### 3.34 $$\int \frac{d x^m+e \log ^{-1+q}(c x^n)}{x} \, dx$$

Optimal. Leaf size=25 $\frac{e \log ^q\left (c x^n\right )}{n q}+\frac{d x^m}{m}$

[Out]

(d*x^m)/m + (e*Log[c*x^n]^q)/(n*q)

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Rubi [A]  time = 0.0338121, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.136, Rules used = {14, 2302, 30} $\frac{e \log ^q\left (c x^n\right )}{n q}+\frac{d x^m}{m}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d*x^m)/m + (e*Log[c*x^n]^q)/(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{d x^m+e \log ^{-1+q}\left (c x^n\right )}{x} \, dx &=\int \left (d x^{-1+m}+\frac{e \log ^{-1+q}\left (c x^n\right )}{x}\right ) \, dx\\ &=\frac{d x^m}{m}+e \int \frac{\log ^{-1+q}\left (c x^n\right )}{x} \, dx\\ &=\frac{d x^m}{m}+\frac{e \operatorname{Subst}\left (\int x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{d x^m}{m}+\frac{e \log ^q\left (c x^n\right )}{n q}\\ \end{align*}

Mathematica [A]  time = 0.0201541, size = 25, normalized size = 1. $\frac{e \log ^q\left (c x^n\right )}{n q}+\frac{d x^m}{m}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(d*x^m)/m + (e*Log[c*x^n]^q)/(n*q)

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Maple [A]  time = 0.007, size = 26, normalized size = 1. \begin{align*}{\frac{d{x}^{m}}{m}}+{\frac{e \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q}}{nq}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x^m/m+e*ln(c*x^n)^q/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((d*x^m+e*log(c*x^n)^(-1+q))/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.86243, size = 107, normalized size = 4.28 \begin{align*} \frac{d n q x^{m} +{\left (e m n \log \left (x\right ) + e m \log \left (c\right )\right )}{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q - 1}}{m n q} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*n*q*x^m + (e*m*n*log(x) + e*m*log(c))*(n*log(x) + log(c))^(q - 1))/(m*n*q)

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Sympy [A]  time = 60.9989, size = 53, normalized size = 2.12 \begin{align*} d \left (\begin{cases} \frac{x^{m}}{m} & \text{for}\: m \neq 0 \\\log{\left (x \right )} & \text{otherwise} \end{cases}\right ) + e \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((d*x**m+e*ln(c*x**n)**(-1+q))/x,x)

[Out]

d*Piecewise((x**m/m, Ne(m, 0)), (log(x), True)) + e*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 = 1.31321, size = 36, normalized size = 1.44 \begin{align*} \frac{d x^{m}}{m} + \frac{{\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} e}{n q} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x^m/m + (n*log(x) + log(c))^q*e/(n*q)