\(\int \frac {1}{(f+g x) (h+i x)^2 (a+b \log (c (d+e x)^n))} \, dx\) [237]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\frac {g^2 \text {Int}\left (\frac {1}{(f+g x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{(g h-f i)^2}-\frac {i \text {Int}\left (\frac {1}{(h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{g h-f i}-\frac {g i \text {Int}\left (\frac {1}{(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )},x\right )}{(g h-f i)^2} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 31, 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 = {} \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.50 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.58 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )} {\left (i x + h\right )}^{2} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{\left (f+g\,x\right )\,{\left (h+i\,x\right )}^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )} \,d x \]

[In]

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

[Out]

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