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

Optimal. Leaf size=161 \[ \frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {4 b d e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac {2 d e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}-\frac {b e^2 n x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r^2}+\frac {e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )^2}{2 r}+\frac {4 b^2 d e n^2 x^r}{r^3}+\frac {b^2 e^2 n^2 x^{2 r}}{4 r^3} \]

[Out]

4*b^2*d*e*n^2*x^r/r^3+1/4*b^2*e^2*n^2*x^(2*r)/r^3-4*b*d*e*n*x^r*(a+b*ln(c*x^n))/r^2-1/2*b*e^2*n*x^(2*r)*(a+b*l
n(c*x^n))/r^2+2*d*e*x^r*(a+b*ln(c*x^n))^2/r+1/2*e^2*x^(2*r)*(a+b*ln(c*x^n))^2/r+1/3*d^2*(a+b*ln(c*x^n))^3/b/n

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Rubi [A]  time = 0.24, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2353, 2302, 30, 2305, 2304} \[ \frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {4 b d e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac {2 d e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}-\frac {b e^2 n x^{2 r} \left (a+b \log \left (c x^n\right )\right )}{2 r^2}+\frac {e^2 x^{2 r} \left (a+b \log \left (c x^n\right )\right )^2}{2 r}+\frac {4 b^2 d e n^2 x^r}{r^3}+\frac {b^2 e^2 n^2 x^{2 r}}{4 r^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

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

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 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
]

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 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
]))

Rubi steps

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

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Mathematica [A]  time = 0.27, size = 179, normalized size = 1.11 \[ \frac {3 e n x^r \left (2 a^2 r^2 \left (4 d+e x^r\right )-2 a b n r \left (8 d+e x^r\right )+b^2 n^2 \left (16 d+e x^r\right )\right )+12 a^2 d^2 n r^3 \log (x)+6 b r^2 \log ^2\left (c x^n\right ) \left (2 a d^2 r+b e n x^r \left (4 d+e x^r\right )\right )-6 b e n r x^r \log \left (c x^n\right ) \left (b n \left (8 d+e x^r\right )-2 a r \left (4 d+e x^r\right )\right )+4 b^2 d^2 r^3 \log ^3\left (c x^n\right )}{12 n r^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*e*n*x^r*(2*a^2*r^2*(4*d + e*x^r) - 2*a*b*n*r*(8*d + e*x^r) + b^2*n^2*(16*d + e*x^r)) + 12*a^2*d^2*n*r^3*Log
[x] - 6*b*e*n*r*x^r*(-2*a*r*(4*d + e*x^r) + b*n*(8*d + e*x^r))*Log[c*x^n] + 6*b*r^2*(2*a*d^2*r + b*e*n*x^r*(4*
d + e*x^r))*Log[c*x^n]^2 + 4*b^2*d^2*r^3*Log[c*x^n]^3)/(12*n*r^3)

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fricas [B]  time = 0.69, size = 353, normalized size = 2.19 \[ \frac {4 \, b^{2} d^{2} n^{2} r^{3} \log \relax (x)^{3} + 12 \, {\left (b^{2} d^{2} n r^{3} \log \relax (c) + a b d^{2} n r^{3}\right )} \log \relax (x)^{2} + 3 \, {\left (2 \, b^{2} e^{2} n^{2} r^{2} \log \relax (x)^{2} + 2 \, b^{2} e^{2} r^{2} \log \relax (c)^{2} + b^{2} e^{2} n^{2} - 2 \, a b e^{2} n r + 2 \, a^{2} e^{2} r^{2} - 2 \, {\left (b^{2} e^{2} n r - 2 \, a b e^{2} r^{2}\right )} \log \relax (c) + 2 \, {\left (2 \, b^{2} e^{2} n r^{2} \log \relax (c) - b^{2} e^{2} n^{2} r + 2 \, a b e^{2} n r^{2}\right )} \log \relax (x)\right )} x^{2 \, r} + 24 \, {\left (b^{2} d e n^{2} r^{2} \log \relax (x)^{2} + b^{2} d e r^{2} \log \relax (c)^{2} + 2 \, b^{2} d e n^{2} - 2 \, a b d e n r + a^{2} d e r^{2} - 2 \, {\left (b^{2} d e n r - a b d e r^{2}\right )} \log \relax (c) + 2 \, {\left (b^{2} d e n r^{2} \log \relax (c) - b^{2} d e n^{2} r + a b d e n r^{2}\right )} \log \relax (x)\right )} x^{r} + 12 \, {\left (b^{2} d^{2} r^{3} \log \relax (c)^{2} + 2 \, a b d^{2} r^{3} \log \relax (c) + a^{2} d^{2} r^{3}\right )} \log \relax (x)}{12 \, r^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(4*b^2*d^2*n^2*r^3*log(x)^3 + 12*(b^2*d^2*n*r^3*log(c) + a*b*d^2*n*r^3)*log(x)^2 + 3*(2*b^2*e^2*n^2*r^2*l
og(x)^2 + 2*b^2*e^2*r^2*log(c)^2 + b^2*e^2*n^2 - 2*a*b*e^2*n*r + 2*a^2*e^2*r^2 - 2*(b^2*e^2*n*r - 2*a*b*e^2*r^
2)*log(c) + 2*(2*b^2*e^2*n*r^2*log(c) - b^2*e^2*n^2*r + 2*a*b*e^2*n*r^2)*log(x))*x^(2*r) + 24*(b^2*d*e*n^2*r^2
*log(x)^2 + b^2*d*e*r^2*log(c)^2 + 2*b^2*d*e*n^2 - 2*a*b*d*e*n*r + a^2*d*e*r^2 - 2*(b^2*d*e*n*r - a*b*d*e*r^2)
*log(c) + 2*(b^2*d*e*n*r^2*log(c) - b^2*d*e*n^2*r + a*b*d*e*n*r^2)*log(x))*x^r + 12*(b^2*d^2*r^3*log(c)^2 + 2*
a*b*d^2*r^3*log(c) + a^2*d^2*r^3)*log(x))/r^3

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giac [B]  time = 0.32, size = 421, normalized size = 2.61 \[ \frac {1}{3} \, b^{2} d^{2} n^{2} \log \relax (x)^{3} + \frac {2 \, b^{2} d n^{2} x^{r} e \log \relax (x)^{2}}{r} + b^{2} d^{2} n \log \relax (c) \log \relax (x)^{2} + \frac {4 \, b^{2} d n x^{r} e \log \relax (c) \log \relax (x)}{r} + b^{2} d^{2} \log \relax (c)^{2} \log \relax (x) + a b d^{2} n \log \relax (x)^{2} + \frac {b^{2} n^{2} x^{2 \, r} e^{2} \log \relax (x)^{2}}{2 \, r} + \frac {2 \, b^{2} d x^{r} e \log \relax (c)^{2}}{r} - \frac {4 \, b^{2} d n^{2} x^{r} e \log \relax (x)}{r^{2}} + \frac {4 \, a b d n x^{r} e \log \relax (x)}{r} + 2 \, a b d^{2} \log \relax (c) \log \relax (x) + \frac {b^{2} n x^{2 \, r} e^{2} \log \relax (c) \log \relax (x)}{r} - \frac {4 \, b^{2} d n x^{r} e \log \relax (c)}{r^{2}} + \frac {4 \, a b d x^{r} e \log \relax (c)}{r} + \frac {b^{2} x^{2 \, r} e^{2} \log \relax (c)^{2}}{2 \, r} + a^{2} d^{2} \log \relax (x) - \frac {b^{2} n^{2} x^{2 \, r} e^{2} \log \relax (x)}{2 \, r^{2}} + \frac {a b n x^{2 \, r} e^{2} \log \relax (x)}{r} + \frac {4 \, b^{2} d n^{2} x^{r} e}{r^{3}} - \frac {4 \, a b d n x^{r} e}{r^{2}} + \frac {2 \, a^{2} d x^{r} e}{r} - \frac {b^{2} n x^{2 \, r} e^{2} \log \relax (c)}{2 \, r^{2}} + \frac {a b x^{2 \, r} e^{2} \log \relax (c)}{r} + \frac {b^{2} n^{2} x^{2 \, r} e^{2}}{4 \, r^{3}} - \frac {a b n x^{2 \, r} e^{2}}{2 \, r^{2}} + \frac {a^{2} x^{2 \, r} e^{2}}{2 \, r} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*d^2*n^2*log(x)^3 + 2*b^2*d*n^2*x^r*e*log(x)^2/r + b^2*d^2*n*log(c)*log(x)^2 + 4*b^2*d*n*x^r*e*log(c)*l
og(x)/r + b^2*d^2*log(c)^2*log(x) + a*b*d^2*n*log(x)^2 + 1/2*b^2*n^2*x^(2*r)*e^2*log(x)^2/r + 2*b^2*d*x^r*e*lo
g(c)^2/r - 4*b^2*d*n^2*x^r*e*log(x)/r^2 + 4*a*b*d*n*x^r*e*log(x)/r + 2*a*b*d^2*log(c)*log(x) + b^2*n*x^(2*r)*e
^2*log(c)*log(x)/r - 4*b^2*d*n*x^r*e*log(c)/r^2 + 4*a*b*d*x^r*e*log(c)/r + 1/2*b^2*x^(2*r)*e^2*log(c)^2/r + a^
2*d^2*log(x) - 1/2*b^2*n^2*x^(2*r)*e^2*log(x)/r^2 + a*b*n*x^(2*r)*e^2*log(x)/r + 4*b^2*d*n^2*x^r*e/r^3 - 4*a*b
*d*n*x^r*e/r^2 + 2*a^2*d*x^r*e/r - 1/2*b^2*n*x^(2*r)*e^2*log(c)/r^2 + a*b*x^(2*r)*e^2*log(c)/r + 1/4*b^2*n^2*x
^(2*r)*e^2/r^3 - 1/2*a*b*n*x^(2*r)*e^2/r^2 + 1/2*a^2*x^(2*r)*e^2/r

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maple [C]  time = 0.53, size = 2844, normalized size = 17.66 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*d^2*ln(c)^2*ln(x)+1/3*b^2*d^2*n^2*ln(x)^3+a^2*d^2*ln(x)+1/2/r*a^2*e^2*(x^r)^2-1/2*I*ln(x)^2*Pi*b^2*d^2*n*c
sgn(I*c*x^n)^2*csgn(I*c)-1/2*I*ln(x)^2*Pi*b^2*d^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I/r*ln(c)*Pi*b^2*e^2*csgn(
I*c*x^n)^3*(x^r)^2-1/2*I/r*Pi*a*b*e^2*csgn(I*c*x^n)^3*(x^r)^2+1/4*I/r^2*Pi*b^2*e^2*n*csgn(I*c*x^n)^3*(x^r)^2-1
/2*b*(-4*I*Pi*b*d*e*r*csgn(I*c*x^n)^2*csgn(I*c)*x^r+4*I*Pi*b*d*e*r*csgn(I*c*x^n)^3*x^r-I*Pi*b*e^2*r*csgn(I*c*x
^n)^2*csgn(I*c)*(x^r)^2+2*I*Pi*ln(x)*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*r^2+I*Pi*b*e^2*r*csgn(I*x^n)*cs
gn(I*c*x^n)*csgn(I*c)*(x^r)^2+I*Pi*b*e^2*r*csgn(I*c*x^n)^3*(x^r)^2-2*I*Pi*ln(x)*b*d^2*csgn(I*x^n)*csgn(I*c*x^n
)^2*r^2+4*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-I*Pi*b*e^2*r*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^
2-4*I*Pi*b*d*e*r*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-2*I*Pi*ln(x)*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)*r^2+2*I*Pi*ln(x)
*b*d^2*csgn(I*c*x^n)^3*r^2+2*b*d^2*n*ln(x)^2*r^2-4*ln(c)*ln(x)*b*d^2*r^2-2*ln(c)*b*e^2*r*(x^r)^2-8*b*d*e*r*x^r
*ln(c)-4*ln(x)*a*d^2*r^2-2*a*e^2*r*(x^r)^2+b*e^2*n*(x^r)^2-8*a*d*e*r*x^r+8*b*d*e*n*x^r)/r^2*ln(x^n)+1/2*b^2*(2
*d^2*r*ln(x)+(x^r)^2*e^2+4*d*e*x^r)/r*ln(x^n)^2+2*a*b*d^2*ln(c)*ln(x)-a*b*d^2*n*ln(x)^2-b^2*d^2*n*ln(c)*ln(x)^
2+I*ln(c)*Pi*ln(x)*b^2*d^2*csgn(I*c*x^n)^2*csgn(I*c)+I*ln(c)*Pi*ln(x)*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2+I*Pi
*ln(x)*a*b*d^2*csgn(I*c*x^n)^2*csgn(I*c)+I*Pi*ln(x)*a*b*d^2*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2/r*Pi^2*b^2*e^2*csg
n(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)*(x^r)^2+1/4/r*Pi^2*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)^3*csgn(I*c)^2*(x^r)^2+
1/4/r*Pi^2*b^2*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)*(x^r)^2-1/8/r*Pi^2*b^2*e^2*csgn(I*x^n)^2*csgn(I*c*x
^n)^2*csgn(I*c)^2*(x^r)^2+1/r*Pi^2*b^2*d*e*csgn(I*c*x^n)^5*csgn(I*c)*x^r+1/r*Pi^2*b^2*d*e*csgn(I*x^n)*csgn(I*c
*x^n)^5*x^r-1/2/r*Pi^2*b^2*d*e*csgn(I*c*x^n)^4*csgn(I*c)^2*x^r-1/2/r*Pi^2*b^2*d*e*csgn(I*x^n)^2*csgn(I*c*x^n)^
4*x^r-2*I/r*ln(c)*Pi*b^2*d*e*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-2*I/r*Pi*a*b*d*e*csgn(I*x^n)*csgn(I*c*x^n
)*csgn(I*c)*x^r+2*I/r^2*Pi*b^2*d*e*n*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*x^r-1/2/r^2*a*b*e^2*n*(x^r)^2-1/8/r*P
i^2*b^2*e^2*csgn(I*c*x^n)^6*(x^r)^2+1/r*ln(c)*a*b*e^2*(x^r)^2-1/2/r^2*ln(c)*b^2*e^2*n*(x^r)^2+2/r*ln(c)^2*b^2*
d*e*x^r-1/4*csgn(I*c*x^n)^4*csgn(I*x^n)^2*d^2*b^2*ln(x)*Pi^2+1/2*csgn(I*c*x^n)^5*csgn(I*x^n)*d^2*b^2*ln(x)*Pi^
2+1/2*csgn(I*c)*csgn(I*c*x^n)^5*d^2*b^2*ln(x)*Pi^2-1/4*csgn(I*c)^2*csgn(I*c*x^n)^4*d^2*b^2*ln(x)*Pi^2-1/4*csgn
(I*c)^2*csgn(I*c*x^n)^2*csgn(I*x^n)^2*d^2*b^2*ln(x)*Pi^2-csgn(I*c)*csgn(I*c*x^n)^4*csgn(I*x^n)*d^2*b^2*ln(x)*P
i^2+1/2*csgn(I*c)^2*csgn(I*c*x^n)^3*csgn(I*x^n)*d^2*b^2*ln(x)*Pi^2+4*b^2*d*e*n^2*x^r/r^3-1/4*csgn(I*c*x^n)^6*d
^2*b^2*ln(x)*Pi^2+1/2/r*ln(c)^2*b^2*e^2*(x^r)^2+1/4/r^3*b^2*e^2*n^2*(x^r)^2+2/r*a^2*d*e*x^r+1/2*csgn(I*c)*csgn
(I*c*x^n)^3*csgn(I*x^n)^2*d^2*b^2*ln(x)*Pi^2-1/2*I/r*ln(c)*Pi*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r
)^2-1/2*I/r*Pi*a*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*(x^r)^2+1/4*I/r^2*Pi*b^2*e^2*n*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*c)*(x^r)^2+2*I/r*ln(c)*Pi*b^2*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r+2*I/r*ln(c)*Pi*b^2*d*e*csgn(I*x^n
)*csgn(I*c*x^n)^2*x^r+2*I/r*Pi*a*b*d*e*csgn(I*c*x^n)^2*csgn(I*c)*x^r-2*I/r^2*Pi*b^2*d*e*n*csgn(I*c*x^n)^2*csgn
(I*c)*x^r+2*I/r*Pi*a*b*d*e*csgn(I*x^n)*csgn(I*c*x^n)^2*x^r-2*I/r^2*Pi*b^2*d*e*n*csgn(I*x^n)*csgn(I*c*x^n)^2*x^
r+1/4/r*Pi^2*b^2*e^2*csgn(I*c*x^n)^5*csgn(I*c)*(x^r)^2+1/4/r*Pi^2*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)^5*(x^r)^2-
1/8/r*Pi^2*b^2*e^2*csgn(I*c*x^n)^4*csgn(I*c)^2*(x^r)^2-1/8/r*Pi^2*b^2*e^2*csgn(I*x^n)^2*csgn(I*c*x^n)^4*(x^r)^
2-1/2/r*Pi^2*b^2*d*e*csgn(I*c*x^n)^6*x^r+4/r*ln(c)*a*b*d*e*x^r-4/r^2*ln(c)*b^2*d*e*n*x^r-4/r^2*a*b*d*e*n*x^r-I
*Pi*ln(x)*a*b*d^2*csgn(I*c*x^n)^3+1/2*I*ln(x)^2*Pi*b^2*d^2*n*csgn(I*c*x^n)^3-I*ln(c)*Pi*ln(x)*b^2*d^2*csgn(I*c
*x^n)^3-I*ln(c)*Pi*ln(x)*b^2*d^2*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)-I*Pi*ln(x)*a*b*d^2*csgn(I*x^n)*csgn(I*c*x
^n)*csgn(I*c)-2/r*Pi^2*b^2*d*e*csgn(I*x^n)*csgn(I*c*x^n)^4*csgn(I*c)*x^r+1/r*Pi^2*b^2*d*e*csgn(I*x^n)*csgn(I*c
*x^n)^3*csgn(I*c)^2*x^r+1/r*Pi^2*b^2*d*e*csgn(I*x^n)^2*csgn(I*c*x^n)^3*csgn(I*c)*x^r-1/2/r*Pi^2*b^2*d*e*csgn(I
*x^n)^2*csgn(I*c*x^n)^2*csgn(I*c)^2*x^r+1/2*I/r*ln(c)*Pi*b^2*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+1/2*I/r*ln(
c)*Pi*b^2*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2+1/2*I/r*Pi*a*b*e^2*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2-1/4*I/r
^2*Pi*b^2*e^2*n*csgn(I*c*x^n)^2*csgn(I*c)*(x^r)^2+1/2*I/r*Pi*a*b*e^2*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-1/4*I
/r^2*Pi*b^2*e^2*n*csgn(I*x^n)*csgn(I*c*x^n)^2*(x^r)^2-2*I/r*ln(c)*Pi*b^2*d*e*csgn(I*c*x^n)^3*x^r-2*I/r*Pi*a*b*
d*e*csgn(I*c*x^n)^3*x^r+2*I/r^2*Pi*b^2*d*e*n*csgn(I*c*x^n)^3*x^r+1/2*I*ln(x)^2*Pi*b^2*d^2*n*csgn(I*x^n)*csgn(I
*c*x^n)*csgn(I*c)

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maxima [A]  time = 1.37, size = 259, normalized size = 1.61 \[ \frac {b^{2} e^{2} x^{2 \, r} \log \left (c x^{n}\right )^{2}}{2 \, r} + \frac {2 \, b^{2} d e x^{r} \log \left (c x^{n}\right )^{2}}{r} + \frac {b^{2} d^{2} \log \left (c x^{n}\right )^{3}}{3 \, n} - \frac {1}{4} \, b^{2} e^{2} {\left (\frac {2 \, n x^{2 \, r} \log \left (c x^{n}\right )}{r^{2}} - \frac {n^{2} x^{2 \, r}}{r^{3}}\right )} - 4 \, b^{2} d e {\left (\frac {n x^{r} \log \left (c x^{n}\right )}{r^{2}} - \frac {n^{2} x^{r}}{r^{3}}\right )} + \frac {a b e^{2} x^{2 \, r} \log \left (c x^{n}\right )}{r} + \frac {4 \, a b d e x^{r} \log \left (c x^{n}\right )}{r} + \frac {a b d^{2} \log \left (c x^{n}\right )^{2}}{n} + a^{2} d^{2} \log \relax (x) - \frac {a b e^{2} n x^{2 \, r}}{2 \, r^{2}} + \frac {a^{2} e^{2} x^{2 \, r}}{2 \, r} - \frac {4 \, a b d e n x^{r}}{r^{2}} + \frac {2 \, a^{2} d e x^{r}}{r} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 33.78, size = 546, normalized size = 3.39 \[ \begin {cases} a^{2} d^{2} \log {\relax (x )} + \frac {2 a^{2} d e x^{r}}{r} + \frac {a^{2} e^{2} x^{2 r}}{2 r} + a b d^{2} n \log {\relax (x )}^{2} + 2 a b d^{2} \log {\relax (c )} \log {\relax (x )} + \frac {4 a b d e n x^{r} \log {\relax (x )}}{r} - \frac {4 a b d e n x^{r}}{r^{2}} + \frac {4 a b d e x^{r} \log {\relax (c )}}{r} + \frac {a b e^{2} n x^{2 r} \log {\relax (x )}}{r} - \frac {a b e^{2} n x^{2 r}}{2 r^{2}} + \frac {a b e^{2} x^{2 r} \log {\relax (c )}}{r} + \frac {b^{2} d^{2} n^{2} \log {\relax (x )}^{3}}{3} + b^{2} d^{2} n \log {\relax (c )} \log {\relax (x )}^{2} + b^{2} d^{2} \log {\relax (c )}^{2} \log {\relax (x )} + \frac {2 b^{2} d e n^{2} x^{r} \log {\relax (x )}^{2}}{r} - \frac {4 b^{2} d e n^{2} x^{r} \log {\relax (x )}}{r^{2}} + \frac {4 b^{2} d e n^{2} x^{r}}{r^{3}} + \frac {4 b^{2} d e n x^{r} \log {\relax (c )} \log {\relax (x )}}{r} - \frac {4 b^{2} d e n x^{r} \log {\relax (c )}}{r^{2}} + \frac {2 b^{2} d e x^{r} \log {\relax (c )}^{2}}{r} + \frac {b^{2} e^{2} n^{2} x^{2 r} \log {\relax (x )}^{2}}{2 r} - \frac {b^{2} e^{2} n^{2} x^{2 r} \log {\relax (x )}}{2 r^{2}} + \frac {b^{2} e^{2} n^{2} x^{2 r}}{4 r^{3}} + \frac {b^{2} e^{2} n x^{2 r} \log {\relax (c )} \log {\relax (x )}}{r} - \frac {b^{2} e^{2} n x^{2 r} \log {\relax (c )}}{2 r^{2}} + \frac {b^{2} e^{2} x^{2 r} \log {\relax (c )}^{2}}{2 r} & \text {for}\: r \neq 0 \\\left (d + e\right )^{2} \left (\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\relax (c )} + b^{2} \log {\relax (c )}^{2}\right ) \log {\relax (x )} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d**2*log(x) + 2*a**2*d*e*x**r/r + a**2*e**2*x**(2*r)/(2*r) + a*b*d**2*n*log(x)**2 + 2*a*b*d**2
*log(c)*log(x) + 4*a*b*d*e*n*x**r*log(x)/r - 4*a*b*d*e*n*x**r/r**2 + 4*a*b*d*e*x**r*log(c)/r + a*b*e**2*n*x**(
2*r)*log(x)/r - a*b*e**2*n*x**(2*r)/(2*r**2) + a*b*e**2*x**(2*r)*log(c)/r + b**2*d**2*n**2*log(x)**3/3 + b**2*
d**2*n*log(c)*log(x)**2 + b**2*d**2*log(c)**2*log(x) + 2*b**2*d*e*n**2*x**r*log(x)**2/r - 4*b**2*d*e*n**2*x**r
*log(x)/r**2 + 4*b**2*d*e*n**2*x**r/r**3 + 4*b**2*d*e*n*x**r*log(c)*log(x)/r - 4*b**2*d*e*n*x**r*log(c)/r**2 +
2*b**2*d*e*x**r*log(c)**2/r + b**2*e**2*n**2*x**(2*r)*log(x)**2/(2*r) - b**2*e**2*n**2*x**(2*r)*log(x)/(2*r**
2) + b**2*e**2*n**2*x**(2*r)/(4*r**3) + b**2*e**2*n*x**(2*r)*log(c)*log(x)/r - b**2*e**2*n*x**(2*r)*log(c)/(2*
r**2) + b**2*e**2*x**(2*r)*log(c)**2/(2*r), Ne(r, 0)), ((d + e)**2*Piecewise(((a**2*log(c*x**n) + a*b*log(c*x*
*n)**2 + b**2*log(c*x**n)**3/3)/n, Ne(n, 0)), ((a**2 + 2*a*b*log(c) + b**2*log(c)**2)*log(x), True)), True))

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