3.3.57 \(\int \frac {f+g x}{(A+B \log (\frac {e (a+b x)}{c+d x}))^2} \, dx\) [257]

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {f+g x}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2},x\right ) \]

[Out]

Unintegrable((g*x+f)/(A+B*ln(e*(b*x+a)/(d*x+c)))^2,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 {f+g x}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {f+g x}{\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.81, size = 0, normalized size = 0.00 \[\int \frac {g x +f}{\left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-(b*d*g*x^3 + a*c*f + (a*d*g + (d*f + c*g)*b)*x^2 + (b*c*f + (d*f + c*g)*a)*x)/((b*c - a*d)*B^2*log(b*x + a) -
(b*c - a*d)*B^2*log(d*x + c) + (b*c - a*d)*A*B + (b*c - a*d)*B^2) + integrate((3*b*d*g*x^2 + b*c*f + (d*f + c
*g)*a + 2*(a*d*g + (d*f + c*g)*b)*x)/((b*c - a*d)*B^2*log(b*x + a) - (b*c - a*d)*B^2*log(d*x + c) + (b*c - a*d
)*A*B + (b*c - a*d)*B^2), 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((g*x+f)/(A+B*log(e*(b*x+a)/(d*x+c)))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {a c f + a c g x + a d f x + a d g x^{2} + b c f x + b c g x^{2} + b d f x^{2} + b d g x^{3}}{A B a d - A B b c + \left (B^{2} a d - B^{2} b c\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}} - \frac {\int \frac {a c g}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {a d f}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {b c f}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {2 a d g x}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {2 b c g x}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {2 b d f x}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx + \int \frac {3 b d g x^{2}}{A + B \log {\left (\frac {a e}{c + d x} + \frac {b e x}{c + d x} \right )}}\, dx}{B \left (a d - b c\right )} \end {gather*}

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)))**2,x)

[Out]

(a*c*f + a*c*g*x + a*d*f*x + a*d*g*x**2 + b*c*f*x + b*c*g*x**2 + b*d*f*x**2 + b*d*g*x**3)/(A*B*a*d - A*B*b*c +
(B**2*a*d - B**2*b*c)*log(e*(a + b*x)/(c + d*x))) - (Integral(a*c*g/(A + B*log(a*e/(c + d*x) + b*e*x/(c + d*x
))), x) + Integral(a*d*f/(A + B*log(a*e/(c + d*x) + b*e*x/(c + d*x))), x) + Integral(b*c*f/(A + B*log(a*e/(c +
d*x) + b*e*x/(c + d*x))), x) + Integral(2*a*d*g*x/(A + B*log(a*e/(c + d*x) + b*e*x/(c + d*x))), x) + Integral
(2*b*c*g*x/(A + B*log(a*e/(c + d*x) + b*e*x/(c + d*x))), x) + Integral(2*b*d*f*x/(A + B*log(a*e/(c + d*x) + b*
e*x/(c + d*x))), x) + Integral(3*b*d*g*x**2/(A + B*log(a*e/(c + d*x) + b*e*x/(c + d*x))), x))/(B*(a*d - b*c))

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

[Out]

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

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

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)))^2,x)

[Out]

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

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