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

Optimal. Leaf size=22 \[ \frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

[Out]

(a + b*Log[c*x^n])^4/(4*b*n)

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Rubi [A]  time = 0.0222569, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2302, 30} \[ \frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])^3/x,x]

[Out]

(a + b*Log[c*x^n])^4/(4*b*n)

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx &=\frac{\operatorname{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]  time = 0.003283, size = 22, normalized size = 1. \[ \frac{\left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])^3/x,x]

[Out]

(a + b*Log[c*x^n])^4/(4*b*n)

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Maple [B]  time = 0.036, size = 75, normalized size = 3.4 \begin{align*}{\frac{{b}^{3} \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{4}}{4\,n}}+{\frac{{b}^{2} \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{3}a}{n}}+{\frac{3\,b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{2}{a}^{2}}{2\,n}}+{\frac{\ln \left ( c{x}^{n} \right ){a}^{3}}{n}}+{\frac{{a}^{4}}{4\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10159, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{4}}{4 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 0.810149, size = 255, normalized size = 11.59 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 37.4416, size = 92, normalized size = 4.18 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [B]  time = 1.2242, size = 154, normalized size = 7. \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)