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

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

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.91, size = 0, normalized size = 0.00 \[ \int \frac {(f+g x)^m}{\left (a+b \log \left (c (d+e x)^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (g x + f\right )}^{m}}{b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 8.96, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right )^{m}}{\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**m/(a+b*ln(c*(e*x+d)**n))**2,x)

[Out]

Exception raised: HeuristicGCDFailed

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