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

Optimal. Leaf size=35 \[ \text {Int}\left (\frac {a g+b g x}{B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.10, 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 = {} \[ \int \frac {a g+b g x}{A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {a g+b g x}{A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b g x + a g}{B \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + A}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b g x + a g}{B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {b g x +a g}{B \ln \left (\frac {\left (d x +c \right )^{2} e}{\left (b x +a \right )^{2}}\right )+A}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b g x + a g}{B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a\,g+b\,g\,x}{A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ g \left (\int \frac {a}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {b x}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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