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*ln(c*x^n)^q/n/q

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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]

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 = 0.03, size = 25, normalized size = 1.00 \[ \frac {e \log ^q\left (c x^n\right )}{n q}+\frac {d x^m}{m} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 42, normalized size = 1.68 \[ \frac {d n q x^{m} + {\left (e m n \log \relax (x) + e m \log \relax (c)\right )} {\left (n \log \relax (x) + \log \relax (c)\right )}^{q - 1}}{m n q} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 27, normalized size = 1.08 \[ \frac {d x^{m}}{m} + \frac {{\left (n \log \relax (x) + \log \relax (c)\right )}^{q} e}{n q} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 26, normalized size = 1.04 \[ \frac {d \,x^{m}}{m}+\frac {e \ln \left (c \,x^{n}\right )^{q}}{n q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 25, normalized size = 1.00 \[ \frac {d x^{m}}{m} + \frac {e \log \left (c x^{n}\right )^{q}}{n q} \]

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]

d*x^m/m + e*log(c*x^n)^q/(n*q)

________________________________________________________________________________________

mupad [B]  time = 0.32, size = 25, normalized size = 1.00 \[ \frac {d\,x^m}{m}+\frac {e\,{\ln \left (c\,x^n\right )}^q}{n\,q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x^m)/m + (e*log(c*x^n)^q)/(n*q)

________________________________________________________________________________________

sympy [A]  time = 30.67, size = 53, normalized size = 2.12 \[ d \left (\begin {cases} \frac {x^{m}}{m} & \text {for}\: m \neq 0 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + e \left (\begin {cases} \frac {\log {\relax (x )}}{\log {\relax (c )}} & \text {for}\: n = 0 \wedge q = 0 \\\frac {\log {\relax (c )}^{q} \log {\relax (x )}}{\log {\relax (c )}} & \text {for}\: n = 0 \\\frac {\log {\left (n \log {\relax (x )} + \log {\relax (c )} \right )}}{n} & \text {for}\: q = 0 \\\frac {\left (n \log {\relax (x )} + \log {\relax (c )}\right )^{q}}{n q} & \text {otherwise} \end {cases}\right ) \]

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))

________________________________________________________________________________________