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

Optimal. Leaf size=31 \[ \text{Unintegrable}\left (\frac{1}{(f+g x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.065483, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.883091, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 1.366, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{2}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x + f\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{A g^{2} x^{2} + 2 \, A f g x + A f^{2} +{\left (B g^{2} x^{2} + 2 \, B f g x + B f^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (g x + f\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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