3.68 \(\int (f x)^{-1-n} \log (c (d+e x^n)^p) \, dx\)

Optimal. Leaf size=80 \[ -\frac{(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac{e p x^n \log (x) (f x)^{-n}}{d f}-\frac{e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n} \]

[Out]

(e*p*x^n*Log[x])/(d*f*(f*x)^n) - (e*p*x^n*Log[d + e*x^n])/(d*f*n*(f*x)^n) - Log[c*(d + e*x^n)^p]/(f*n*(f*x)^n)

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Rubi [A]  time = 0.0356827, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2455, 19, 266, 36, 29, 31} \[ -\frac{(f x)^{-n} \log \left (c \left (d+e x^n\right )^p\right )}{f n}+\frac{e p x^n \log (x) (f x)^{-n}}{d f}-\frac{e p x^n (f x)^{-n} \log \left (d+e x^n\right )}{d f n} \]

Antiderivative was successfully verified.

[In]

Int[(f*x)^(-1 - n)*Log[c*(d + e*x^n)^p],x]

[Out]

(e*p*x^n*Log[x])/(d*f*(f*x)^n) - (e*p*x^n*Log[d + e*x^n])/(d*f*n*(f*x)^n) - Log[c*(d + e*x^n)^p]/(f*n*(f*x)^n)

Rule 2455

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[((f*x)^(m
+ 1)*(a + b*Log[c*(d + e*x^n)^p]))/(f*(m + 1)), x] - Dist[(b*e*n*p)/(f*(m + 1)), Int[(x^(n - 1)*(f*x)^(m + 1))
/(d + e*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 19

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + n)*(b*v)^n)/(a*v)^n, Int[u*v^(m + n),
x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[m + n]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0175413, size = 57, normalized size = 0.71 \[ -\frac{(f x)^{-n} \left (d \log \left (c \left (d+e x^n\right )^p\right )+e p x^n \log \left (d+e x^n\right )-e n p x^n \log (x)\right )}{d f n} \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^(-1 - n)*Log[c*(d + e*x^n)^p],x]

[Out]

-((-(e*n*p*x^n*Log[x]) + e*p*x^n*Log[d + e*x^n] + d*Log[c*(d + e*x^n)^p])/(d*f*n*(f*x)^n))

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Maple [F]  time = 1.819, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1-n}\ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^(-1-n)*ln(c*(d+e*x^n)^p),x)

[Out]

int((f*x)^(-1-n)*ln(c*(d+e*x^n)^p),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1-n)*log(c*(d+e*x^n)^p),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.0769, size = 159, normalized size = 1.99 \begin{align*} \frac{e f^{-n - 1} n p x^{n} \log \left (x\right ) - d f^{-n - 1} \log \left (c\right ) -{\left (e f^{-n - 1} p x^{n} + d f^{-n - 1} p\right )} \log \left (e x^{n} + d\right )}{d n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1-n)*log(c*(d+e*x^n)^p),x, algorithm="fricas")

[Out]

(e*f^(-n - 1)*n*p*x^n*log(x) - d*f^(-n - 1)*log(c) - (e*f^(-n - 1)*p*x^n + d*f^(-n - 1)*p)*log(e*x^n + d))/(d*
n*x^n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**(-1-n)*ln(c*(d+e*x**n)**p),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{-n - 1} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1-n)*log(c*(d+e*x^n)^p),x, algorithm="giac")

[Out]

integrate((f*x)^(-n - 1)*log((e*x^n + d)^p*c), x)