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

Optimal. Leaf size=245 \[ -\frac{6 b d^2 e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac{3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac{3 b d e^2 n x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r^2}+\frac{3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )^2}{2 r}-\frac{2 b e^3 n x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{9 r^2}+\frac{e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )^2}{3 r}+\frac{6 b^2 d^2 e n^2 x^r}{r^3}+\frac{3 b^2 d e^2 n^2 x^{2 r}}{4 r^3}+\frac{2 b^2 e^3 n^2 x^{3 r}}{27 r^3} \]

[Out]

(6*b^2*d^2*e*n^2*x^r)/r^3 + (3*b^2*d*e^2*n^2*x^(2*r))/(4*r^3) + (2*b^2*e^3*n^2*x^(3*r))/(27*r^3) - (6*b*d^2*e*
n*x^r*(a + b*Log[c*x^n]))/r^2 - (3*b*d*e^2*n*x^(2*r)*(a + b*Log[c*x^n]))/(2*r^2) - (2*b*e^3*n*x^(3*r)*(a + b*L
og[c*x^n]))/(9*r^2) + (3*d^2*e*x^r*(a + b*Log[c*x^n])^2)/r + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n])^2)/(2*r) + (e
^3*x^(3*r)*(a + b*Log[c*x^n])^2)/(3*r) + (d^3*(a + b*Log[c*x^n])^3)/(3*b*n)

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Rubi [A]  time = 0.303403, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2353, 2302, 30, 2305, 2304} \[ -\frac{6 b d^2 e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac{3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac{3 b d e^2 n x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r^2}+\frac{3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )^2}{2 r}-\frac{2 b e^3 n x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{9 r^2}+\frac{e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )^2}{3 r}+\frac{6 b^2 d^2 e n^2 x^r}{r^3}+\frac{3 b^2 d e^2 n^2 x^{2 r}}{4 r^3}+\frac{2 b^2 e^3 n^2 x^{3 r}}{27 r^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b^2*d^2*e*n^2*x^r)/r^3 + (3*b^2*d*e^2*n^2*x^(2*r))/(4*r^3) + (2*b^2*e^3*n^2*x^(3*r))/(27*r^3) - (6*b*d^2*e*
n*x^r*(a + b*Log[c*x^n]))/r^2 - (3*b*d*e^2*n*x^(2*r)*(a + b*Log[c*x^n]))/(2*r^2) - (2*b*e^3*n*x^(3*r)*(a + b*L
og[c*x^n]))/(9*r^2) + (3*d^2*e*x^r*(a + b*Log[c*x^n])^2)/r + (3*d*e^2*x^(2*r)*(a + b*Log[c*x^n])^2)/(2*r) + (e
^3*x^(3*r)*(a + b*Log[c*x^n])^2)/(3*r) + (d^3*(a + b*Log[c*x^n])^3)/(3*b*n)

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

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

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int \frac{\left (d+e x^r\right )^3 \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx &=\int \left (\frac{d^3 \left (a+b \log \left (c x^n\right )\right )^2}{x}+3 d^2 e x^{-1+r} \left (a+b \log \left (c x^n\right )\right )^2+3 d e^2 x^{-1+2 r} \left (a+b \log \left (c x^n\right )\right )^2+e^3 x^{-1+3 r} \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=d^3 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\left (3 d^2 e\right ) \int x^{-1+r} \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\left (3 d e^2\right ) \int x^{-1+2 r} \left (a+b \log \left (c x^n\right )\right )^2 \, dx+e^3 \int x^{-1+3 r} \left (a+b \log \left (c x^n\right )\right )^2 \, dx\\ &=\frac{3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac{3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )^2}{2 r}+\frac{e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )^2}{3 r}+\frac{d^3 \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}-\frac{\left (6 b d^2 e n\right ) \int x^{-1+r} \left (a+b \log \left (c x^n\right )\right ) \, dx}{r}-\frac{\left (3 b d e^2 n\right ) \int x^{-1+2 r} \left (a+b \log \left (c x^n\right )\right ) \, dx}{r}-\frac{\left (2 b e^3 n\right ) \int x^{-1+3 r} \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 r}\\ &=\frac{6 b^2 d^2 e n^2 x^r}{r^3}+\frac{3 b^2 d e^2 n^2 x^{2 r}}{4 r^3}+\frac{2 b^2 e^3 n^2 x^{3 r}}{27 r^3}-\frac{6 b d^2 e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}-\frac{3 b d e^2 n x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r^2}-\frac{2 b e^3 n x^{3 r} \left (a+b \log \left (c x^n\right )\right )}{9 r^2}+\frac{3 d^2 e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac{3 d e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )^2}{2 r}+\frac{e^3 x^{3 r} \left (a+b \log \left (c x^n\right )\right )^2}{3 r}+\frac{d^3 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}\\ \end{align*}

Mathematica [A]  time = 0.433927, size = 262, normalized size = 1.07 \[ \frac{e n x^r \left (18 a^2 r^2 \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )-6 a b n r \left (108 d^2+27 d e x^r+4 e^2 x^{2 r}\right )+b^2 n^2 \left (648 d^2+81 d e x^r+8 e^2 x^{2 r}\right )\right )+108 a^2 d^3 n r^3 \log (x)+18 b r^2 \log ^2\left (c x^n\right ) \left (6 a d^3 r+b e n x^r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )\right )-6 b e n r x^r \log \left (c x^n\right ) \left (b n \left (108 d^2+27 d e x^r+4 e^2 x^{2 r}\right )-6 a r \left (18 d^2+9 d e x^r+2 e^2 x^{2 r}\right )\right )+36 b^2 d^3 r^3 \log ^3\left (c x^n\right )}{108 n r^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*n*x^r*(18*a^2*r^2*(18*d^2 + 9*d*e*x^r + 2*e^2*x^(2*r)) - 6*a*b*n*r*(108*d^2 + 27*d*e*x^r + 4*e^2*x^(2*r)) +
b^2*n^2*(648*d^2 + 81*d*e*x^r + 8*e^2*x^(2*r))) + 108*a^2*d^3*n*r^3*Log[x] - 6*b*e*n*r*x^r*(-6*a*r*(18*d^2 +
9*d*e*x^r + 2*e^2*x^(2*r)) + b*n*(108*d^2 + 27*d*e*x^r + 4*e^2*x^(2*r)))*Log[c*x^n] + 18*b*r^2*(6*a*d^3*r + b*
e*n*x^r*(18*d^2 + 9*d*e*x^r + 2*e^2*x^(2*r)))*Log[c*x^n]^2 + 36*b^2*d^3*r^3*Log[c*x^n]^3)/(108*n*r^3)

________________________________________________________________________________________

Maple [C]  time = 0.378, size = 3984, normalized size = 16.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/r*a^2*e^3*(x^r)^3+1/3*b^2*d^3*n^2*ln(x)^3+ln(c)^2*ln(x)*b^2*d^3+1/6/r*Pi^2*b^2*e^3*csgn(I*x^n)^2*csgn(I*c*
x^n)^3*csgn(I*c)*(x^r)^3-1/12/r*Pi^2*b^2*e^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*(x^r)^3-1/3/r*Pi^2*b^2*
e^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)*(x^r)^3+1/6/r*Pi^2*b^2*e^3*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2*(
x^r)^3-3/8/r*Pi^2*b^2*d*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4*(x^r)^2-1/12/r*Pi^2*b^2*e^3*csgn(I*x^n)^2*csgn(I*c*x
^n)^4*(x^r)^3+1/6/r*Pi^2*b^2*e^3*csgn(I*x^n)*csgn(I*c*x^n)^5*(x^r)^3+1/6/r*Pi^2*b^2*e^3*csgn(I*c*x^n)^5*csgn(I
*c)*(x^r)^3-1/12/r*Pi^2*b^2*e^3*csgn(I*c*x^n)^4*csgn(I*c)^2*(x^r)^3+6*b^2*d^2*e*n^2*x^r/r^3-1/18*b*(27*I*Pi*b*
d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+54*I*Pi*b*d^2*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-
36*ln(x)*a*d^3*r^2+4*b*e^3*n*(x^r)^3-12*a*e^3*r*(x^r)^3-36*ln(c)*ln(x)*b*d^3*r^2+18*b*d^3*n*ln(x)^2*r^2-54*a*d
*e^2*r*(x^r)^2-108*a*d^2*e*r*x^r-12*ln(c)*b*e^3*r*(x^r)^3+27*b*d*e^2*n*(x^r)^2+108*b*d^2*e*n*x^r-54*I*Pi*b*d^2
*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-54*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r-6*I*Pi*b*e^3*r*csgn(I*x^n
)*csgn(I*c*x^n)^2*(x^r)^3-6*I*Pi*b*e^3*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3+27*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^3*(
x^r)^2-108*ln(c)*b*d^2*e*r*x^r-54*ln(c)*b*d*e^2*r*(x^r)^2+6*I*Pi*b*e^3*r*csgn(I*c*x^n)^3*(x^r)^3+18*I*Pi*ln(x)
*b*d^3*csgn(I*c*x^n)^3*r^2+54*I*Pi*b*d^2*e*r*csgn(I*c*x^n)^3*x^r-18*I*Pi*ln(x)*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)
^2*r^2-18*I*Pi*ln(x)*b*d^3*csgn(I*c*x^n)^2*csgn(I*c)*r^2+6*I*Pi*b*e^3*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x
^r)^3-27*I*Pi*b*d*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-27*I*Pi*b*d*e^2*r*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^
2+18*I*Pi*ln(x)*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*r^2)/r^2*ln(x^n)+1/6*b^2*(2*e^3*(x^r)^3+6*d^3*ln(x)*
r+9*d*e^2*(x^r)^2+18*d^2*e*x^r)/r*ln(x^n)^2+1/9*I/r^2*Pi*b^2*e^3*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3
+3/2*I/r*ln(c)*Pi*b^2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+3/2*I/r*ln(c)*Pi*b^2*d*e^2*csgn(I*c*x^n)^2*csg
n(I*c)*(x^r)^2+3/2*I/r*Pi*a*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-3/4*I/r^2*Pi*b^2*d*e^2*n*csgn(I*x^n)*c
sgn(I*c*x^n)^2*(x^r)^2+3/2*I/r*Pi*a*b*d*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-3/4*I/r^2*Pi*b^2*d*e^2*n*csgn(I*
c*x^n)^2*csgn(I*c)*(x^r)^2+3*I/r*ln(c)*Pi*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r+3*I/r*ln(c)*Pi*b^2*d^2*e*c
sgn(I*c*x^n)^2*csgn(I*c)*x^r+3*I/r*Pi*a*b*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-3*I/r^2*Pi*b^2*d^2*e*n*csgn(I*
x^n)*csgn(I*c*x^n)^2*x^r+3*I/r*Pi*a*b*d^2*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-3*I/r^2*Pi*b^2*d^2*e*n*csgn(I*c*x^n)
^2*csgn(I*c)*x^r-1/3*I/r*ln(c)*Pi*b^2*e^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-1/3*I/r*Pi*a*b*e^3*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^3-1/4*Pi^2*ln(x)*b^2*d^3*csgn(I*x^n)^2*csgn(I*c*x^n)^4+1/2*Pi^2*ln(x)*b^2
*d^3*csgn(I*x^n)*csgn(I*c*x^n)^5+1/3/r*ln(c)^2*b^2*e^3*(x^r)^3+2/27/r^3*b^2*e^3*n^2*(x^r)^3+3/2/r*a^2*d*e^2*(x
^r)^2+3/r*a^2*d^2*e*x^r+2*ln(c)*ln(x)*a*b*d^3-ln(x)^2*ln(c)*b^2*d^3*n-ln(x)^2*a*b*d^3*n+ln(x)*a^2*d^3-1/4*Pi^2
*ln(x)*b^2*d^3*csgn(I*c*x^n)^6-3/8/r*Pi^2*b^2*d*e^2*csgn(I*c*x^n)^6*(x^r)^2-3/4/r*Pi^2*b^2*d^2*e*csgn(I*c*x^n)
^6*x^r+3/r*ln(c)*a*b*d*e^2*(x^r)^2-3/2/r^2*ln(c)*b^2*d*e^2*n*(x^r)^2-3/2/r^2*a*b*d*e^2*n*(x^r)^2+6/r*ln(c)*a*b
*d^2*e*x^r-6/r^2*ln(c)*b^2*d^2*e*n*x^r-6/r^2*a*b*d^2*e*n*x^r+1/2*I*ln(x)^2*Pi*b^2*d^3*n*csgn(I*c*x^n)^3-I*ln(c
)*Pi*ln(x)*b^2*d^3*csgn(I*c*x^n)^3-I*Pi*ln(x)*a*b*d^3*csgn(I*c*x^n)^3+3/4*I/r^2*Pi*b^2*d*e^2*n*csgn(I*c*x^n)^3
*(x^r)^2-3*I/r*ln(c)*Pi*b^2*d^2*e*csgn(I*c*x^n)^3*x^r-3*I/r*Pi*a*b*d^2*e*csgn(I*c*x^n)^3*x^r+3*I/r^2*Pi*b^2*d^
2*e*n*csgn(I*c*x^n)^3*x^r+3/4/r*Pi^2*b^2*d*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)*(x^r)^2-3/8/r*Pi^2*b^2*
d*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*(x^r)^2-3/2/r*Pi^2*b^2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(
I*c)*(x^r)^2+3/4/r*Pi^2*b^2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2*(x^r)^2+3/2/r*Pi^2*b^2*d^2*e*csgn(I*
x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)*x^r-3/4/r*Pi^2*b^2*d^2*e*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*x^r-3/r*Pi
^2*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)*x^r+3/2/r*Pi^2*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I
*c)^2*x^r+1/3*I/r*ln(c)*Pi*b^2*e^3*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+1/3*I/r*ln(c)*Pi*b^2*e^3*csgn(I*c*x^n)^
2*csgn(I*c)*(x^r)^3-I*ln(c)*Pi*ln(x)*b^2*d^3*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*ln(x)*a*b*d^3*csgn(I*x^n
)*csgn(I*c*x^n)*csgn(I*c)+1/2*I*ln(x)^2*Pi*b^2*d^3*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)+1/3*I/r*Pi*a*b*e^3*cs
gn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3-1/9*I/r^2*Pi*b^2*e^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^3+1/3*I/r*Pi*a*b*e^
3*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-1/9*I/r^2*Pi*b^2*e^3*n*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^3-3/2*I/r*ln(c)*Pi*
b^2*d*e^2*csgn(I*c*x^n)^3*(x^r)^2-3/2*I/r*Pi*a*b*d*e^2*csgn(I*c*x^n)^3*(x^r)^2+1/2*Pi^2*ln(x)*b^2*d^3*csgn(I*x
^n)^2*csgn(I*c*x^n)^3*csgn(I*c)-1/4*Pi^2*ln(x)*b^2*d^3*csgn(I*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2-Pi^2*ln(x)*b^
2*d^3*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)+1/2*Pi^2*ln(x)*b^2*d^3*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2+1/2
*Pi^2*ln(x)*b^2*d^3*csgn(I*c*x^n)^5*csgn(I*c)-1/4*Pi^2*ln(x)*b^2*d^3*csgn(I*c*x^n)^4*csgn(I*c)^2+3/4/r*Pi^2*b^
2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)^5*(x^r)^2+3/4/r*Pi^2*b^2*d*e^2*csgn(I*c*x^n)^5*csgn(I*c)*(x^r)^2-3/8/r*Pi^2*
b^2*d*e^2*csgn(I*c*x^n)^4*csgn(I*c)^2*(x^r)^2-3/4/r*Pi^2*b^2*d^2*e*csgn(I*x^n)^2*csgn(I*c*x^n)^4*x^r+3/2/r*Pi^
2*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)^5*x^r+3/2/r*Pi^2*b^2*d^2*e*csgn(I*c*x^n)^5*csgn(I*c)*x^r-3/4/r*Pi^2*b^2*
d^2*e*csgn(I*c*x^n)^4*csgn(I*c)^2*x^r+I*ln(c)*Pi*ln(x)*b^2*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+I*ln(c)*Pi*ln(x)*b^
2*d^3*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*ln(x)*a*b*d^3*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi*ln(x)*a*b*d^3*csgn(I*c*x^n
)^2*csgn(I*c)-1/3*I/r*ln(c)*Pi*b^2*e^3*csgn(I*c*x^n)^3*(x^r)^3-1/3*I/r*Pi*a*b*e^3*csgn(I*c*x^n)^3*(x^r)^3+1/9*
I/r^2*Pi*b^2*e^3*n*csgn(I*c*x^n)^3*(x^r)^3-1/2*I*ln(x)^2*Pi*b^2*d^3*n*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*ln(x)^
2*Pi*b^2*d^3*n*csgn(I*c*x^n)^2*csgn(I*c)-3/2*I/r*ln(c)*Pi*b^2*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^
2-3/2*I/r*Pi*a*b*d*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+3/4*I/r^2*Pi*b^2*d*e^2*n*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)*(x^r)^2-3*I/r*ln(c)*Pi*b^2*d^2*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-3*I/r*Pi*a*b*d^2*e*
csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r+3*I/r^2*Pi*b^2*d^2*e*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-1/12/r*
Pi^2*b^2*e^3*csgn(I*c*x^n)^6*(x^r)^3+2/3/r*ln(c)*a*b*e^3*(x^r)^3-2/9/r^2*ln(c)*b^2*e^3*n*(x^r)^3+3/2/r*ln(c)^2
*b^2*d*e^2*(x^r)^2-2/9/r^2*a*b*e^3*n*(x^r)^3+3/r*ln(c)^2*b^2*d^2*e*x^r+3/4/r^3*b^2*d*e^2*n^2*(x^r)^2

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.64984, size = 1172, normalized size = 4.78 \begin{align*} \frac{36 \, b^{2} d^{3} n^{2} r^{3} \log \left (x\right )^{3} + 108 \,{\left (b^{2} d^{3} n r^{3} \log \left (c\right ) + a b d^{3} n r^{3}\right )} \log \left (x\right )^{2} + 4 \,{\left (9 \, b^{2} e^{3} n^{2} r^{2} \log \left (x\right )^{2} + 9 \, b^{2} e^{3} r^{2} \log \left (c\right )^{2} + 2 \, b^{2} e^{3} n^{2} - 6 \, a b e^{3} n r + 9 \, a^{2} e^{3} r^{2} - 6 \,{\left (b^{2} e^{3} n r - 3 \, a b e^{3} r^{2}\right )} \log \left (c\right ) + 6 \,{\left (3 \, b^{2} e^{3} n r^{2} \log \left (c\right ) - b^{2} e^{3} n^{2} r + 3 \, a b e^{3} n r^{2}\right )} \log \left (x\right )\right )} x^{3 \, r} + 81 \,{\left (2 \, b^{2} d e^{2} n^{2} r^{2} \log \left (x\right )^{2} + 2 \, b^{2} d e^{2} r^{2} \log \left (c\right )^{2} + b^{2} d e^{2} n^{2} - 2 \, a b d e^{2} n r + 2 \, a^{2} d e^{2} r^{2} - 2 \,{\left (b^{2} d e^{2} n r - 2 \, a b d e^{2} r^{2}\right )} \log \left (c\right ) + 2 \,{\left (2 \, b^{2} d e^{2} n r^{2} \log \left (c\right ) - b^{2} d e^{2} n^{2} r + 2 \, a b d e^{2} n r^{2}\right )} \log \left (x\right )\right )} x^{2 \, r} + 324 \,{\left (b^{2} d^{2} e n^{2} r^{2} \log \left (x\right )^{2} + b^{2} d^{2} e r^{2} \log \left (c\right )^{2} + 2 \, b^{2} d^{2} e n^{2} - 2 \, a b d^{2} e n r + a^{2} d^{2} e r^{2} - 2 \,{\left (b^{2} d^{2} e n r - a b d^{2} e r^{2}\right )} \log \left (c\right ) + 2 \,{\left (b^{2} d^{2} e n r^{2} \log \left (c\right ) - b^{2} d^{2} e n^{2} r + a b d^{2} e n r^{2}\right )} \log \left (x\right )\right )} x^{r} + 108 \,{\left (b^{2} d^{3} r^{3} \log \left (c\right )^{2} + 2 \, a b d^{3} r^{3} \log \left (c\right ) + a^{2} d^{3} r^{3}\right )} \log \left (x\right )}{108 \, r^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*(36*b^2*d^3*n^2*r^3*log(x)^3 + 108*(b^2*d^3*n*r^3*log(c) + a*b*d^3*n*r^3)*log(x)^2 + 4*(9*b^2*e^3*n^2*r^
2*log(x)^2 + 9*b^2*e^3*r^2*log(c)^2 + 2*b^2*e^3*n^2 - 6*a*b*e^3*n*r + 9*a^2*e^3*r^2 - 6*(b^2*e^3*n*r - 3*a*b*e
^3*r^2)*log(c) + 6*(3*b^2*e^3*n*r^2*log(c) - b^2*e^3*n^2*r + 3*a*b*e^3*n*r^2)*log(x))*x^(3*r) + 81*(2*b^2*d*e^
2*n^2*r^2*log(x)^2 + 2*b^2*d*e^2*r^2*log(c)^2 + b^2*d*e^2*n^2 - 2*a*b*d*e^2*n*r + 2*a^2*d*e^2*r^2 - 2*(b^2*d*e
^2*n*r - 2*a*b*d*e^2*r^2)*log(c) + 2*(2*b^2*d*e^2*n*r^2*log(c) - b^2*d*e^2*n^2*r + 2*a*b*d*e^2*n*r^2)*log(x))*
x^(2*r) + 324*(b^2*d^2*e*n^2*r^2*log(x)^2 + b^2*d^2*e*r^2*log(c)^2 + 2*b^2*d^2*e*n^2 - 2*a*b*d^2*e*n*r + a^2*d
^2*e*r^2 - 2*(b^2*d^2*e*n*r - a*b*d^2*e*r^2)*log(c) + 2*(b^2*d^2*e*n*r^2*log(c) - b^2*d^2*e*n^2*r + a*b*d^2*e*
n*r^2)*log(x))*x^r + 108*(b^2*d^3*r^3*log(c)^2 + 2*a*b*d^3*r^3*log(c) + a^2*d^3*r^3)*log(x))/r^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.33084, size = 856, normalized size = 3.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*d^3*n^2*log(x)^3 + 3*b^2*d^2*n^2*x^r*e*log(x)^2/r + b^2*d^3*n*log(c)*log(x)^2 + 6*b^2*d^2*n*x^r*e*log(
c)*log(x)/r + b^2*d^3*log(c)^2*log(x) + a*b*d^3*n*log(x)^2 + 3/2*b^2*d*n^2*x^(2*r)*e^2*log(x)^2/r + 3*b^2*d^2*
x^r*e*log(c)^2/r - 6*b^2*d^2*n^2*x^r*e*log(x)/r^2 + 6*a*b*d^2*n*x^r*e*log(x)/r + 2*a*b*d^3*log(c)*log(x) + 3*b
^2*d*n*x^(2*r)*e^2*log(c)*log(x)/r + 1/3*b^2*n^2*x^(3*r)*e^3*log(x)^2/r - 6*b^2*d^2*n*x^r*e*log(c)/r^2 + 6*a*b
*d^2*x^r*e*log(c)/r + 3/2*b^2*d*x^(2*r)*e^2*log(c)^2/r + a^2*d^3*log(x) - 3/2*b^2*d*n^2*x^(2*r)*e^2*log(x)/r^2
+ 3*a*b*d*n*x^(2*r)*e^2*log(x)/r + 2/3*b^2*n*x^(3*r)*e^3*log(c)*log(x)/r + 6*b^2*d^2*n^2*x^r*e/r^3 - 6*a*b*d^
2*n*x^r*e/r^2 + 3*a^2*d^2*x^r*e/r - 3/2*b^2*d*n*x^(2*r)*e^2*log(c)/r^2 + 3*a*b*d*x^(2*r)*e^2*log(c)/r + 1/3*b^
2*x^(3*r)*e^3*log(c)^2/r - 2/9*b^2*n^2*x^(3*r)*e^3*log(x)/r^2 + 2/3*a*b*n*x^(3*r)*e^3*log(x)/r + 3/4*b^2*d*n^2
*x^(2*r)*e^2/r^3 - 3/2*a*b*d*n*x^(2*r)*e^2/r^2 + 3/2*a^2*d*x^(2*r)*e^2/r - 2/9*b^2*n*x^(3*r)*e^3*log(c)/r^2 +
2/3*a*b*x^(3*r)*e^3*log(c)/r + 2/27*b^2*n^2*x^(3*r)*e^3/r^3 - 2/9*a*b*n*x^(3*r)*e^3/r^2 + 1/3*a^2*x^(3*r)*e^3/
r