3.2.92 \(\int \frac {a g+b g x}{A+B \log (\frac {e (c+d x)}{a+b x})} \, dx\) [192]

Optimal. Leaf size=33 \[ \text {Int}\left (\frac {a g+b g x}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, 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 = {} \begin {gather*} \int \frac {a g+b g x}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a g+b g x}{A+B \log \left (\frac {e (c+d x)}{a+b x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.69, size = 0, normalized size = 0.00 \[\int \frac {b g x +a g}{A +B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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