3.1.99 \(\int \frac {1}{(f+g x)^2 (a+b \log (c (d+e x)^n))^2} \, dx\) [99]

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, 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 = {} \begin {gather*} \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 2.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(f+g x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \end {gather*}

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 = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (g x +f \right )^{2} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{2} \left (f + g x\right )^{2}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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((x*e + d)^n*c) + a)^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (f+g\,x\right )}^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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