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

Optimal. Leaf size=34 \[ \text{Unintegrable}\left (\frac{a g+b g x}{B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.101449, 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 = {} \[ \int \frac{a g+b g x}{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.115246, size = 0, normalized size = 0. \[ \int \frac{a g+b g x}{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.857, size = 0, normalized size = 0. \begin{align*} \int{(bgx+ag) \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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