\(\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx\) [215]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx=\text {Int}\left (\frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx=\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx \]

[In]

Int[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^2),x]

[Out]

Defer[Int][(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^2), 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)^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx=\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx \]

[In]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^2),x]

[Out]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^2), x]

Maple [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

\[\int \frac {\left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{\left (d f x +d e \right ) \left (i x +h \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2}} \,d x } \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^2,x, algorithm="fricas")

[Out]

integral((b*log(c*f*x + c*e) + a)^p/(d*f*i^2*x^3 + d*e*h^2 + (2*d*f*h*i + d*e*i^2)*x^2 + (d*f*h^2 + 2*d*e*h*i)
*x), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2}} \,d x } \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^2,x, algorithm="maxima")

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1,[0,1,1,0,0,0,0]%%%} / %%%{1,[0,0,1,1,1,0,0]%%%}+%%%{-1
,[0,0,0,1,0

Mupad [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{{\left (h+i\,x\right )}^2\,\left (d\,e+d\,f\,x\right )} \,d x \]

[In]

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

[Out]

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