3.383 \(\int \frac {(d+e x^r)^2 (a+b \log (c x^n))}{x^3} \, dx\)

Optimal. Leaf size=135 \[ -\frac {d^2 \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {2 d e x^{r-2} \left (a+b \log \left (c x^n\right )\right )}{2-r}-\frac {e^2 x^{-2 (1-r)} \left (a+b \log \left (c x^n\right )\right )}{2 (1-r)}-\frac {b d^2 n}{4 x^2}-\frac {2 b d e n x^{r-2}}{(2-r)^2}-\frac {b e^2 n x^{-2 (1-r)}}{4 (1-r)^2} \]

[Out]

-1/4*b*d^2*n/x^2-1/4*b*e^2*n/(1-r)^2/(x^(2-2*r))-2*b*d*e*n*x^(-2+r)/(2-r)^2-1/2*d^2*(a+b*ln(c*x^n))/x^2-1/2*e^
2*(a+b*ln(c*x^n))/(1-r)/(x^(2-2*r))-2*d*e*x^(-2+r)*(a+b*ln(c*x^n))/(2-r)

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Rubi [A]  time = 0.16, antiderivative size = 114, normalized size of antiderivative = 0.84, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {270, 2334, 12, 14} \[ -\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {4 d e x^{r-2}}{2-r}+\frac {e^2 x^{-2 (1-r)}}{1-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b d^2 n}{4 x^2}-\frac {2 b d e n x^{r-2}}{(2-r)^2}-\frac {b e^2 n x^{-2 (1-r)}}{4 (1-r)^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^3,x]

[Out]

-(b*d^2*n)/(4*x^2) - (b*e^2*n)/(4*(1 - r)^2*x^(2*(1 - r))) - (2*b*d*e*n*x^(-2 + r))/(2 - r)^2 - ((d^2/x^2 + e^
2/((1 - r)*x^(2*(1 - r))) + (4*d*e*x^(-2 + r))/(2 - r))*(a + b*Log[c*x^n]))/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rubi steps

\begin {align*} \int \frac {\left (d+e x^r\right )^2 \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=-\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {e^2 x^{-2 (1-r)}}{1-r}+\frac {4 d e x^{-2+r}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-d^2 \left (2-3 r+r^2\right )+4 d e (-1+r) x^r+e^2 (-2+r) x^{2 r}}{2 (1-r) (2-r) x^3} \, dx\\ &=-\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {e^2 x^{-2 (1-r)}}{1-r}+\frac {4 d e x^{-2+r}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \frac {-d^2 \left (2-3 r+r^2\right )+4 d e (-1+r) x^r+e^2 (-2+r) x^{2 r}}{x^3} \, dx}{2 \left (2-3 r+r^2\right )}\\ &=-\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {e^2 x^{-2 (1-r)}}{1-r}+\frac {4 d e x^{-2+r}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(b n) \int \left (-\frac {d^2 (-2+r) (-1+r)}{x^3}+4 d e (-1+r) x^{-3+r}+e^2 (-2+r) x^{-3+2 r}\right ) \, dx}{2 \left (2-3 r+r^2\right )}\\ &=-\frac {b d^2 n}{4 x^2}-\frac {b e^2 n x^{-2 (1-r)}}{4 (1-r)^2}-\frac {2 b d e n x^{-2+r}}{(2-r)^2}-\frac {1}{2} \left (\frac {d^2}{x^2}+\frac {e^2 x^{-2 (1-r)}}{1-r}+\frac {4 d e x^{-2+r}}{2-r}\right ) \left (a+b \log \left (c x^n\right )\right )\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 120, normalized size = 0.89 \[ \frac {a \left (-2 d^2+\frac {8 d e x^r}{r-2}+\frac {2 e^2 x^{2 r}}{r-1}\right )+2 b \log \left (c x^n\right ) \left (-d^2+\frac {4 d e x^r}{r-2}+\frac {e^2 x^{2 r}}{r-1}\right )+b n \left (-d^2-\frac {8 d e x^r}{(r-2)^2}-\frac {e^2 x^{2 r}}{(r-1)^2}\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x^r)^2*(a + b*Log[c*x^n]))/x^3,x]

[Out]

(b*n*(-d^2 - (8*d*e*x^r)/(-2 + r)^2 - (e^2*x^(2*r))/(-1 + r)^2) + a*(-2*d^2 + (8*d*e*x^r)/(-2 + r) + (2*e^2*x^
(2*r))/(-1 + r)) + 2*b*(-d^2 + (4*d*e*x^r)/(-2 + r) + (e^2*x^(2*r))/(-1 + r))*Log[c*x^n])/(4*x^2)

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fricas [B]  time = 0.49, size = 457, normalized size = 3.39 \[ -\frac {{\left (b d^{2} n + 2 \, a d^{2}\right )} r^{4} + 4 \, b d^{2} n - 6 \, {\left (b d^{2} n + 2 \, a d^{2}\right )} r^{3} + 8 \, a d^{2} + 13 \, {\left (b d^{2} n + 2 \, a d^{2}\right )} r^{2} - 12 \, {\left (b d^{2} n + 2 \, a d^{2}\right )} r - {\left (2 \, a e^{2} r^{3} - 4 \, b e^{2} n - 8 \, a e^{2} - {\left (b e^{2} n + 10 \, a e^{2}\right )} r^{2} + 4 \, {\left (b e^{2} n + 4 \, a e^{2}\right )} r + 2 \, {\left (b e^{2} r^{3} - 5 \, b e^{2} r^{2} + 8 \, b e^{2} r - 4 \, b e^{2}\right )} \log \relax (c) + 2 \, {\left (b e^{2} n r^{3} - 5 \, b e^{2} n r^{2} + 8 \, b e^{2} n r - 4 \, b e^{2} n\right )} \log \relax (x)\right )} x^{2 \, r} - 8 \, {\left (a d e r^{3} - b d e n - 2 \, a d e - {\left (b d e n + 4 \, a d e\right )} r^{2} + {\left (2 \, b d e n + 5 \, a d e\right )} r + {\left (b d e r^{3} - 4 \, b d e r^{2} + 5 \, b d e r - 2 \, b d e\right )} \log \relax (c) + {\left (b d e n r^{3} - 4 \, b d e n r^{2} + 5 \, b d e n r - 2 \, b d e n\right )} \log \relax (x)\right )} x^{r} + 2 \, {\left (b d^{2} r^{4} - 6 \, b d^{2} r^{3} + 13 \, b d^{2} r^{2} - 12 \, b d^{2} r + 4 \, b d^{2}\right )} \log \relax (c) + 2 \, {\left (b d^{2} n r^{4} - 6 \, b d^{2} n r^{3} + 13 \, b d^{2} n r^{2} - 12 \, b d^{2} n r + 4 \, b d^{2} n\right )} \log \relax (x)}{4 \, {\left (r^{4} - 6 \, r^{3} + 13 \, r^{2} - 12 \, r + 4\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n))/x^3,x, algorithm="fricas")

[Out]

-1/4*((b*d^2*n + 2*a*d^2)*r^4 + 4*b*d^2*n - 6*(b*d^2*n + 2*a*d^2)*r^3 + 8*a*d^2 + 13*(b*d^2*n + 2*a*d^2)*r^2 -
12*(b*d^2*n + 2*a*d^2)*r - (2*a*e^2*r^3 - 4*b*e^2*n - 8*a*e^2 - (b*e^2*n + 10*a*e^2)*r^2 + 4*(b*e^2*n + 4*a*e
^2)*r + 2*(b*e^2*r^3 - 5*b*e^2*r^2 + 8*b*e^2*r - 4*b*e^2)*log(c) + 2*(b*e^2*n*r^3 - 5*b*e^2*n*r^2 + 8*b*e^2*n*
r - 4*b*e^2*n)*log(x))*x^(2*r) - 8*(a*d*e*r^3 - b*d*e*n - 2*a*d*e - (b*d*e*n + 4*a*d*e)*r^2 + (2*b*d*e*n + 5*a
*d*e)*r + (b*d*e*r^3 - 4*b*d*e*r^2 + 5*b*d*e*r - 2*b*d*e)*log(c) + (b*d*e*n*r^3 - 4*b*d*e*n*r^2 + 5*b*d*e*n*r
- 2*b*d*e*n)*log(x))*x^r + 2*(b*d^2*r^4 - 6*b*d^2*r^3 + 13*b*d^2*r^2 - 12*b*d^2*r + 4*b*d^2)*log(c) + 2*(b*d^2
*n*r^4 - 6*b*d^2*n*r^3 + 13*b*d^2*n*r^2 - 12*b*d^2*n*r + 4*b*d^2*n)*log(x))/((r^4 - 6*r^3 + 13*r^2 - 12*r + 4)
*x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{r} + d\right )}^{2} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n))/x^3,x, algorithm="giac")

[Out]

integrate((e*x^r + d)^2*(b*log(c*x^n) + a)/x^3, x)

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maple [C]  time = 0.33, size = 1923, normalized size = 14.24 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+e*x^r)^2*(b*ln(c*x^n)+a)/x^3,x)

[Out]

-1/2*b*(-e^2*(x^r)^2*r+d^2*r^2-4*d*e*r*x^r+2*(x^r)^2*e^2-3*d^2*r+4*d*e*x^r+2*d^2)/x^2/(r-1)/(r-2)*ln(x^n)-1/4*
(4*I*Pi*b*d*e*r^3*x^r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+8*ln(c)*b*e^2*(x^r)^2-2*a*e^2*r^3*(x^r)^2+16*a*d*e*x
^r+10*a*e^2*r^2*(x^r)^2-16*a*e^2*r*(x^r)^2+4*b*e^2*n*(x^r)^2+2*b*d^2*r^4*ln(c)-12*b*d^2*r^3*ln(c)+26*b*d^2*r^2
*ln(c)-24*b*d^2*r*ln(c)+8*a*d^2+b*d^2*n*r^4-6*b*d^2*n*r^3+4*b*d^2*n+8*a*e^2*(x^r)^2+8*b*d^2*ln(c)+2*a*d^2*r^4+
13*b*d^2*n*r^2-12*b*d^2*n*r+26*a*d^2*r^2-24*a*d^2*r-12*a*d^2*r^3+12*I*Pi*b*d^2*r*csgn(I*c*x^n)^3+5*I*Pi*b*e^2*
r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-4*I*Pi*b*e^2*csgn(I
*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-5*I*Pi*b*e^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+16*I*Pi*b*d
*e*r^2*csgn(I*c*x^n)^2*csgn(I*c)*x^r-5*I*Pi*b*e^2*r^2*csgn(I*c*x^n)^3*(x^r)^2-8*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)*x^r+16*I*Pi*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+6*I*Pi*b*d^2*r^3*csgn(I*c*x^n)^3-16*I*P
i*b*d*e*r^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+20*I*Pi*b*d*e*r*x^r*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+b*
e^2*n*r^2*(x^r)^2-8*a*d*e*r^3*x^r+32*a*d*e*r^2*x^r-40*a*d*e*r*x^r-4*b*e^2*n*r*(x^r)^2+8*b*d*e*n*x^r+10*ln(c)*b
*e^2*r^2*(x^r)^2-16*ln(c)*b*e^2*r*(x^r)^2+16*b*d*e*x^r*ln(c)-2*ln(c)*b*e^2*r^3*(x^r)^2+I*Pi*b*e^2*r^3*csgn(I*c
*x^n)^3*(x^r)^2-6*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*c*x^n)^2-16*b*d*e*n*r*x^r+8*b*d*e*n*r^2*x^r+32*b*d*e*r^2*x^r
*ln(c)-40*b*d*e*r*x^r*ln(c)-8*b*d*e*r^3*x^r*ln(c)+13*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^n)^2+13*I*Pi*b*d^2*
r^2*csgn(I*c*x^n)^2*csgn(I*c)-8*I*Pi*b*d*e*csgn(I*c*x^n)^3*x^r+4*I*Pi*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^
2+8*I*Pi*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+8*I*Pi*b*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+5*I*Pi*b*e^2*r^2*csg
n(I*c*x^n)^2*csgn(I*c)*(x^r)^2-16*I*Pi*b*d*e*r^2*csgn(I*c*x^n)^3*x^r-13*I*Pi*b*d^2*r^2*csgn(I*x^n)*csgn(I*c*x^
n)*csgn(I*c)+6*I*Pi*b*d^2*r^3*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-8*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(
x^r)^2-4*I*Pi*b*d^2*csgn(I*c*x^n)^3-I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+12*I*Pi*b*d^2*r*csgn(I*
c)*csgn(I*x^n)*csgn(I*c*x^n)-I*Pi*b*e^2*r^3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+4*I*Pi*b*d*e*r^3*x^r*csgn(I*c*x^
n)^3+20*I*Pi*b*d*e*r*x^r*csgn(I*c*x^n)^3-8*I*Pi*b*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-6*I*Pi*b*d^2*r^3*csg
n(I*x^n)*csgn(I*c*x^n)^2-12*I*Pi*b*d^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2-12*I*Pi*b*d^2*r*csgn(I*c)*csgn(I*c*x^n)^2
+I*Pi*b*e^2*r^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+I*Pi*b*d^2*r^4*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi*b*
d^2*r^4*csgn(I*c*x^n)^2*csgn(I*c)+4*I*Pi*b*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-4*I*Pi*b*d^2*csgn(I*x^n)*csgn
(I*c*x^n)*csgn(I*c)-13*I*Pi*b*d^2*r^2*csgn(I*c*x^n)^3+4*I*Pi*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+8*I*Pi*b*e^2*r*
csgn(I*c*x^n)^3*(x^r)^2+8*I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2-20*I*Pi*b*d*e*r*x^r*csgn(I*
x^n)*csgn(I*c*x^n)^2-20*I*Pi*b*d*e*r*x^r*csgn(I*c)*csgn(I*c*x^n)^2+4*I*Pi*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)-I*Pi
*b*d^2*r^4*csgn(I*c*x^n)^3-4*I*Pi*b*d*e*r^3*x^r*csgn(I*x^n)*csgn(I*c*x^n)^2-4*I*Pi*b*d*e*r^3*x^r*csgn(I*c)*csg
n(I*c*x^n)^2-4*I*Pi*b*e^2*csgn(I*c*x^n)^3*(x^r)^2)/(r-1)^2/x^2/(r-2)^2

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x^r)^2*(a+b*log(c*x^n))/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(r-3>0)', see `assume?` for mor
e details)Is r-3 equal to -1?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d+e\,x^r\right )}^2\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^r)^2*(a + b*log(c*x^n)))/x^3,x)

[Out]

int(((d + e*x^r)^2*(a + b*log(c*x^n)))/x^3, x)

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sympy [A]  time = 15.65, size = 2807, normalized size = 20.79 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+e*x**r)**2*(a+b*ln(c*x**n))/x**3,x)

[Out]

Piecewise((-a*d**2/(2*x**2) - 2*a*d*e/x + a*e**2*log(x) + b*d**2*(-n/(4*x**2) - log(c*x**n)/(2*x**2)) + 2*b*d*
e*(-n/x - log(c*x**n)/x) - b*e**2*Piecewise((-log(c)*log(x), Eq(n, 0)), (-log(c*x**n)**2/(2*n), True)), Eq(r,
1)), (-a*d**2/(2*x**2) + 2*a*d*e*log(x) + a*e**2*x**2/2 - b*d**2*n*log(x)/(2*x**2) - b*d**2*n/(4*x**2) - b*d**
2*log(c)/(2*x**2) + b*d*e*n*log(x)**2 + 2*b*d*e*log(c)*log(x) + b*e**2*n*x**2*log(x)/2 - b*e**2*n*x**2/4 + b*e
**2*x**2*log(c)/2, Eq(r, 2)), (-2*a*d**2*r**4/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2
) + 12*a*d**2*r**3/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 26*a*d**2*r**2/(4*r**4*
x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 24*a*d**2*r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*
x**2 - 48*r*x**2 + 16*x**2) - 8*a*d**2/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 8*a
*d*e*r**3*x**r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 32*a*d*e*r**2*x**r/(4*r**4*
x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 40*a*d*e*r*x**r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r
**2*x**2 - 48*r*x**2 + 16*x**2) - 16*a*d*e*x**r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x*
*2) + 2*a*e**2*r**3*x**(2*r)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 10*a*e**2*r**
2*x**(2*r)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 16*a*e**2*r*x**(2*r)/(4*r**4*x*
*2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 8*a*e**2*x**(2*r)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r
**2*x**2 - 48*r*x**2 + 16*x**2) - 2*b*d**2*n*r**4*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**
2 + 16*x**2) - b*d**2*n*r**4/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 12*b*d**2*n*r
**3*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 6*b*d**2*n*r**3/(4*r**4*x**2 -
24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 26*b*d**2*n*r**2*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*
r**2*x**2 - 48*r*x**2 + 16*x**2) - 13*b*d**2*n*r**2/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 1
6*x**2) + 24*b*d**2*n*r*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 12*b*d**2*n
*r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 8*b*d**2*n*log(x)/(4*r**4*x**2 - 24*r**
3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 4*b*d**2*n/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x*
*2 + 16*x**2) - 2*b*d**2*r**4*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 12*b*
d**2*r**3*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 26*b*d**2*r**2*log(c)/(4*
r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 24*b*d**2*r*log(c)/(4*r**4*x**2 - 24*r**3*x**
2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 8*b*d**2*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x*
*2 + 16*x**2) + 8*b*d*e*n*r**3*x**r*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) -
32*b*d*e*n*r**2*x**r*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 8*b*d*e*n*r**
2*x**r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 40*b*d*e*n*r*x**r*log(x)/(4*r**4*x*
*2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 16*b*d*e*n*r*x**r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r
**2*x**2 - 48*r*x**2 + 16*x**2) - 16*b*d*e*n*x**r*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**
2 + 16*x**2) - 8*b*d*e*n*x**r/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 8*b*d*e*r**3
*x**r*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 32*b*d*e*r**2*x**r*log(c)/(4*
r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 40*b*d*e*r*x**r*log(c)/(4*r**4*x**2 - 24*r**3
*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 16*b*d*e*x**r*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2
- 48*r*x**2 + 16*x**2) + 2*b*e**2*n*r**3*x**(2*r)*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**
2 + 16*x**2) - 10*b*e**2*n*r**2*x**(2*r)*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x*
*2) - b*e**2*n*r**2*x**(2*r)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 16*b*e**2*n*r
*x**(2*r)*log(x)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) + 4*b*e**2*n*r*x**(2*r)/(4*
r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 8*b*e**2*n*x**(2*r)*log(x)/(4*r**4*x**2 - 24*
r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 4*b*e**2*n*x**(2*r)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x*
*2 - 48*r*x**2 + 16*x**2) + 2*b*e**2*r**3*x**(2*r)*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x*
*2 + 16*x**2) - 10*b*e**2*r**2*x**(2*r)*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**
2) + 16*b*e**2*r*x**(2*r)*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2) - 8*b*e**2*
x**(2*r)*log(c)/(4*r**4*x**2 - 24*r**3*x**2 + 52*r**2*x**2 - 48*r*x**2 + 16*x**2), True))

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