3.2.64 \(\int \frac {(f+g x)^m}{(a+b \log (c (d+e x)^n))^2} \, dx\) [164]

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 = 0.02, 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 {(f+g x)^m}{\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[(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 = 2.98, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(f+g x)^m}{\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[(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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Maple [A]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (g x +f \right )^{m}}{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}

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 >> no luck

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

[Out]

integrate((g*x + f)^m/(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 {{\left (f+g\,x\right )}^m}{{\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((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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