\(\int \frac {1}{(f+g x)^3 (A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))} \, dx\) [286]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\text {Int}\left (\frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )},x\right ) \]

[Out]

Unintegrable(1/(g*x+f)^3/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2)),x)

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx \]

[In]

Int[1/((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])),x]

[Out]

Defer[Int][1/((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx \]

[In]

Integrate[1/((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])),x]

[Out]

Integrate[1/((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])), x]

Maple [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (g x +f \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}d x\]

[In]

int(1/(g*x+f)^3/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2)),x)

[Out]

int(1/(g*x+f)^3/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2)),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.52 \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}} \,d x } \]

[In]

integrate(1/(g*x+f)^3/(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/(g*x+f)**3/(A+B*ln(e*(b*x+a)**2/(d*x+c)**2)),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}} \,d x } \]

[In]

integrate(1/(g*x+f)^3/(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="maxima")

[Out]

integrate(1/((g*x + f)^3*(B*log((b*x + a)^2*e/(d*x + c)^2) + A)), x)

Giac [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\int { \frac {1}{{\left (g x + f\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}} \,d x } \]

[In]

integrate(1/(g*x+f)^3/(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="giac")

[Out]

integrate(1/((g*x + f)^3*(B*log((b*x + a)^2*e/(d*x + c)^2) + A)), x)

Mupad [N/A]

Not integrable

Time = 12.41 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx=\int \frac {1}{{\left (f+g\,x\right )}^3\,\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )} \,d x \]

[In]

int(1/((f + g*x)^3*(A + B*log((e*(a + b*x)^2)/(c + d*x)^2))),x)

[Out]

int(1/((f + g*x)^3*(A + B*log((e*(a + b*x)^2)/(c + d*x)^2))), x)