3.1423 \(\int (b d+2 c d x)^m (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=141 \[ \frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac{(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)} \]

[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))

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Rubi [A]  time = 0.0786644, antiderivative size = 141, normalized size of antiderivative = 1., 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} \[ \frac{3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{m+3}}{128 c^4 d^3 (m+3)}-\frac{3 \left (b^2-4 a c\right ) (b d+2 c d x)^{m+5}}{128 c^4 d^5 (m+5)}-\frac{\left (b^2-4 a c\right )^3 (b d+2 c d x)^{m+1}}{128 c^4 d (m+1)}+\frac{(b d+2 c d x)^{m+7}}{128 c^4 d^7 (m+7)} \]

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*}

Mathematica [A]  time = 0.0910315, size = 103, normalized size = 0.73 \[ \frac{(b+2 c x) \left (-\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^4}{m+5}+\frac{3 \left (b^2-4 a c\right )^2 (b+2 c x)^2}{m+3}-\frac{\left (b^2-4 a c\right )^3}{m+1}+\frac{(b+2 c x)^6}{m+7}\right ) (d (b+2 c x))^m}{128 c^4} \]

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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Maple [B]  time = 0.048, size = 653, normalized size = 4.6 \begin{align*}{\frac{ \left ( 2\,cdx+bd \right ) ^{m} \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\,ab{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\,{x}^{6}{c}^{6}+12\,{a}^{2}{c}^{4}{m}^{3}{x}^{2}+12\,a{b}^{2}{c}^{3}{m}^{3}{x}^{2}+264\,ab{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\,{x}^{5}b{c}^{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\,ab{c}^{4}m{x}^{3}+252\,{x}^{4}a{c}^{5}-6\,{b}^{4}{c}^{2}{m}^{2}{x}^{2}+44\,{b}^{3}{c}^{3}m{x}^{3}+162\,{x}^{4}{b}^{2}{c}^{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\,{x}^{3}ab{c}^{4}-18\,{b}^{4}{c}^{2}m{x}^{2}+24\,{x}^{3}{b}^{3}{c}^{3}+60\,{a}^{3}{c}^{3}{m}^{2}-6\,{a}^{2}{b}^{2}{c}^{2}{m}^{2}+564\,{a}^{2}b{c}^{3}mx+420\,{x}^{2}{a}^{2}{c}^{4}-96\,a{b}^{3}{c}^{2}mx+168\,{x}^{2}a{b}^{2}{c}^{3}+6\,{b}^{5}cmx-12\,{x}^{2}{b}^{4}{c}^{2}+284\,{a}^{3}{c}^{3}m-72\,{a}^{2}{b}^{2}{c}^{2}m+420\,x{a}^{2}b{c}^{3}+6\,a{b}^{4}cm-84\,xa{b}^{3}{c}^{2}+6\,x{b}^{5}c+420\,{a}^{3}{c}^{3}-210\,{a}^{2}{b}^{2}{c}^{2}+42\,a{b}^{4}c-3\,{b}^{6} \right ) \left ( 2\,cx+b \right ) }{8\,{c}^{4} \left ({m}^{4}+16\,{m}^{3}+86\,{m}^{2}+176\,m+105 \right ) }} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 2.19783, size = 1474, normalized size = 10.45 \begin{align*} \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{align*}

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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Sympy [A]  time = 28.5141, size = 10271, normalized size = 72.84 \begin{align*} \text{result too large to display} \end{align*}

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)), (-320*a**3*b**4*c**3/(3
840*b**10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 92160
0*b**6*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) - 96*a**2*b**6*c**2/(3840*b*
*10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6
*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) - 1152*a**2*b**5*c**3*x/(3840*b**1
0*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c
**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 3840*a**2*b**3*c**5*x**3/(3840*b**
10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*
c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 5760*a**2*b**2*c**6*x**4/(3840*b*
*10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6
*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 4608*a**2*b*c**7*x**5/(3840*b**1
0*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c
**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 1536*a**2*c**8*x**6/(3840*b**10*c*
*4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*
d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 3840*a*b**5*c**4*x**3/(3840*b**10*c**4
*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d*
*7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 11520*a*b**4*c**5*x**4/(3840*b**10*c**4*
d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**
7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 11520*a*b**3*c**6*x**5/(3840*b**10*c**4*d
**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**7
*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 3840*a*b**2*c**7*x**6/(3840*b**10*c**4*d**
7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**7*x
**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 30*b**10*log(b/(2*c) + x)/(3840*b**10*c**4*d
**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**7
*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 11*b**10/(3840*b**10*c**4*d**7 + 46080*b**
9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**7*x**4 + 737280*
b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 360*b**9*c*x*log(b/(2*c) + x)/(3840*b**10*c**4*d**7 + 460
80*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**7*x**4 + 7
37280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 72*b**9*c*x/(3840*b**10*c**4*d**7 + 46080*b**9*c**5
*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**7*x**4 + 737280*b**5*c
**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 1800*b**8*c**2*x**2*log(b/(2*c) + x)/(3840*b**10*c**4*d**7 + 46
080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**7*x**4 +
737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 4800*b**7*c**3*x**3*log(b/(2*c) + x)/(3840*b**10*c
**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8
*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) - 1200*b**7*c**3*x**3/(3840*b**10*c**4*
d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6*c**8*d**
7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 7200*b**6*c**4*x**4*log(b/(2*c) + x)/(384
0*b**10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*
b**6*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) - 3240*b**6*c**4*x**4/(3840*b*
*10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 921600*b**6
*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 5760*b**5*c**5*x**5*log(b/(2*c)
+ x)/(3840*b**10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3
+ 921600*b**6*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) - 3168*b**5*c**5*x**5
/(3840*b**10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*x**3 + 92
1600*b**6*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) + 1920*b**4*c**6*x**6*log
(b/(2*c) + x)/(3840*b**10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d
**7*x**3 + 921600*b**6*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6) - 1056*b**4*
c**6*x**6/(3840*b**10*c**4*d**7 + 46080*b**9*c**5*d**7*x + 230400*b**8*c**6*d**7*x**2 + 614400*b**7*c**7*d**7*
x**3 + 921600*b**6*c**8*d**7*x**4 + 737280*b**5*c**9*d**7*x**5 + 245760*b**4*c**10*d**7*x**6), Eq(m, -7)), (-6
4*a**3*b**2*c**3/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**
5*x**3 + 8192*b**2*c**8*d**5*x**4) - 32*a**2*b**4*c**2/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**
4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) - 256*a**2*b**3*c**3*x/(512*b**6*c**4
*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x*
*4) + 512*a**2*b*c**5*x**3/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**
3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) + 256*a**2*c**6*x**4/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x
+ 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) + 48*a*b**6*c*log(b/(2*c)
+ x)/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 819
2*b**2*c**8*d**5*x**4) + 28*a*b**6*c/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 +
16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) + 384*a*b**5*c**2*x*log(b/(2*c) + x)/(512*b**6*c**4*d**
5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4)
+ 128*a*b**5*c**2*x/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*
d**5*x**3 + 8192*b**2*c**8*d**5*x**4) + 1152*a*b**4*c**3*x**2*log(b/(2*c) + x)/(512*b**6*c**4*d**5 + 4096*b**5
*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) + 1536*a*b**3
*c**4*x**3*log(b/(2*c) + x)/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b*
*3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) - 256*a*b**3*c**4*x**3/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5
*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) + 768*a*b**2*c**5*x**4*
log(b/(2*c) + x)/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**
5*x**3 + 8192*b**2*c**8*d**5*x**4) - 128*a*b**2*c**5*x**4/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*
b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) - 12*b**8*log(b/(2*c) + x)/(512*b*
*6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*
d**5*x**4) - 7*b**8/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*
d**5*x**3 + 8192*b**2*c**8*d**5*x**4) - 96*b**7*c*x*log(b/(2*c) + x)/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5
*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) - 32*b**7*c*x/(512*b**6
*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d*
*5*x**4) - 288*b**6*c**2*x**2*log(b/(2*c) + x)/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d
**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) - 384*b**5*c**3*x**3*log(b/(2*c) + x)/(512*b*
*6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*
d**5*x**4) + 192*b**5*c**3*x**3/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 1638
4*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) - 192*b**4*c**4*x**4*log(b/(2*c) + x)/(512*b**6*c**4*d**5 +
4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) + 41
6*b**4*c**4*x**4/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**
5*x**3 + 8192*b**2*c**8*d**5*x**4) + 384*b**3*c**5*x**5/(512*b**6*c**4*d**5 + 4096*b**5*c**5*d**5*x + 12288*b*
*4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*c**8*d**5*x**4) + 128*b**2*c**6*x**6/(512*b**6*c**4*
d**5 + 4096*b**5*c**5*d**5*x + 12288*b**4*c**6*d**5*x**2 + 16384*b**3*c**7*d**5*x**3 + 8192*b**2*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) - 19
2*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 + 102
4*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) + 28
4*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) + 42
0*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 + 1
408*c**4*m + 840*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 + 68
8*c**4*m**2 + 1408*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 + 688*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 + 688*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 + 84
0*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 + 84
0*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 + 1408*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 + 68
8*c**4*m**2 + 1408*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 + 1408*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 + 1408*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))

________________________________________________________________________________________

Giac [B]  time = 1.19964, size = 2033, normalized size = 14.42 \begin{align*} \text{result too large to display} \end{align*}

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)