\(\int \frac {(a m x^m+b n q \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))^p}{x} \, dx\) [16]

Optimal result

Integrand size = 43, antiderivative size = 26 \[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{1+p} \]

[Out]

(a*x^m+b*ln(c*x^n)^q)^(p+1)/(p+1)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2624} \[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{p+1}}{p+1} \]

[In]

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

[Out]

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

Rule 2624

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x
_)^(m_.)))/(x_), x_Symbol] :> Simp[e*((a*x^m + b*Log[c*x^n]^q)^(p + 1)/(b*n*q*(p + 1))), x] /; FreeQ[{a, b, c,
d, e, m, n, p, q, r}, x] && EqQ[r, q - 1] && NeQ[p, -1] && EqQ[a*e*m - b*d*n*q, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{1+p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\frac {\left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{1+p} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {\left (a m \,x^{m}+b n q \ln \left (c \,x^{n}\right )^{-1+q}\right ) \left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{p}}{x}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\frac {{\left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\right )} {\left ({\left (n \log \left (x\right ) + \log \left (c\right )\right )}^{q} b + a x^{m}\right )}^{p}}{p + 1} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to divide,
perhaps due to rounding error%%%{1,[0,0,2,5,2,0,5,0,3,1,2,3]%%%}+%%%{-2,[0,0,2,4,2,1,5,0,2,1,2,3]%%%}+%%%{5,[
0,0,2,4,2,0,4,

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a m x^m+b n q \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {\left (a\,m\,x^m+b\,n\,q\,{\ln \left (c\,x^n\right )}^{q-1}\right )\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^p}{x} \,d x \]

[In]

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

[Out]

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