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

Optimal. Leaf size=32 \[ \text{Unintegrable}\left (\frac{f+g x}{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.113645, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{f+g x}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.695821, size = 0, normalized size = 0. \[ \int \frac{f+g x}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.253, size = 0, normalized size = 0. \begin{align*} \int{(gx+f) \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-2}}\, dx \end{align*}

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))^n))^2,x)

[Out]

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

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

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))^n))^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*n - a*d*n)*B^2*log((b*x +
a)^n) - (b*c*n - a*d*n)*B^2*log((d*x + c)^n) + (b*c*n - a*d*n)*A*B + (b*c*n*log(e) - a*d*n*log(e))*B^2) + int
egrate((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*n - a*d*n)*B^2*log((b*x + a)^
n) - (b*c*n - a*d*n)*B^2*log((d*x + c)^n) + (b*c*n - a*d*n)*A*B + (b*c*n*log(e) - a*d*n*log(e))*B^2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{g x + f}{B^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A^{2}}, x\right ) \end{align*}

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))^n))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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))**n))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \end{align*}

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))^n))^2,x, algorithm="giac")

[Out]

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