3.26 \(\int \frac {1}{(a g+b g x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2} \, dx\)

Optimal. Leaf size=38 \[ \text {Int}\left (\frac {1}{(a g+b g x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ) \]

[Out]

Unintegrable(1/(b*g*x+a*g)/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,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 = {} \[ \int \frac {1}{(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {1}{(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{A^{2} b g x + A^{2} a g + {\left (B^{2} b g x + B^{2} a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b g x + A B a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

integral(1/(A^2*b*g*x + A^2*a*g + (B^2*b*g*x + B^2*a*g)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b*g*x + A*B*
a*g)*log(e*((b*x + a)/(d*x + c))^n)), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b g x + a g\right )} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

integrate(1/((b*g*x + a*g)*(B*log(e*((b*x + a)/(d*x + c))^n) + A)^2), x)

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maple [A]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b g x +a g \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*g*x+a*g)/(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2,x)

[Out]

int(1/(b*g*x+a*g)/(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ d \int \frac {1}{{\left (b c g n - a d g n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c g n - a d g n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left (b c g n - a d g n\right )} A B + {\left (b c g n \log \relax (e) - a d g n \log \relax (e)\right )} B^{2}}\,{d x} - \frac {d x + c}{{\left (b c g n - a d g n\right )} B^{2} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c g n - a d g n\right )} B^{2} \log \left ({\left (d x + c\right )}^{n}\right ) + {\left (b c g n - a d g n\right )} A B + {\left (b c g n \log \relax (e) - a d g n \log \relax (e)\right )} B^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*g*x+a*g)/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

d*integrate(1/((b*c*g*n - a*d*g*n)*B^2*log((b*x + a)^n) - (b*c*g*n - a*d*g*n)*B^2*log((d*x + c)^n) + (b*c*g*n
- a*d*g*n)*A*B + (b*c*g*n*log(e) - a*d*g*n*log(e))*B^2), x) - (d*x + c)/((b*c*g*n - a*d*g*n)*B^2*log((b*x + a)
^n) - (b*c*g*n - a*d*g*n)*B^2*log((d*x + c)^n) + (b*c*g*n - a*d*g*n)*A*B + (b*c*g*n*log(e) - a*d*g*n*log(e))*B
^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a*g + b*g*x)*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2),x)

[Out]

int(1/((a*g + b*g*x)*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2), x)

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sympy [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(1/(b*g*x+a*g)/(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

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