\(\int \frac {(d+e x^r) (a+b \log (c x^n))^2}{x} \, dx\) [429]

Optimal result

Integrand size = 23, antiderivative size = 80 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {2 b^2 e n^2 x^r}{r^3}-\frac {2 b e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac {d \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2395, 2339, 30, 2342, 2341} \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {d \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {2 b e n x^r \left (a+b \log \left (c x^n\right )\right )}{r^2}+\frac {e x^r \left (a+b \log \left (c x^n\right )\right )^2}{r}+\frac {2 b^2 e n^2 x^r}{r^3} \]

[In]

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

[Out]

(2*b^2*e*n^2*x^r)/r^3 - (2*b*e*n*x^r*(a + b*Log[c*x^n]))/r^2 + (e*x^r*(a + b*Log[c*x^n])^2)/r + (d*(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 2339

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 2341

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 2342

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 2395

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.36 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {e \left (2 b^2 n^2-2 a b n r+a^2 r^2\right ) x^r}{r^3}+a^2 d \log (x)-\frac {2 b e (b n-a r) x^r \log \left (c x^n\right )}{r^2}+\frac {b \left (a d r+b e n x^r\right ) \log ^2\left (c x^n\right )}{n r}+\frac {b^2 d \log ^3\left (c x^n\right )}{3 n} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.86

method	result	size
parallelrisch	\(\frac {3 x^{r} \ln \left (c \,x^{n}\right )^{2} b^{2} e \,r^{2} n +b^{2} d \ln \left (c \,x^{n}\right )^{3} r^{3}+3 \ln \left (x \right ) a^{2} d n \,r^{3}+6 x^{r} \ln \left (c \,x^{n}\right ) a b e n \,r^{2}-6 x^{r} \ln \left (c \,x^{n}\right ) b^{2} e \,n^{2} r +3 a b d \ln \left (c \,x^{n}\right )^{2} r^{3}+3 x^{r} a^{2} e n \,r^{2}-6 x^{r} a b e \,n^{2} r +6 x^{r} b^{2} e \,n^{3}}{3 r^{3} n}\)	\(149\)
risch	\(\text {Expression too large to display}\)	\(1712\)

[In]

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

[Out]

1/3*(3*x^r*ln(c*x^n)^2*b^2*e*r^2*n+b^2*d*ln(c*x^n)^3*r^3+3*ln(x)*a^2*d*n*r^3+6*x^r*ln(c*x^n)*a*b*e*n*r^2-6*x^r
*ln(c*x^n)*b^2*e*n^2*r+3*a*b*d*ln(c*x^n)^2*r^3+3*x^r*a^2*e*n*r^2-6*x^r*a*b*e*n^2*r+6*x^r*b^2*e*n^3)/r^3/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (78) = 156\).

Time = 0.31 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.41 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {b^{2} d n^{2} r^{3} \log \left (x\right )^{3} + 3 \, {\left (b^{2} d n r^{3} \log \left (c\right ) + a b d n r^{3}\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{2} e n^{2} r^{2} \log \left (x\right )^{2} + b^{2} e r^{2} \log \left (c\right )^{2} + 2 \, b^{2} e n^{2} - 2 \, a b e n r + a^{2} e r^{2} - 2 \, {\left (b^{2} e n r - a b e r^{2}\right )} \log \left (c\right ) + 2 \, {\left (b^{2} e n r^{2} \log \left (c\right ) - b^{2} e n^{2} r + a b e n r^{2}\right )} \log \left (x\right )\right )} x^{r} + 3 \, {\left (b^{2} d r^{3} \log \left (c\right )^{2} + 2 \, a b d r^{3} \log \left (c\right ) + a^{2} d r^{3}\right )} \log \left (x\right )}{3 \, r^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (76) = 152\).

Time = 6.87 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.06 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\begin {cases} \left (a + b \log {\left (c \right )}\right )^{2} \left (d + e\right ) \log {\left (x \right )} & \text {for}\: n = 0 \wedge r = 0 \\\left (a + b \log {\left (c \right )}\right )^{2} \left (d \log {\left (x \right )} + \frac {e x^{r}}{r}\right ) & \text {for}\: n = 0 \\\left (d + e\right ) \left (\begin {cases} \frac {a^{2} \log {\left (c x^{n} \right )} + a b \log {\left (c x^{n} \right )}^{2} + \frac {b^{2} \log {\left (c x^{n} \right )}^{3}}{3}}{n} & \text {for}\: n \neq 0 \\\left (a^{2} + 2 a b \log {\left (c \right )} + b^{2} \log {\left (c \right )}^{2}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) & \text {for}\: r = 0 \\\frac {a^{2} d \log {\left (c x^{n} \right )}}{n} + \frac {a^{2} e x^{r}}{r} + \frac {a b d \log {\left (c x^{n} \right )}^{2}}{n} - \frac {2 a b e n x^{r}}{r^{2}} + \frac {2 a b e x^{r} \log {\left (c x^{n} \right )}}{r} + \frac {b^{2} d \log {\left (c x^{n} \right )}^{3}}{3 n} + \frac {2 b^{2} e n^{2} x^{r}}{r^{3}} - \frac {2 b^{2} e n x^{r} \log {\left (c x^{n} \right )}}{r^{2}} + \frac {b^{2} e x^{r} \log {\left (c x^{n} \right )}^{2}}{r} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise(((a + b*log(c))**2*(d + e)*log(x), Eq(n, 0) & Eq(r, 0)), ((a + b*log(c))**2*(d*log(x) + e*x**r/r), E
q(n, 0)), ((d + e)*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)), Eq(r, 0)), (a**2*d*log(c*x**n)/n + a**2*e*x**r/r + a*b*d
*log(c*x**n)**2/n - 2*a*b*e*n*x**r/r**2 + 2*a*b*e*x**r*log(c*x**n)/r + b**2*d*log(c*x**n)**3/(3*n) + 2*b**2*e*
n**2*x**r/r**3 - 2*b**2*e*n*x**r*log(c*x**n)/r**2 + b**2*e*x**r*log(c*x**n)**2/r, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.64 \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\frac {b^{2} e x^{r} \log \left (c x^{n}\right )^{2}}{r} + \frac {b^{2} d \log \left (c x^{n}\right )^{3}}{3 \, n} - 2 \, b^{2} e {\left (\frac {n x^{r} \log \left (c x^{n}\right )}{r^{2}} - \frac {n^{2} x^{r}}{r^{3}}\right )} + \frac {2 \, a b e x^{r} \log \left (c x^{n}\right )}{r} + \frac {a b d \log \left (c x^{n}\right )^{2}}{n} + a^{2} d \log \left (x\right ) - \frac {2 \, a b e n x^{r}}{r^{2}} + \frac {a^{2} e x^{r}}{r} \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\int { \frac {{\left (e x^{r} + d\right )} {\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^r\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx=\int \frac {\left (d+e\,x^r\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]

[In]

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

[Out]

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