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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0334217, 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 = {} \[ \int \frac{1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.54796, size = 0, normalized size = 0. \[ \int \frac{1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 3.427, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{2} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} g^{2} x^{2} + 2 \, a^{2} f g x + a^{2} f^{2} +{\left (b^{2} g^{2} x^{2} + 2 \, b^{2} f g x + b^{2} f^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \,{\left (a b g^{2} x^{2} + 2 \, a b f g x + a b f^{2}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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