3.69 \(\int (f x)^{-1-2 n} \log (c (d+e x^n)^p) \, dx\)

Optimal. Leaf size=120 \[ -\frac{(f x)^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{e^2 p x^{2 n} \log (x) (f x)^{-2 n}}{2 d^2 f}+\frac{e^2 p x^{2 n} (f x)^{-2 n} \log \left (d+e x^n\right )}{2 d^2 f n}-\frac{e p x^n (f x)^{-2 n}}{2 d f n} \]

[Out]

-(e*p*x^n)/(2*d*f*n*(f*x)^(2*n)) - (e^2*p*x^(2*n)*Log[x])/(2*d^2*f*(f*x)^(2*n)) + (e^2*p*x^(2*n)*Log[d + e*x^n
])/(2*d^2*f*n*(f*x)^(2*n)) - Log[c*(d + e*x^n)^p]/(2*f*n*(f*x)^(2*n))

________________________________________________________________________________________

Rubi [A]  time = 0.0578244, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2455, 20, 266, 44} \[ -\frac{(f x)^{-2 n} \log \left (c \left (d+e x^n\right )^p\right )}{2 f n}-\frac{e^2 p x^{2 n} \log (x) (f x)^{-2 n}}{2 d^2 f}+\frac{e^2 p x^{2 n} (f x)^{-2 n} \log \left (d+e x^n\right )}{2 d^2 f n}-\frac{e p x^n (f x)^{-2 n}}{2 d f n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*p*x^n)/(2*d*f*n*(f*x)^(2*n)) - (e^2*p*x^(2*n)*Log[x])/(2*d^2*f*(f*x)^(2*n)) + (e^2*p*x^(2*n)*Log[d + e*x^n
])/(2*d^2*f*n*(f*x)^(2*n)) - Log[c*(d + e*x^n)^p]/(2*f*n*(f*x)^(2*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 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(b^IntPart[n]*(b*v)^FracPart[n])/(a^IntPart[n
]*(a*v)^FracPart[n]), Int[u*(a*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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0481357, size = 76, normalized size = 0.63 \[ -\frac{(f x)^{-2 n} \left (d \left (d \log \left (c \left (d+e x^n\right )^p\right )+e p x^n\right )-e^2 p x^{2 n} \log \left (d+e x^n\right )+e^2 n p x^{2 n} \log (x)\right )}{2 d^2 f n} \]

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^(-1 - 2*n)*Log[c*(d + e*x^n)^p],x]

[Out]

-(e^2*n*p*x^(2*n)*Log[x] - e^2*p*x^(2*n)*Log[d + e*x^n] + d*(e*p*x^n + d*Log[c*(d + e*x^n)^p]))/(2*d^2*f*n*(f*
x)^(2*n))

________________________________________________________________________________________

Maple [F]  time = 1.854, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1-2\,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)^(-1-2*n)*ln(c*(d+e*x^n)^p),x)

[Out]

int((f*x)^(-1-2*n)*ln(c*(d+e*x^n)^p),x)

________________________________________________________________________________________

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)^(-1-2*n)*log(c*(d+e*x^n)^p),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.07245, size = 240, normalized size = 2. \begin{align*} -\frac{e^{2} f^{-2 \, n - 1} n p x^{2 \, n} \log \left (x\right ) + d e f^{-2 \, n - 1} p x^{n} + d^{2} f^{-2 \, n - 1} \log \left (c\right ) -{\left (e^{2} f^{-2 \, n - 1} p x^{2 \, n} - d^{2} f^{-2 \, n - 1} p\right )} \log \left (e x^{n} + d\right )}{2 \, d^{2} n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^(-1-2*n)*log(c*(d+e*x^n)^p),x, algorithm="fricas")

[Out]

-1/2*(e^2*f^(-2*n - 1)*n*p*x^(2*n)*log(x) + d*e*f^(-2*n - 1)*p*x^n + d^2*f^(-2*n - 1)*log(c) - (e^2*f^(-2*n -
1)*p*x^(2*n) - d^2*f^(-2*n - 1)*p)*log(e*x^n + d))/(d^2*n*x^(2*n))

________________________________________________________________________________________

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)**(-1-2*n)*ln(c*(d+e*x**n)**p),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (f x\right )^{-2 \, 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)^(-1-2*n)*log(c*(d+e*x^n)^p),x, algorithm="giac")

[Out]

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