\(\int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx\) [196]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx=\frac {\log (a+b \log (c (e+f x)))}{b d (f h-e i)}-\frac {i \text {Int}\left (\frac {1}{(h+i x) (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)} \]

[Out]

ln(a+b*ln(c*(f*x+e)))/b/d/(-e*i+f*h)-i*Unintegrable(1/(i*x+h)/(a+b*ln(c*(f*x+e))),x)/d/(-e*i+f*h)

Rubi [N/A]

Not integrable

Time = 0.16 (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 = {} \[ \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx=\int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx \]

[In]

Int[1/((d*e + d*f*x)*(h + i*x)*(a + b*Log[c*(e + f*x)])),x]

[Out]

Log[a + b*Log[c*(e + f*x)]]/(b*d*(f*h - e*i)) - (i*Defer[Int][1/((h + i*x)*(a + b*Log[c*(e + f*x)])), x])/(d*(
f*h - e*i))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {f}{d (f h-e i) (e+f x) (a+b \log (c (e+f x)))}-\frac {i}{d (f h-e i) (h+i x) (a+b \log (c (e+f x)))}\right ) \, dx \\ & = \frac {f \int \frac {1}{(e+f x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)}-\frac {i \int \frac {1}{(h+i x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d (f h-e i)}-\frac {i \int \frac {1}{(h+i x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d (f h-e i)}-\frac {i \int \frac {1}{(h+i x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)} \\ & = \frac {\log (a+b \log (c (e+f x)))}{b d (f h-e i)}-\frac {i \int \frac {1}{(h+i x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx=\int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx \]

[In]

Integrate[1/((d*e + d*f*x)*(h + i*x)*(a + b*Log[c*(e + f*x)])),x]

[Out]

Integrate[1/((d*e + d*f*x)*(h + i*x)*(a + b*Log[c*(e + f*x)])), x]

Maple [N/A]

Not integrable

Time = 0.82 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d f x +d e \right ) \left (i x +h \right ) \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}d x\]

[In]

int(1/(d*f*x+d*e)/(i*x+h)/(a+b*ln(c*(f*x+e))),x)

[Out]

int(1/(d*f*x+d*e)/(i*x+h)/(a+b*ln(c*(f*x+e))),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.12 \[ \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx=\int { \frac {1}{{\left (d f x + d e\right )} {\left (i x + h\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*f*x+d*e)/(i*x+h)/(a+b*log(c*(f*x+e))),x, algorithm="fricas")

[Out]

integral(1/(a*d*f*i*x^2 + a*d*e*h + (a*d*f*h + a*d*e*i)*x + (b*d*f*i*x^2 + b*d*e*h + (b*d*f*h + b*d*e*i)*x)*lo
g(c*f*x + c*e)), x)

Sympy [N/A]

Not integrable

Time = 2.61 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.09 \[ \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx=\frac {\int \frac {1}{a e h + a e i x + a f h x + a f i x^{2} + b e h \log {\left (c e + c f x \right )} + b e i x \log {\left (c e + c f x \right )} + b f h x \log {\left (c e + c f x \right )} + b f i x^{2} \log {\left (c e + c f x \right )}}\, dx}{d} \]

[In]

integrate(1/(d*f*x+d*e)/(i*x+h)/(a+b*ln(c*(f*x+e))),x)

[Out]

Integral(1/(a*e*h + a*e*i*x + a*f*h*x + a*f*i*x**2 + b*e*h*log(c*e + c*f*x) + b*e*i*x*log(c*e + c*f*x) + b*f*h
*x*log(c*e + c*f*x) + b*f*i*x**2*log(c*e + c*f*x)), x)/d

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx=\int { \frac {1}{{\left (d f x + d e\right )} {\left (i x + h\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*f*x+d*e)/(i*x+h)/(a+b*log(c*(f*x+e))),x, algorithm="maxima")

[Out]

integrate(1/((d*f*x + d*e)*(i*x + h)*(b*log((f*x + e)*c) + a)), x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx=\int { \frac {1}{{\left (d f x + d e\right )} {\left (i x + h\right )} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*f*x+d*e)/(i*x+h)/(a+b*log(c*(f*x+e))),x, algorithm="giac")

[Out]

integrate(1/((d*f*x + d*e)*(i*x + h)*(b*log((f*x + e)*c) + a)), x)

Mupad [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d e+d f x) (h+i x) (a+b \log (c (e+f x)))} \, dx=\int \frac {1}{\left (h+i\,x\right )\,\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \]

[In]

int(1/((h + i*x)*(d*e + d*f*x)*(a + b*log(c*(e + f*x)))),x)

[Out]

int(1/((h + i*x)*(d*e + d*f*x)*(a + b*log(c*(e + f*x)))), x)