\(\int \frac {(a+b \log (c x^n))^3}{x} \, dx\) [61]

Optimal result

Integrand size = 16, antiderivative size = 22 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

[Out]

1/4*(a+b*ln(c*x^n))^4/b/n

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2339, 30} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

[In]

Int[(a + b*Log[c*x^n])^3/x,x]

[Out]

(a + b*Log[c*x^n])^4/(4*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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{b n} \\ & = \frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

[In]

Integrate[(a + b*Log[c*x^n])^3/x,x]

[Out]

(a + b*Log[c*x^n])^4/(4*b*n)

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4 b n}\)	\(21\)
default	\(\frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4 b n}\)	\(21\)
parallelrisch	\(\frac {b^{3} \ln \left (c \,x^{n}\right )^{4}+4 a \,b^{2} \ln \left (c \,x^{n}\right )^{3}+4 \ln \left (x \right ) a^{3} n +6 a^{2} b \ln \left (c \,x^{n}\right )^{2}}{4 n}\)	\(55\)
parts	\(\ln \left (x \right ) a^{3}+\frac {b^{3} \ln \left (c \,x^{n}\right )^{4}}{4 n}+\frac {a \,b^{2} \ln \left (c \,x^{n}\right )^{3}}{n}+\frac {3 a^{2} b \ln \left (c \,x^{n}\right )^{2}}{2 n}\)	\(57\)
risch	\(\text {Expression too large to display}\)	\(2945\)

[In]

int((a+b*ln(c*x^n))^3/x,x,method=_RETURNVERBOSE)

[Out]

1/4*(a+b*ln(c*x^n))^4/b/n

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (20) = 40\).

Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.55 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\frac {1}{4} \, b^{3} n^{3} \log \left (x\right )^{4} + {\left (b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )} \log \left (x\right )^{3} + \frac {3}{2} \, {\left (b^{3} n \log \left (c\right )^{2} + 2 \, a b^{2} n \log \left (c\right ) + a^{2} b n\right )} \log \left (x\right )^{2} + {\left (b^{3} \log \left (c\right )^{3} + 3 \, a b^{2} \log \left (c\right )^{2} + 3 \, a^{2} b \log \left (c\right ) + a^{3}\right )} \log \left (x\right ) \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (15) = 30\).

Time = 7.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.18 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\begin {cases} \frac {a^{3} \log {\left (c x^{n} \right )} + \frac {3 a^{2} b \log {\left (c x^{n} \right )}^{2}}{2} + a b^{2} \log {\left (c x^{n} \right )}^{3} + \frac {b^{3} \log {\left (c x^{n} \right )}^{4}}{4}}{n} & \text {for}\: n \neq 0 \\\left (a^{3} + 3 a^{2} b \log {\left (c \right )} + 3 a b^{2} \log {\left (c \right )}^{2} + b^{3} \log {\left (c \right )}^{3}\right ) \log {\left (x \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((a+b*ln(c*x**n))**3/x,x)

[Out]

Piecewise(((a**3*log(c*x**n) + 3*a**2*b*log(c*x**n)**2/2 + a*b**2*log(c*x**n)**3 + b**3*log(c*x**n)**4/4)/n, N
e(n, 0)), ((a**3 + 3*a**2*b*log(c) + 3*a*b**2*log(c)**2 + b**3*log(c)**3)*log(x), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{4}}{4 \, b n} \]

[In]

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

[Out]

1/4*(b*log(c*x^n) + a)^4/(b*n)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (20) = 40\).

Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 5.18 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\frac {1}{4} \, b^{3} n^{3} \log \left (x\right )^{4} + b^{3} n^{2} \log \left (c\right ) \log \left (x\right )^{3} + \frac {3}{2} \, b^{3} n \log \left (c\right )^{2} \log \left (x\right )^{2} + a b^{2} n^{2} \log \left (x\right )^{3} + b^{3} \log \left (c\right )^{3} \log \left (x\right ) + 3 \, a b^{2} n \log \left (c\right ) \log \left (x\right )^{2} + 3 \, a b^{2} \log \left (c\right )^{2} \log \left (x\right ) + \frac {3}{2} \, a^{2} b n \log \left (x\right )^{2} + 3 \, a^{2} b \log \left (c\right ) \log \left (x\right ) + a^{3} \log \left (x\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=a^3\,\ln \left (x\right )+\frac {b^3\,{\ln \left (c\,x^n\right )}^4}{4\,n}+\frac {3\,a^2\,b\,{\ln \left (c\,x^n\right )}^2}{2\,n}+\frac {a\,b^2\,{\ln \left (c\,x^n\right )}^3}{n} \]

[In]

int((a + b*log(c*x^n))^3/x,x)

[Out]

a^3*log(x) + (b^3*log(c*x^n)^4)/(4*n) + (3*a^2*b*log(c*x^n)^2)/(2*n) + (a*b^2*log(c*x^n)^3)/n