3.627 \(\int \frac {a+b \log (c (d+e x^m)^n)}{x \log ^3(f x^p)} \, dx\)

Optimal. Leaf size=69 \[ \frac {b e m n \text {Int}\left (\frac {x^{m-1}}{\left (d+e x^m\right ) \log ^2\left (f x^p\right )},x\right )}{2 p}-\frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{2 p \log ^2\left (f x^p\right )} \]

[Out]

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

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Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3),x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx &=-\frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{2 p \log ^2\left (f x^p\right )}+\frac {(b e m n) \int \frac {x^{-1+m}}{\left (d+e x^m\right ) \log ^2\left (f x^p\right )} \, dx}{2 p}\\ \end {align*}

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Mathematica [A]  time = 11.06, size = 0, normalized size = 0.00 \[ \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3),x]

[Out]

Integrate[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3), x]

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fricas [A]  time = 1.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a}{x \log \left (f x^{p}\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^m)^n))/x/log(f*x^p)^3,x, algorithm="fricas")

[Out]

integral((b*log((e*x^m + d)^n*c) + a)/(x*log(f*x^p)^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^m)^n))/x/log(f*x^p)^3,x, algorithm="giac")

[Out]

integrate((b*log((e*x^m + d)^n*c) + a)/(x*log(f*x^p)^3), x)

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maple [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \,x^{m}+d \right )^{n}\right )+a}{x \ln \left (f \,x^{p}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*ln(c*(e*x^m+d)^n)+a)/x/ln(f*x^p)^3,x)

[Out]

int((b*ln(c*(e*x^m+d)^n)+a)/x/ln(f*x^p)^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e*x^m)^n))/x/log(f*x^p)^3,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )}{x\,{\ln \left (f\,x^p\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*(d + e*x^m)^n))/(x*log(f*x^p)^3),x)

[Out]

int((a + b*log(c*(d + e*x^m)^n))/(x*log(f*x^p)^3), x)

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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((a+b*ln(c*(d+e*x**m)**n))/x/ln(f*x**p)**3,x)

[Out]

Timed out

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