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

Optimal. Leaf size=37 \[ \text{Unintegrable}\left (\frac{(a g+b g x)^2}{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.237531, 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 = {} \[ \int \frac{(a g+b g x)^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{(a g+b g x)^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \left (\frac{a^2 g^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}+\frac{2 a b g^2 x}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}+\frac{b^2 g^2 x^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}\right ) \, dx\\ &=\left (a^2 g^2\right ) \int \frac{1}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+\left (2 a b g^2\right ) \int \frac{x}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx+\left (b^2 g^2\right ) \int \frac{x^2}{\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 = 0.983394, size = 0, normalized size = 0. \[ \int \frac{(a g+b g x)^2}{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.395, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bgx+ag \right ) ^{2} \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((b*g*x+a*g)^2/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{3} d g^{2} x^{4} + a^{3} c g^{2} +{\left (b^{3} c g^{2} + 3 \, a b^{2} d g^{2}\right )} x^{3} + 3 \,{\left (a b^{2} c g^{2} + a^{2} b d g^{2}\right )} x^{2} +{\left (3 \, a^{2} b c g^{2} + a^{3} d g^{2}\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{4 \, b^{3} d g^{2} x^{3} + 3 \, a^{2} b c g^{2} + a^{3} d g^{2} + 3 \,{\left (b^{3} c g^{2} + 3 \, a b^{2} d g^{2}\right )} x^{2} + 6 \,{\left (a b^{2} c g^{2} + a^{2} b d g^{2}\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((b*g*x+a*g)^2/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

-(b^3*d*g^2*x^4 + a^3*c*g^2 + (b^3*c*g^2 + 3*a*b^2*d*g^2)*x^3 + 3*(a*b^2*c*g^2 + a^2*b*d*g^2)*x^2 + (3*a^2*b*c
*g^2 + a^3*d*g^2)*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) + integrate((4*b^3*d*g^2*x^3 + 3*a^2*b*c*g^2 + a^3*d*g^2 + 3*(b
^3*c*g^2 + 3*a*b^2*d*g^2)*x^2 + 6*(a*b^2*c*g^2 + a^2*b*d*g^2)*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)

________________________________________________________________________________________

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

[Out]

integral((b^2*g^2*x^2 + 2*a*b*g^2*x + a^2*g^2)/(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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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