3.7.27 \(\int \frac {a+b \log (c (d+e x^m)^n)}{x \log ^3(f x^p)} \, dx\) [627]

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

[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 = 0.07, 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 {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \end {gather*}

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]

-1/2*(a + b*Log[c*(d + e*x^m)^n])/(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 = 7.95, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \end {gather*}

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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Maple [A]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{m}\right )^{n}\right )}{x \ln \left (f \,x^{p}\right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

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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