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

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {f+g x}{B \log \left (\frac {e (a+b x)}{c+d x}\right )+A},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {f+g x}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {f+g x}{A+B \log \left (\frac {e (a+b x)}{c+d x}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {g x + f}{B \log \left (\frac {b e x + a e}{d x + c}\right ) + A}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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maple [A]  time = 0.99, size = 0, normalized size = 0.00 \[ \int \frac {g x +f}{B \ln \left (\frac {\left (b x +a \right ) e}{d x +c}\right )+A}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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