\(\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx\) [214]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx=\text {Int}\left (\frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x)

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 32, 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 = {} \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx=\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx \]

[In]

Int[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)),x]

[Out]

Defer[Int][(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx=\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx \]

[In]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)),x]

[Out]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)), x]

Maple [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

\[\int \frac {\left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{\left (d f x +d e \right ) \left (i x +h \right )}d x\]

[In]

int((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x)

[Out]

int((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{{\left (d f x + d e\right )} {\left (i x + h\right )}} \,d x } \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x, algorithm="fricas")

[Out]

integral((b*log(c*f*x + c*e) + a)^p/(d*f*i*x^2 + d*e*h + (d*f*h + d*e*i)*x), x)

Sympy [N/A]

Not integrable

Time = 154.49 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx=\frac {\int \frac {\left (a + b \log {\left (c e + c f x \right )}\right )^{p}}{e h + e i x + f h x + f i x^{2}}\, dx}{d} \]

[In]

integrate((a+b*ln(c*(f*x+e)))**p/(d*f*x+d*e)/(i*x+h),x)

[Out]

Integral((a + b*log(c*e + c*f*x))**p/(e*h + e*i*x + f*h*x + f*i*x**2), x)/d

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{{\left (d f x + d e\right )} {\left (i x + h\right )}} \,d x } \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x, algorithm="maxima")

[Out]

integrate((b*log((f*x + e)*c) + a)^p/((d*f*x + d*e)*(i*x + h)), x)

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{{\left (d f x + d e\right )} {\left (i x + h\right )}} \,d x } \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h),x, algorithm="giac")

[Out]

integrate((b*log((f*x + e)*c) + a)^p/((d*f*x + d*e)*(i*x + h)), x)

Mupad [N/A]

Not integrable

Time = 1.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{\left (h+i\,x\right )\,\left (d\,e+d\,f\,x\right )} \,d x \]

[In]

int((a + b*log(c*(e + f*x)))^p/((h + i*x)*(d*e + d*f*x)),x)

[Out]

int((a + b*log(c*(e + f*x)))^p/((h + i*x)*(d*e + d*f*x)), x)