### 3.19 $$\int \frac{a m x^m+b n q \log ^{-1+q}(c x^n)}{x} \, dx$$

Optimal. Leaf size=16 $a x^m+b \log ^q\left (c x^n\right )$

[Out]

a*x^m + b*Log[c*x^n]^q

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Rubi [A]  time = 0.034799, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.12, Rules used = {14, 2302, 30} $a x^m+b \log ^q\left (c x^n\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[(a*m*x^m + b*n*q*Log[c*x^n]^(-1 + q))/x,x]

[Out]

a*x^m + b*Log[c*x^n]^q

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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{a m x^m+b n q \log ^{-1+q}\left (c x^n\right )}{x} \, dx &=\int \left (a m x^{-1+m}+\frac{b n q \log ^{-1+q}\left (c x^n\right )}{x}\right ) \, dx\\ &=a x^m+(b n q) \int \frac{\log ^{-1+q}\left (c x^n\right )}{x} \, dx\\ &=a x^m+(b q) \operatorname{Subst}\left (\int x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )\\ &=a x^m+b \log ^q\left (c x^n\right )\\ \end{align*}

Mathematica [A]  time = 0.0146627, size = 16, normalized size = 1. $a x^m+b \log ^q\left (c x^n\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a*m*x^m + b*n*q*Log[c*x^n]^(-1 + q))/x,x]

[Out]

a*x^m + b*Log[c*x^n]^q

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Maple [A]  time = 0.003, size = 17, normalized size = 1.1 \begin{align*} a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((a*m*x^m+b*n*q*ln(c*x^n)^(-1+q))/x,x)

[Out]

a*x^m+b*ln(c*x^n)^q

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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((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.9989, size = 82, normalized size = 5.12 \begin{align*}{\left (b n \log \left (x\right ) + b \log \left (c\right )\right )}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1} + a x^{m} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))/x,x, algorithm="fricas")

[Out]

(b*n*log(x) + b*log(c))*(n*log(x) + log(c))^(q - 1) + a*x^m

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Sympy [A]  time = 61.9015, size = 58, normalized size = 3.62 \begin{align*} a m \left (\begin{cases} \frac{x^{m}}{m} & \text{for}\: m \neq 0 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) + b n q \left (\begin{cases} \frac{\log{\left (x \right )}}{\log{\left (c \right )}} & \text{for}\: n = 0 \wedge q = 0 \\\frac{\log{\left (c \right )}^{q} \log{\left (x \right )}}{\log{\left (c \right )}} & \text{for}\: n = 0 \\\frac{\log{\left (n \log{\left (x \right )} + \log{\left (c \right )} \right )}}{n} & \text{for}\: q = 0 \\\frac{\left (n \log{\left (x \right )} + \log{\left (c \right )}\right )^{q}}{n q} & \text{otherwise} \end{cases}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x**m+b*n*q*ln(c*x**n)**(-1+q))/x,x)

[Out]

a*m*Piecewise((x**m/m, Ne(m, 0)), (log(x), True)) + b*n*q*Piecewise((log(x)/log(c), Eq(n, 0) & Eq(q, 0)), (log
(c)**q*log(x)/log(c), Eq(n, 0)), (log(n*log(x) + log(c))/n, Eq(q, 0)), ((n*log(x) + log(c))**q/(n*q), True))

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Giac [A]  time = 1.31518, size = 23, normalized size = 1.44 \begin{align*}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*m*x^m+b*n*q*log(c*x^n)^(-1+q))/x,x, algorithm="giac")

[Out]

(n*log(x) + log(c))^q*b + a*x^m