\(\int \frac {(h+i x)^q (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx\) [209]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {(h+i x)^q (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int \frac {(h+i x)^q (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(h+i x)^q (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int \frac {(h+i x)^q (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {(h+i x)^q (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int { \frac {{\left (i x + h\right )}^{q} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{d f x + d e} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(h+i x)^q (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int { \frac {{\left (i x + h\right )}^{q} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{d f x + d e} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

integrate((i*x+h)^q*(a+b*log(c*(f*x+e)))^p/(d*f*x+d*e),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,[0,0,5,0,2,0,5,0,3,0,2,0]%%%}+%%%{5,[0,0,4,0,2,0,4,1,3,0,2,0]%%%}+%%%{10,[
0,0,3,0,2,0,3,

Mupad [N/A]

Not integrable

Time = 1.63 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(h+i x)^q (a+b \log (c (e+f x)))^p}{d e+d f x} \, dx=\int \frac {{\left (h+i\,x\right )}^q\,{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{d\,e+d\,f\,x} \,d x \]

[In]

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

[Out]

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