*d) + c**2*x**6/(12*d), Eq(m, -1)), (4*a**3*b*c**3*m**3*(
b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 60*a**3*b*c**3*m**2
*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 284*a**3*b*c**3*m
*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 420*a**3*b*c**3*(
b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 8*a**3*c**4*m**3*x*
(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 120*a**3*c**4*m**2
*x*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 568*a**3*c**4*m
*x*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 840*a**3*c**4*x
*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) - 6*a**2*b**3*c**2*
m**2*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) - 72*a**2*b**3*
c**2*m*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) - 210*a**2*b*
*3*c**2*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 12*a**2*b*
*2*c**3*m**3*x*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 144
*a**2*b**2*c**3*m**2*x*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**
4) + 420*a**2*b**2*c**3*m*x*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 84
0*c**4) + 36*a**2*b*c**4*m**3*x**2*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4
*m + 840*c**4) + 468*a**2*b*c**4*m**2*x**2*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1
408*c**4*m + 840*c**4) + 1692*a**2*b*c**4*m*x**2*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m*
*2 + 1408*c**4*m + 840*c**4) + 1260*a**2*b*c**4*x**2*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**
4*m**2 + 1408*c**4*m + 840*c**4) + 24*a**2*c**5*m**3*x**3*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 68
8*c**4*m**2 + 1408*c**4*m + 840*c**4) + 312*a**2*c**5*m**2*x**3*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**
3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 1128*a**2*c**5*m*x**3*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4
*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 840*a**2*c**5*x**3*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**
4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 6*a*b**5*c*m*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**
3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 42*a*b**5*c*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 68
8*c**4*m**2 + 1408*c**4*m + 840*c**4) - 12*a*b**4*c**2*m**2*x*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3
+ 688*c**4*m**2 + 1408*c**4*m + 840*c**4) - 84*a*b**4*c**2*m*x*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3
+ 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 12*a*b**3*c**3*m**3*x**2*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c*
*4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 96*a*b**3*c**3*m**2*x**2*(b*d + 2*c*d*x)**m/(8*c**4*m**4 +
128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 84*a*b**3*c**3*m*x**2*(b*d + 2*c*d*x)**m/(8*c**4*m*
*4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 48*a*b**2*c**4*m**3*x**3*(b*d + 2*c*d*x)**m/(8*
c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 504*a*b**2*c**4*m**2*x**3*(b*d + 2*c*d*x
)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 1296*a*b**2*c**4*m*x**3*(b*d + 2
*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 840*a*b**2*c**4*x**3*(b*d
+ 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 60*a*b*c**5*m**3*x**4*(
b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 660*a*b*c**5*m**2*x
**4*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 1860*a*b*c**5*
m*x**4*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 1260*a*b*c*
*5*x**4*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 24*a*c**6*
m**3*x**5*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 264*a*c*
*6*m**2*x**5*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 744*a
*c**6*m*x**5*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 504*a
*c**6*x**5*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) - 3*b**7*
(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 6*b**6*c*m*x*(b*d
+ 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) - 6*b**5*c**2*m**2*x**2*(
b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) - 6*b**5*c**2*m*x**2*
(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 4*b**4*c**3*m**3*x
**3*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 12*b**4*c**3*m
**2*x**3*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 8*b**4*c*
*3*m*x**3*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 20*b**3*
c**4*m**3*x**4*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4) + 150
*b**3*c**4*m**2*x**4*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c**4)
+ 340*b**3*c**4*m*x**4*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c*
*4) + 210*b**3*c**4*x**4*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m + 840*c
**4) + 36*b**2*c**5*m**3*x**5*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**4*m +
840*c**4) + 312*b**2*c**5*m**2*x**5*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**
4*m + 840*c**4) + 780*b**2*c**5*m*x**5*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*
c**4*m + 840*c**4) + 504*b**2*c**5*x**5*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408
*c**4*m + 840*c**4) + 28*b*c**6*m**3*x**6*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 14
08*c**4*m + 840*c**4) + 252*b*c**6*m**2*x**6*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 +
1408*c**4*m + 840*c**4) + 644*b*c**6*m*x**6*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 +
1408*c**4*m + 840*c**4) + 420*b*c**6*x**6*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1
408*c**4*m + 840*c**4) + 8*c**7*m**3*x**7*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 14
08*c**4*m + 840*c**4) + 72*c**7*m**2*x**7*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 14
08*c**4*m + 840*c**4) + 184*c**7*m*x**7*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408
*c**4*m + 840*c**4) + 120*c**7*x**7*(b*d + 2*c*d*x)**m/(8*c**4*m**4 + 128*c**4*m**3 + 688*c**4*m**2 + 1408*c**
4*m + 840*c**4), True))
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