3.522 \(\int (d x)^m (a^2+2 a b x^n+b^2 x^{2 n})^{3/2} \, dx\)

Optimal. Leaf size=238 \[ \frac{3 a^2 b^2 x^{n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+2 n+1) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+3 n+1) \left (a b+b^2 x^n\right )}+\frac{a^3 (d x)^{m+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left (a+b x^n\right )} \]

[Out]

(a^3*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(d*(1 + m)*(a + b*x^n)) + (3*a^2*b^2*x^(1 + n)*(d*x)^m
*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/((1 + m + n)*(a*b + b^2*x^n)) + (3*a*b^3*x^(1 + 2*n)*(d*x)^m*Sqrt[a^2 +
2*a*b*x^n + b^2*x^(2*n)])/((1 + m + 2*n)*(a*b + b^2*x^n)) + (b^4*x^(1 + 3*n)*(d*x)^m*Sqrt[a^2 + 2*a*b*x^n + b^
2*x^(2*n)])/((1 + m + 3*n)*(a*b + b^2*x^n))

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Rubi [A]  time = 0.0980303, antiderivative size = 238, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1355, 270, 20, 30} \[ \frac{3 a^2 b^2 x^{n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+n+1) \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{2 n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+2 n+1) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{3 n+1} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(m+3 n+1) \left (a b+b^2 x^n\right )}+\frac{a^3 (d x)^{m+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (m+1) \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d*x)^m*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

(a^3*(d*x)^(1 + m)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(d*(1 + m)*(a + b*x^n)) + (3*a^2*b^2*x^(1 + n)*(d*x)^m
*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/((1 + m + n)*(a*b + b^2*x^n)) + (3*a*b^3*x^(1 + 2*n)*(d*x)^m*Sqrt[a^2 +
2*a*b*x^n + b^2*x^(2*n)])/((1 + m + 2*n)*(a*b + b^2*x^n)) + (b^4*x^(1 + 3*n)*(d*x)^m*Sqrt[a^2 + 2*a*b*x^n + b^
2*x^(2*n)])/((1 + m + 3*n)*(a*b + b^2*x^n))

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int (d x)^m \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int (d x)^m \left (a b+b^2 x^n\right )^3 \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \left (a^3 b^3 (d x)^m+3 a^2 b^4 x^n (d x)^m+3 a b^5 x^{2 n} (d x)^m+b^6 x^{3 n} (d x)^m\right ) \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=\frac{a^3 (d x)^{1+m} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac{\left (3 a^2 b^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^n (d x)^m \, dx}{a b+b^2 x^n}+\frac{\left (3 a b^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^{2 n} (d x)^m \, dx}{a b+b^2 x^n}+\frac{\left (b^4 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^{3 n} (d x)^m \, dx}{a b+b^2 x^n}\\ &=\frac{a^3 (d x)^{1+m} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac{\left (3 a^2 b^2 x^{-m} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^{m+n} \, dx}{a b+b^2 x^n}+\frac{\left (3 a b^3 x^{-m} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^{m+2 n} \, dx}{a b+b^2 x^n}+\frac{\left (b^4 x^{-m} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}\right ) \int x^{m+3 n} \, dx}{a b+b^2 x^n}\\ &=\frac{a^3 (d x)^{1+m} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{d (1+m) \left (a+b x^n\right )}+\frac{3 a^2 b^2 x^{1+n} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1+m+n) \left (a b+b^2 x^n\right )}+\frac{3 a b^3 x^{1+2 n} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1+m+2 n) \left (a b+b^2 x^n\right )}+\frac{b^4 x^{1+3 n} (d x)^m \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1+m+3 n) \left (a b+b^2 x^n\right )}\\ \end{align*}

Mathematica [A]  time = 0.109328, size = 90, normalized size = 0.38 \[ \frac{x (d x)^m \left (\left (a+b x^n\right )^2\right )^{3/2} \left (\frac{3 a^2 b x^n}{m+n+1}+\frac{a^3}{m+1}+\frac{3 a b^2 x^{2 n}}{m+2 n+1}+\frac{b^3 x^{3 n}}{m+3 n+1}\right )}{\left (a+b x^n\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d*x)^m*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

(x*(d*x)^m*((a + b*x^n)^2)^(3/2)*(a^3/(1 + m) + (3*a^2*b*x^n)/(1 + m + n) + (3*a*b^2*x^(2*n))/(1 + m + 2*n) +
(b^3*x^(3*n))/(1 + m + 3*n)))/(a + b*x^n)^3

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Maple [C]  time = 0.055, size = 532, normalized size = 2.2 \begin{align*}{\frac{x \left ({a}^{3}+3\,{b}^{3}{m}^{2}n \left ({x}^{n} \right ) ^{3}+2\,{b}^{3}m{n}^{2} \left ({x}^{n} \right ) ^{3}+3\,a{b}^{2}{m}^{3} \left ({x}^{n} \right ) ^{2}+6\,{b}^{3}mn \left ({x}^{n} \right ) ^{3}+3\,{a}^{2}b{m}^{3}{x}^{n}+9\,a{b}^{2}{m}^{2} \left ({x}^{n} \right ) ^{2}+9\,a{b}^{2}{n}^{2} \left ({x}^{n} \right ) ^{2}+9\,{a}^{2}b{m}^{2}{x}^{n}+18\,{a}^{2}b{n}^{2}{x}^{n}+9\,ma{b}^{2} \left ({x}^{n} \right ) ^{2}+12\,a{b}^{2} \left ({x}^{n} \right ) ^{2}n+9\,m{a}^{2}b{x}^{n}+15\,{a}^{2}bn{x}^{n}+{b}^{3} \left ({x}^{n} \right ) ^{3}+{a}^{3}{m}^{3}+3\,{a}^{3}{m}^{2}+11\,{a}^{3}{n}^{2}+6\,{a}^{3}n+6\,{a}^{3}{n}^{3}+3\,m{a}^{3}+6\,{a}^{3}{m}^{2}n+11\,{a}^{3}m{n}^{2}+12\,{a}^{3}mn+{b}^{3}{m}^{3} \left ({x}^{n} \right ) ^{3}+3\,{b}^{3}{m}^{2} \left ({x}^{n} \right ) ^{3}+2\,{b}^{3}{n}^{2} \left ({x}^{n} \right ) ^{3}+3\,m{b}^{3} \left ({x}^{n} \right ) ^{3}+3\,{b}^{3} \left ({x}^{n} \right ) ^{3}n+3\,{a}^{2}b{x}^{n}+3\,a{b}^{2} \left ({x}^{n} \right ) ^{2}+12\,a{b}^{2}{m}^{2}n \left ({x}^{n} \right ) ^{2}+9\,a{b}^{2}m{n}^{2} \left ({x}^{n} \right ) ^{2}+15\,{a}^{2}b{m}^{2}n{x}^{n}+18\,{a}^{2}bm{n}^{2}{x}^{n}+24\,a{b}^{2}mn \left ({x}^{n} \right ) ^{2}+30\,{a}^{2}bmn{x}^{n} \right ) }{ \left ( a+b{x}^{n} \right ) \left ( 1+m \right ) \left ( 1+m+n \right ) \left ( 1+m+2\,n \right ) \left ( 1+m+3\,n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( idx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( id \right ) \pi +i \left ({\it csgn} \left ( idx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( idx \right ){\it csgn} \left ( id \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( x \right ) +2\,\ln \left ( d \right ) \right ) }{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x)

[Out]

((a+b*x^n)^2)^(1/2)/(a+b*x^n)*x*(a^3+3*b^3*m^2*n*(x^n)^3+2*b^3*m*n^2*(x^n)^3+3*a*b^2*m^3*(x^n)^2+6*b^3*m*n*(x^
n)^3+3*a^2*b*m^3*x^n+9*a*b^2*m^2*(x^n)^2+9*a*b^2*n^2*(x^n)^2+9*a^2*b*m^2*x^n+18*a^2*b*n^2*x^n+9*m*a*b^2*(x^n)^
2+12*a*b^2*(x^n)^2*n+9*m*a^2*b*x^n+15*a^2*b*n*x^n+b^3*(x^n)^3+a^3*m^3+3*a^3*m^2+11*a^3*n^2+6*a^3*n+6*a^3*n^3+3
*m*a^3+6*a^3*m^2*n+11*a^3*m*n^2+12*a^3*m*n+b^3*m^3*(x^n)^3+3*b^3*m^2*(x^n)^3+2*b^3*n^2*(x^n)^3+3*m*b^3*(x^n)^3
+3*b^3*(x^n)^3*n+3*a^2*b*x^n+3*a*b^2*(x^n)^2+12*a*b^2*m^2*n*(x^n)^2+9*a*b^2*m*n^2*(x^n)^2+15*a^2*b*m^2*n*x^n+1
8*a^2*b*m*n^2*x^n+24*a*b^2*m*n*(x^n)^2+30*a^2*b*m*n*x^n)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)*exp(1/2*m*(-I*csgn(
I*d*x)^3*Pi+I*csgn(I*d*x)^2*csgn(I*d)*Pi+I*csgn(I*d*x)^2*csgn(I*x)*Pi-I*csgn(I*d*x)*csgn(I*d)*csgn(I*x)*Pi+2*l
n(x)+2*ln(d)))

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Maxima [A]  time = 1.01157, size = 373, normalized size = 1.57 \begin{align*} \frac{{\left (m^{3} + 3 \, m^{2}{\left (2 \, n + 1\right )} + 6 \, n^{3} +{\left (11 \, n^{2} + 12 \, n + 3\right )} m + 11 \, n^{2} + 6 \, n + 1\right )} a^{3} d^{m} x x^{m} +{\left (m^{3} + 3 \, m^{2}{\left (n + 1\right )} +{\left (2 \, n^{2} + 6 \, n + 3\right )} m + 2 \, n^{2} + 3 \, n + 1\right )} b^{3} d^{m} x e^{\left (m \log \left (x\right ) + 3 \, n \log \left (x\right )\right )} + 3 \,{\left (m^{3} + m^{2}{\left (4 \, n + 3\right )} +{\left (3 \, n^{2} + 8 \, n + 3\right )} m + 3 \, n^{2} + 4 \, n + 1\right )} a b^{2} d^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )} + 3 \,{\left (m^{3} + m^{2}{\left (5 \, n + 3\right )} +{\left (6 \, n^{2} + 10 \, n + 3\right )} m + 6 \, n^{2} + 5 \, n + 1\right )} a^{2} b d^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}}{m^{4} + 2 \, m^{3}{\left (3 \, n + 2\right )} +{\left (11 \, n^{2} + 18 \, n + 6\right )} m^{2} + 6 \, n^{3} + 2 \,{\left (3 \, n^{3} + 11 \, n^{2} + 9 \, n + 2\right )} m + 11 \, n^{2} + 6 \, n + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x, algorithm="maxima")

[Out]

((m^3 + 3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1)*a^3*d^m*x*x^m + (m^3 + 3*m^2*(n +
1) + (2*n^2 + 6*n + 3)*m + 2*n^2 + 3*n + 1)*b^3*d^m*x*e^(m*log(x) + 3*n*log(x)) + 3*(m^3 + m^2*(4*n + 3) + (3*
n^2 + 8*n + 3)*m + 3*n^2 + 4*n + 1)*a*b^2*d^m*x*e^(m*log(x) + 2*n*log(x)) + 3*(m^3 + m^2*(5*n + 3) + (6*n^2 +
10*n + 3)*m + 6*n^2 + 5*n + 1)*a^2*b*d^m*x*e^(m*log(x) + n*log(x)))/(m^4 + 2*m^3*(3*n + 2) + (11*n^2 + 18*n +
6)*m^2 + 6*n^3 + 2*(3*n^3 + 11*n^2 + 9*n + 2)*m + 11*n^2 + 6*n + 1)

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Fricas [A]  time = 1.63259, size = 887, normalized size = 3.73 \begin{align*} \frac{{\left (b^{3} m^{3} + 3 \, b^{3} m^{2} + 3 \, b^{3} m + b^{3} + 2 \,{\left (b^{3} m + b^{3}\right )} n^{2} + 3 \,{\left (b^{3} m^{2} + 2 \, b^{3} m + b^{3}\right )} n\right )} x x^{3 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 3 \,{\left (a b^{2} m^{3} + 3 \, a b^{2} m^{2} + 3 \, a b^{2} m + a b^{2} + 3 \,{\left (a b^{2} m + a b^{2}\right )} n^{2} + 4 \,{\left (a b^{2} m^{2} + 2 \, a b^{2} m + a b^{2}\right )} n\right )} x x^{2 \, n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} + 3 \,{\left (a^{2} b m^{3} + 3 \, a^{2} b m^{2} + 3 \, a^{2} b m + a^{2} b + 6 \,{\left (a^{2} b m + a^{2} b\right )} n^{2} + 5 \,{\left (a^{2} b m^{2} + 2 \, a^{2} b m + a^{2} b\right )} n\right )} x x^{n} e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )} +{\left (a^{3} m^{3} + 6 \, a^{3} n^{3} + 3 \, a^{3} m^{2} + 3 \, a^{3} m + a^{3} + 11 \,{\left (a^{3} m + a^{3}\right )} n^{2} + 6 \,{\left (a^{3} m^{2} + 2 \, a^{3} m + a^{3}\right )} n\right )} x e^{\left (m \log \left (d\right ) + m \log \left (x\right )\right )}}{m^{4} + 6 \,{\left (m + 1\right )} n^{3} + 4 \, m^{3} + 11 \,{\left (m^{2} + 2 \, m + 1\right )} n^{2} + 6 \, m^{2} + 6 \,{\left (m^{3} + 3 \, m^{2} + 3 \, m + 1\right )} n + 4 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x, algorithm="fricas")

[Out]

((b^3*m^3 + 3*b^3*m^2 + 3*b^3*m + b^3 + 2*(b^3*m + b^3)*n^2 + 3*(b^3*m^2 + 2*b^3*m + b^3)*n)*x*x^(3*n)*e^(m*lo
g(d) + m*log(x)) + 3*(a*b^2*m^3 + 3*a*b^2*m^2 + 3*a*b^2*m + a*b^2 + 3*(a*b^2*m + a*b^2)*n^2 + 4*(a*b^2*m^2 + 2
*a*b^2*m + a*b^2)*n)*x*x^(2*n)*e^(m*log(d) + m*log(x)) + 3*(a^2*b*m^3 + 3*a^2*b*m^2 + 3*a^2*b*m + a^2*b + 6*(a
^2*b*m + a^2*b)*n^2 + 5*(a^2*b*m^2 + 2*a^2*b*m + a^2*b)*n)*x*x^n*e^(m*log(d) + m*log(x)) + (a^3*m^3 + 6*a^3*n^
3 + 3*a^3*m^2 + 3*a^3*m + a^3 + 11*(a^3*m + a^3)*n^2 + 6*(a^3*m^2 + 2*a^3*m + a^3)*n)*x*e^(m*log(d) + m*log(x)
))/(m^4 + 6*(m + 1)*n^3 + 4*m^3 + 11*(m^2 + 2*m + 1)*n^2 + 6*m^2 + 6*(m^3 + 3*m^2 + 3*m + 1)*n + 4*m + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 1.39822, size = 3671, normalized size = 15.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x, algorithm="giac")

[Out]

(b^3*m^3*x*x^(3*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m^2*n*x*x^(3*n)*e^(m*log(d) + m*log(x))*sgn(
b*x^n + a) + 2*b^3*m*n^2*x*x^(3*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*a*b^2*m^3*x*x^(2*n)*e^(m*log(d)
+ m*log(x))*sgn(b*x^n + a) + b^3*m^3*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 12*a*b^2*m^2*n*x*x^(2*
n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m^2*n*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a
*b^2*m*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 2*b^3*m*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x))*sg
n(b*x^n + a) + 3*a^2*b*m^3*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*a*b^2*m^3*x*x^n*e^(m*log(d) + m*lo
g(x))*sgn(b*x^n + a) + b^3*m^3*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 15*a^2*b*m^2*n*x*x^n*e^(m*log(d)
+ m*log(x))*sgn(b*x^n + a) + 12*a*b^2*m^2*n*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m^2*n*x*x^n*
e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 18*a^2*b*m*n^2*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a*b^2
*m*n^2*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 2*b^3*m*n^2*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a)
+ a^3*m^3*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*a^2*b*m^3*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3
*a*b^2*m^3*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + b^3*m^3*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 6*a^3
*m^2*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 15*a^2*b*m^2*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 12
*a*b^2*m^2*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m^2*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) +
11*a^3*m*n^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 18*a^2*b*m*n^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n +
a) + 9*a*b^2*m*n^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 2*b^3*m*n^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n
+ a) + 6*a^3*n^3*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m^2*x*x^(3*n)*e^(m*log(d) + m*log(x))*sgn(b*
x^n + a) + 6*b^3*m*n*x*x^(3*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 2*b^3*n^2*x*x^(3*n)*e^(m*log(d) + m*lo
g(x))*sgn(b*x^n + a) + 9*a*b^2*m^2*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m^2*x*x^(2*n)*e^(m
*log(d) + m*log(x))*sgn(b*x^n + a) + 24*a*b^2*m*n*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 6*b^3*m*n
*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a*b^2*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n +
a) + 2*b^3*n^2*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a^2*b*m^2*x*x^n*e^(m*log(d) + m*log(x))*sg
n(b*x^n + a) + 9*a*b^2*m^2*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m^2*x*x^n*e^(m*log(d) + m*log(
x))*sgn(b*x^n + a) + 30*a^2*b*m*n*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 24*a*b^2*m*n*x*x^n*e^(m*log(d
) + m*log(x))*sgn(b*x^n + a) + 6*b^3*m*n*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 18*a^2*b*n^2*x*x^n*e^(
m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a*b^2*n^2*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 2*b^3*n^2*x*x
^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*a^3*m^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a^2*b*m^2*x
*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a*b^2*m^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m^2*x*e
^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 12*a^3*m*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 30*a^2*b*m*n*x*e
^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 24*a*b^2*m*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 6*b^3*m*n*x*e^
(m*log(d) + m*log(x))*sgn(b*x^n + a) + 11*a^3*n^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 18*a^2*b*n^2*x*e^
(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a*b^2*n^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 2*b^3*n^2*x*e^(m
*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m*x*x^(3*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*n*x*x^(3
*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a*b^2*m*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^
3*m*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 12*a*b^2*n*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n
+ a) + 3*b^3*n*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a^2*b*m*x*x^n*e^(m*log(d) + m*log(x))*sgn(
b*x^n + a) + 9*a*b^2*m*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*b^3*m*x*x^n*e^(m*log(d) + m*log(x))*sg
n(b*x^n + a) + 15*a^2*b*n*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 12*a*b^2*n*x*x^n*e^(m*log(d) + m*log(
x))*sgn(b*x^n + a) + 3*b^3*n*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*a^3*m*x*e^(m*log(d) + m*log(x))*
sgn(b*x^n + a) + 9*a^2*b*m*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 9*a*b^2*m*x*e^(m*log(d) + m*log(x))*sgn(
b*x^n + a) + 3*b^3*m*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 6*a^3*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n +
a) + 15*a^2*b*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 12*a*b^2*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a)
+ 3*b^3*n*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + b^3*x*x^(3*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3
*a*b^2*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + b^3*x*x^(2*n)*e^(m*log(d) + m*log(x))*sgn(b*x^n + a)
+ 3*a^2*b*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*a*b^2*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a)
+ b^3*x*x^n*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + a^3*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*a^2*b*x
*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + 3*a*b^2*x*e^(m*log(d) + m*log(x))*sgn(b*x^n + a) + b^3*x*e^(m*log(d)
+ m*log(x))*sgn(b*x^n + a))/(m^4 + 6*m^3*n + 11*m^2*n^2 + 6*m*n^3 + 4*m^3 + 18*m^2*n + 22*m*n^2 + 6*n^3 + 6*m
^2 + 18*m*n + 11*n^2 + 4*m + 6*n + 1)