\(\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx\) [216]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx=\text {Int}\left (\frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^3,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 = {} \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx=\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx \]

[In]

Int[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^3),x]

[Out]

Defer[Int][(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^3), 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)^3} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx=\int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx \]

[In]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^3),x]

[Out]

Integrate[(a + b*Log[c*(e + f*x)])^p/((d*e + d*f*x)*(h + i*x)^3), x]

Maple [N/A]

Not integrable

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

\[\int \frac {\left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{p}}{\left (d f x +d e \right ) \left (i x +h \right )^{3}}d x\]

[In]

int((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^3,x)

[Out]

int((a+b*ln(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^3,x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.88 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^3,x, algorithm="fricas")

[Out]

integral((b*log(c*f*x + c*e) + a)^p/(d*f*i^3*x^4 + d*e*h^3 + (3*d*f*h*i^2 + d*e*i^3)*x^3 + 3*(d*f*h^2*i + d*e*
h*i^2)*x^2 + (d*f*h^3 + 3*d*e*h^2*i)*x), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx=\text {Timed out} \]

[In]

integrate((a+b*ln(c*(f*x+e)))**p/(d*f*x+d*e)/(i*x+h)**3,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^3,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{p}}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{3}} \,d x } \]

[In]

integrate((a+b*log(c*(f*x+e)))^p/(d*f*x+d*e)/(i*x+h)^3,x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \log (c (e+f x)))^p}{(d e+d f x) (h+i x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^p}{{\left (h+i\,x\right )}^3\,\left (d\,e+d\,f\,x\right )} \,d x \]

[In]

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

[Out]

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