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=36 \[ \text {Int}\left (\frac {c g+d g x}{B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {c g+d g x}{A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]

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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fricas [A]  time = 1.32, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d g x + c g}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}, x\right ) \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [A]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {d g x +c g}{B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {d g x + c g}{B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {c\,g+d\,g\,x}{A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ g \left (\int \frac {c}{A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}\, dx + \int \frac {d x}{A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}\, dx\right ) \]

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]

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

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