3.87 \(\int \frac {1}{(f+g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2} \, dx\)

Optimal. Leaf size=35 \[ \text {Int}\left (\frac {1}{(f+g x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2},x\right ) \]

[Out]

Unintegrable(1/(g*x+f)^3/(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

________________________________________________________________________________________

Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2),x]

[Out]

Defer[Int][1/((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx &=\int \frac {1}{(f+g x)^3 \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 = 35.46, size = 0, normalized size = 0.00 \[ \int \frac {1}{(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2),x]

[Out]

Integrate[1/((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2), x]

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{A^{2} g^{3} x^{3} + 3 \, A^{2} f g^{2} x^{2} + 3 \, A^{2} f^{2} g x + A^{2} f^{3} + {\left (B^{2} g^{3} x^{3} + 3 \, B^{2} f g^{2} x^{2} + 3 \, B^{2} f^{2} g x + B^{2} f^{3}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B g^{3} x^{3} + 3 \, A B f g^{2} x^{2} + 3 \, A B f^{2} g x + A B f^{3}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

integral(1/(A^2*g^3*x^3 + 3*A^2*f*g^2*x^2 + 3*A^2*f^2*g*x + A^2*f^3 + (B^2*g^3*x^3 + 3*B^2*f*g^2*x^2 + 3*B^2*f
^2*g*x + B^2*f^3)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*g^3*x^3 + 3*A*B*f*g^2*x^2 + 3*A*B*f^2*g*x + A*B*f^
3)*log(e*((b*x + a)/(d*x + c))^n)), x)

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (g x + f\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

integrate(1/((g*x + f)^3*(B*log(e*((b*x + a)/(d*x + c))^n) + A)^2), x)

________________________________________________________________________________________

maple [A]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (g x +f \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(g*x+f)^3/(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2,x)

[Out]

int(1/(g*x+f)^3/(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2,x)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b d x^{2} + a c + {\left (b c + a d\right )} x}{{\left ({\left (b c g^{3} n - a d g^{3} n\right )} A B + {\left (b c g^{3} n \log \relax (e) - a d g^{3} n \log \relax (e)\right )} B^{2}\right )} x^{3} + {\left (b c f^{3} n - a d f^{3} n\right )} A B + {\left (b c f^{3} n \log \relax (e) - a d f^{3} n \log \relax (e)\right )} B^{2} + 3 \, {\left ({\left (b c f g^{2} n - a d f g^{2} n\right )} A B + {\left (b c f g^{2} n \log \relax (e) - a d f g^{2} n \log \relax (e)\right )} B^{2}\right )} x^{2} + 3 \, {\left ({\left (b c f^{2} g n - a d f^{2} g n\right )} A B + {\left (b c f^{2} g n \log \relax (e) - a d f^{2} g n \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b c g^{3} n - a d g^{3} n\right )} B^{2} x^{3} + 3 \, {\left (b c f g^{2} n - a d f g^{2} n\right )} B^{2} x^{2} + 3 \, {\left (b c f^{2} g n - a d f^{2} g n\right )} B^{2} x + {\left (b c f^{3} n - a d f^{3} n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b c g^{3} n - a d g^{3} n\right )} B^{2} x^{3} + 3 \, {\left (b c f g^{2} n - a d f g^{2} n\right )} B^{2} x^{2} + 3 \, {\left (b c f^{2} g n - a d f^{2} g n\right )} B^{2} x + {\left (b c f^{3} n - a d f^{3} n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )} - \int \frac {b d g x^{2} - b c f - {\left (d f - 3 \, c g\right )} a + 2 \, {\left (a d g - {\left (d f - c g\right )} b\right )} x}{{\left ({\left (b c g^{4} n - a d g^{4} n\right )} A B + {\left (b c g^{4} n \log \relax (e) - a d g^{4} n \log \relax (e)\right )} B^{2}\right )} x^{4} + 4 \, {\left ({\left (b c f g^{3} n - a d f g^{3} n\right )} A B + {\left (b c f g^{3} n \log \relax (e) - a d f g^{3} n \log \relax (e)\right )} B^{2}\right )} x^{3} + {\left (b c f^{4} n - a d f^{4} n\right )} A B + {\left (b c f^{4} n \log \relax (e) - a d f^{4} n \log \relax (e)\right )} B^{2} + 6 \, {\left ({\left (b c f^{2} g^{2} n - a d f^{2} g^{2} n\right )} A B + {\left (b c f^{2} g^{2} n \log \relax (e) - a d f^{2} g^{2} n \log \relax (e)\right )} B^{2}\right )} x^{2} + 4 \, {\left ({\left (b c f^{3} g n - a d f^{3} g n\right )} A B + {\left (b c f^{3} g n \log \relax (e) - a d f^{3} g n \log \relax (e)\right )} B^{2}\right )} x + {\left ({\left (b c g^{4} n - a d g^{4} n\right )} B^{2} x^{4} + 4 \, {\left (b c f g^{3} n - a d f g^{3} n\right )} B^{2} x^{3} + 6 \, {\left (b c f^{2} g^{2} n - a d f^{2} g^{2} n\right )} B^{2} x^{2} + 4 \, {\left (b c f^{3} g n - a d f^{3} g n\right )} B^{2} x + {\left (b c f^{4} n - a d f^{4} n\right )} B^{2}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b c g^{4} n - a d g^{4} n\right )} B^{2} x^{4} + 4 \, {\left (b c f g^{3} n - a d f g^{3} n\right )} B^{2} x^{3} + 6 \, {\left (b c f^{2} g^{2} n - a d f^{2} g^{2} n\right )} B^{2} x^{2} + 4 \, {\left (b c f^{3} g n - a d f^{3} g n\right )} B^{2} x + {\left (b c f^{4} n - a d f^{4} n\right )} B^{2}\right )} \log \left ({\left (d x + c\right )}^{n}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(g*x+f)^3/(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

-(b*d*x^2 + a*c + (b*c + a*d)*x)/(((b*c*g^3*n - a*d*g^3*n)*A*B + (b*c*g^3*n*log(e) - a*d*g^3*n*log(e))*B^2)*x^
3 + (b*c*f^3*n - a*d*f^3*n)*A*B + (b*c*f^3*n*log(e) - a*d*f^3*n*log(e))*B^2 + 3*((b*c*f*g^2*n - a*d*f*g^2*n)*A
*B + (b*c*f*g^2*n*log(e) - a*d*f*g^2*n*log(e))*B^2)*x^2 + 3*((b*c*f^2*g*n - a*d*f^2*g*n)*A*B + (b*c*f^2*g*n*lo
g(e) - a*d*f^2*g*n*log(e))*B^2)*x + ((b*c*g^3*n - a*d*g^3*n)*B^2*x^3 + 3*(b*c*f*g^2*n - a*d*f*g^2*n)*B^2*x^2 +
3*(b*c*f^2*g*n - a*d*f^2*g*n)*B^2*x + (b*c*f^3*n - a*d*f^3*n)*B^2)*log((b*x + a)^n) - ((b*c*g^3*n - a*d*g^3*n
)*B^2*x^3 + 3*(b*c*f*g^2*n - a*d*f*g^2*n)*B^2*x^2 + 3*(b*c*f^2*g*n - a*d*f^2*g*n)*B^2*x + (b*c*f^3*n - a*d*f^3
*n)*B^2)*log((d*x + c)^n)) - integrate((b*d*g*x^2 - b*c*f - (d*f - 3*c*g)*a + 2*(a*d*g - (d*f - c*g)*b)*x)/(((
b*c*g^4*n - a*d*g^4*n)*A*B + (b*c*g^4*n*log(e) - a*d*g^4*n*log(e))*B^2)*x^4 + 4*((b*c*f*g^3*n - a*d*f*g^3*n)*A
*B + (b*c*f*g^3*n*log(e) - a*d*f*g^3*n*log(e))*B^2)*x^3 + (b*c*f^4*n - a*d*f^4*n)*A*B + (b*c*f^4*n*log(e) - a*
d*f^4*n*log(e))*B^2 + 6*((b*c*f^2*g^2*n - a*d*f^2*g^2*n)*A*B + (b*c*f^2*g^2*n*log(e) - a*d*f^2*g^2*n*log(e))*B
^2)*x^2 + 4*((b*c*f^3*g*n - a*d*f^3*g*n)*A*B + (b*c*f^3*g*n*log(e) - a*d*f^3*g*n*log(e))*B^2)*x + ((b*c*g^4*n
- a*d*g^4*n)*B^2*x^4 + 4*(b*c*f*g^3*n - a*d*f*g^3*n)*B^2*x^3 + 6*(b*c*f^2*g^2*n - a*d*f^2*g^2*n)*B^2*x^2 + 4*(
b*c*f^3*g*n - a*d*f^3*g*n)*B^2*x + (b*c*f^4*n - a*d*f^4*n)*B^2)*log((b*x + a)^n) - ((b*c*g^4*n - a*d*g^4*n)*B^
2*x^4 + 4*(b*c*f*g^3*n - a*d*f*g^3*n)*B^2*x^3 + 6*(b*c*f^2*g^2*n - a*d*f^2*g^2*n)*B^2*x^2 + 4*(b*c*f^3*g*n - a
*d*f^3*g*n)*B^2*x + (b*c*f^4*n - a*d*f^4*n)*B^2)*log((d*x + c)^n)), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\left (f+g\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((f + g*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2),x)

[Out]

int(1/((f + g*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2), x)

________________________________________________________________________________________

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/(g*x+f)**3/(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

________________________________________________________________________________________