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

Optimal. Leaf size=35 \[ \text{Unintegrable}\left (\frac{c g+d g x}{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.114506, 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 = {} \[ \int \frac{c g+d g x}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.2834, size = 0, normalized size = 0. \[ \int \frac{c g+d g x}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.303, size = 0, normalized size = 0. \begin{align*} \int{(dgx+cg) \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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