\(\int \frac {(d x^m+e \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))^p}{x} \, dx\) [30]

Optimal result

Integrand size = 40, antiderivative size = 40 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\frac {e \left (a x^m+b \log ^q\left (c x^n\right )\right )^{1+p}}{b n (1+p) q}+\left (d-\frac {a e m}{b n q}\right ) \text {Int}\left (x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right )^p,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.46 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int { \frac {{\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )} {\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{p}}{x} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[In]

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

[In]

integrate((d*x^m+e*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,[0,0,2,5,2,0,5,0,2,1,2,2,1]%%%}+%%%{-2,[0,0,2,4,2,1,5,0,1,1,2,2,1]%%%}+%%%
{5,[0,0,2,4,2,

Mupad [N/A]

Not integrable

Time = 1.89 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.05 \[ \int \frac {\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^p}{x} \, dx=\int \frac {{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^p\,\left (d\,x^m+e\,{\ln \left (c\,x^n\right )}^{q-1}\right )}{x} \,d x \]

[In]

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

[Out]

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