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*ln(c*x^n)^q

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Rubi [A]  time = 0.03, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, 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 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 {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*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 1.00 \[ a x^m+b \log ^q\left (c x^n\right ) \]

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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fricas [A]  time = 0.44, size = 28, normalized size = 1.75 \[ {\left (b n \log \relax (x) + b \log \relax (c)\right )} {\left (n \log \relax (x) + \log \relax (c)\right )}^{q - 1} + a x^{m} \]

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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giac [A]  time = 0.20, size = 17, normalized size = 1.06 \[ {\left (n \log \relax (x) + \log \relax (c)\right )}^{q} b + a x^{m} \]

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

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maple [A]  time = 0.08, size = 17, normalized size = 1.06 \[ a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.45, size = 16, normalized size = 1.00 \[ a x^{m} + b \log \left (c x^{n}\right )^{q} \]

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]

a*x^m + b*log(c*x^n)^q

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mupad [B]  time = 0.30, size = 16, normalized size = 1.00 \[ a\,x^m+b\,{\ln \left (c\,x^n\right )}^q \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x^m + b*log(c*x^n)^q

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sympy [A]  time = 30.63, size = 58, normalized size = 3.62 \[ a m \left (\begin {cases} \frac {x^{m}}{m} & \text {for}\: m \neq 0 \\\log {\relax (x )} & \text {otherwise} \end {cases}\right ) + b n q \left (\begin {cases} \frac {\log {\relax (x )}}{\log {\relax (c )}} & \text {for}\: n = 0 \wedge q = 0 \\\frac {\log {\relax (c )}^{q} \log {\relax (x )}}{\log {\relax (c )}} & \text {for}\: n = 0 \\\frac {\log {\left (n \log {\relax (x )} + \log {\relax (c )} \right )}}{n} & \text {for}\: q = 0 \\\frac {\left (n \log {\relax (x )} + \log {\relax (c )}\right )^{q}}{n q} & \text {otherwise} \end {cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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