\(\int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx\) [197]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx=\frac {f \log (a+b \log (c (e+f x)))}{b d (f h-e i)^2}-\frac {i \text {Int}\left (\frac {1}{(h+i x)^2 (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)}-\frac {f i \text {Int}\left (\frac {1}{(h+i x) (a+b \log (c (e+f x)))},x\right )}{d (f h-e i)^2} \]

[Out]

f*ln(a+b*ln(c*(f*x+e)))/b/d/(-e*i+f*h)^2-i*Unintegrable(1/(i*x+h)^2/(a+b*ln(c*(f*x+e))),x)/d/(-e*i+f*h)-f*i*Un
integrable(1/(i*x+h)/(a+b*ln(c*(f*x+e))),x)/d/(-e*i+f*h)^2

Rubi [N/A]

Not integrable

Time = 0.20 (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 = {} \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx=\int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx \]

[In]

Int[1/((d*e + d*f*x)*(h + i*x)^2*(a + b*Log[c*(e + f*x)])),x]

[Out]

(f*Log[a + b*Log[c*(e + f*x)]])/(b*d*(f*h - e*i)^2) - (i*Defer[Int][1/((h + i*x)^2*(a + b*Log[c*(e + f*x)])),
x])/(d*(f*h - e*i)) - (f*i*Defer[Int][1/((h + i*x)*(a + b*Log[c*(e + f*x)])), x])/(d*(f*h - e*i)^2)

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {f^2}{d (f h-e i)^2 (e+f x) (a+b \log (c (e+f x)))}-\frac {i}{d (f h-e i) (h+i x)^2 (a+b \log (c (e+f x)))}-\frac {f i}{d (f h-e i)^2 (h+i x) (a+b \log (c (e+f x)))}\right ) \, dx \\ & = \frac {f^2 \int \frac {1}{(e+f x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)^2}-\frac {(f i) \int \frac {1}{(h+i x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)^2}-\frac {i \int \frac {1}{(h+i x)^2 (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)} \\ & = \frac {f \text {Subst}\left (\int \frac {1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d (f h-e i)^2}-\frac {(f i) \int \frac {1}{(h+i x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)^2}-\frac {i \int \frac {1}{(h+i x)^2 (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)} \\ & = \frac {f \text {Subst}\left (\int \frac {1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d (f h-e i)^2}-\frac {(f i) \int \frac {1}{(h+i x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)^2}-\frac {i \int \frac {1}{(h+i x)^2 (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)} \\ & = \frac {f \log (a+b \log (c (e+f x)))}{b d (f h-e i)^2}-\frac {(f i) \int \frac {1}{(h+i x) (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)^2}-\frac {i \int \frac {1}{(h+i x)^2 (a+b \log (c (e+f x)))} \, dx}{d (f h-e i)} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx=\int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx \]

[In]

Integrate[1/((d*e + d*f*x)*(h + i*x)^2*(a + b*Log[c*(e + f*x)])),x]

[Out]

Integrate[1/((d*e + d*f*x)*(h + i*x)^2*(a + b*Log[c*(e + f*x)])), x]

Maple [N/A]

Not integrable

Time = 0.62 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d f x +d e \right ) \left (i x +h \right )^{2} \left (a +b \ln \left (c \left (f x +e \right )\right )\right )}d x\]

[In]

int(1/(d*f*x+d*e)/(i*x+h)^2/(a+b*ln(c*(f*x+e))),x)

[Out]

int(1/(d*f*x+d*e)/(i*x+h)^2/(a+b*ln(c*(f*x+e))),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.81 \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx=\int { \frac {1}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*f*x+d*e)/(i*x+h)^2/(a+b*log(c*(f*x+e))),x, algorithm="fricas")

[Out]

integral(1/(a*d*f*i^2*x^3 + a*d*e*h^2 + (2*a*d*f*h*i + a*d*e*i^2)*x^2 + (a*d*f*h^2 + 2*a*d*e*h*i)*x + (b*d*f*i
^2*x^3 + b*d*e*h^2 + (2*b*d*f*h*i + b*d*e*i^2)*x^2 + (b*d*f*h^2 + 2*b*d*e*h*i)*x)*log(c*f*x + c*e)), x)

Sympy [N/A]

Not integrable

Time = 4.98 (sec) , antiderivative size = 180, normalized size of antiderivative = 5.62 \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx=\frac {\int \frac {1}{a e h^{2} + 2 a e h i x + a e i^{2} x^{2} + a f h^{2} x + 2 a f h i x^{2} + a f i^{2} x^{3} + b e h^{2} \log {\left (c e + c f x \right )} + 2 b e h i x \log {\left (c e + c f x \right )} + b e i^{2} x^{2} \log {\left (c e + c f x \right )} + b f h^{2} x \log {\left (c e + c f x \right )} + 2 b f h i x^{2} \log {\left (c e + c f x \right )} + b f i^{2} x^{3} \log {\left (c e + c f x \right )}}\, dx}{d} \]

[In]

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

[Out]

Integral(1/(a*e*h**2 + 2*a*e*h*i*x + a*e*i**2*x**2 + a*f*h**2*x + 2*a*f*h*i*x**2 + a*f*i**2*x**3 + b*e*h**2*lo
g(c*e + c*f*x) + 2*b*e*h*i*x*log(c*e + c*f*x) + b*e*i**2*x**2*log(c*e + c*f*x) + b*f*h**2*x*log(c*e + c*f*x) +
2*b*f*h*i*x**2*log(c*e + c*f*x) + b*f*i**2*x**3*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 \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx=\int { \frac {1}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*f*x+d*e)/(i*x+h)^2/(a+b*log(c*(f*x+e))),x, algorithm="maxima")

[Out]

integrate(1/((d*f*x + d*e)*(i*x + h)^2*(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 \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx=\int { \frac {1}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*f*x+d*e)/(i*x+h)^2/(a+b*log(c*(f*x+e))),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(d e+d f x) (h+i x)^2 (a+b \log (c (e+f x)))} \, dx=\int \frac {1}{{\left (h+i\,x\right )}^2\,\left (d\,e+d\,f\,x\right )\,\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )} \,d x \]

[In]

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

[Out]

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