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

Optimal. Leaf size=255 \[ \frac{x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right )^2 \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 e^2 n}-\frac{d x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{e^2 n}-\frac{p x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 e^2 n}+\frac{2 d p x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^2 n}+\frac{p^2 x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right )^2}{4 e^2 n}-\frac{2 d p^2 x^{1-n} (f x)^{2 n-1}}{e n} \]

[Out]

(-2*d*p^2*x^(1 - n)*(f*x)^(-1 + 2*n))/(e*n) + (p^2*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)^2)/(4*e^2*n) + (2*
d*p*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e^2*n) - (p*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(
d + e*x^n)^2*Log[c*(d + e*x^n)^p])/(2*e^2*n) - (d*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)*Log[c*(d + e*x^n)^p
]^2)/(e^2*n) + (x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)^2*Log[c*(d + e*x^n)^p]^2)/(2*e^2*n)

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Rubi [A]  time = 0.187219, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2456, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right )^2 \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 e^2 n}-\frac{d x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{e^2 n}-\frac{p x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 e^2 n}+\frac{2 d p x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^2 n}+\frac{p^2 x^{1-2 n} (f x)^{2 n-1} \left (d+e x^n\right )^2}{4 e^2 n}-\frac{2 d p^2 x^{1-n} (f x)^{2 n-1}}{e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*p^2*x^(1 - n)*(f*x)^(-1 + 2*n))/(e*n) + (p^2*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)^2)/(4*e^2*n) + (2*
d*p*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)*Log[c*(d + e*x^n)^p])/(e^2*n) - (p*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(
d + e*x^n)^2*Log[c*(d + e*x^n)^p])/(2*e^2*n) - (d*x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)*Log[c*(d + e*x^n)^p
]^2)/(e^2*n) + (x^(1 - 2*n)*(f*x)^(-1 + 2*n)*(d + e*x^n)^2*Log[c*(d + e*x^n)^p]^2)/(2*e^2*n)

Rule 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_)*(x_))^(m_), x_Symbol] :> Dist[(f*x)^
m/x^m, Int[x^m*(a + b*Log[c*(d + e*x^n)^p])^q, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p, q}, x] && IntegerQ[
Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int (f x)^{-1+2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx &=\left (x^{1-2 n} (f x)^{-1+2 n}\right ) \int x^{-1+2 n} \log ^2\left (c \left (d+e x^n\right )^p\right ) \, dx\\ &=\frac{\left (x^{1-2 n} (f x)^{-1+2 n}\right ) \operatorname{Subst}\left (\int x \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{\left (x^{1-2 n} (f x)^{-1+2 n}\right ) \operatorname{Subst}\left (\int \left (-\frac{d \log ^2\left (c (d+e x)^p\right )}{e}+\frac{(d+e x) \log ^2\left (c (d+e x)^p\right )}{e}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{\left (x^{1-2 n} (f x)^{-1+2 n}\right ) \operatorname{Subst}\left (\int (d+e x) \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{e n}-\frac{\left (d x^{1-2 n} (f x)^{-1+2 n}\right ) \operatorname{Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{1-2 n} (f x)^{-1+2 n}\right ) \operatorname{Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^n\right )}{e^2 n}-\frac{\left (d x^{1-2 n} (f x)^{-1+2 n}\right ) \operatorname{Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^n\right )}{e^2 n}\\ &=-\frac{d x^{1-2 n} (f x)^{-1+2 n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{e^2 n}+\frac{x^{1-2 n} (f x)^{-1+2 n} \left (d+e x^n\right )^2 \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 e^2 n}-\frac{\left (p x^{1-2 n} (f x)^{-1+2 n}\right ) \operatorname{Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^n\right )}{e^2 n}+\frac{\left (2 d p x^{1-2 n} (f x)^{-1+2 n}\right ) \operatorname{Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^n\right )}{e^2 n}\\ &=-\frac{2 d p^2 x^{1-n} (f x)^{-1+2 n}}{e n}+\frac{p^2 x^{1-2 n} (f x)^{-1+2 n} \left (d+e x^n\right )^2}{4 e^2 n}+\frac{2 d p x^{1-2 n} (f x)^{-1+2 n} \left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )}{e^2 n}-\frac{p x^{1-2 n} (f x)^{-1+2 n} \left (d+e x^n\right )^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 e^2 n}-\frac{d x^{1-2 n} (f x)^{-1+2 n} \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{e^2 n}+\frac{x^{1-2 n} (f x)^{-1+2 n} \left (d+e x^n\right )^2 \log ^2\left (c \left (d+e x^n\right )^p\right )}{2 e^2 n}\\ \end{align*}

Mathematica [A]  time = 0.110963, size = 140, normalized size = 0.55 \[ \frac{x^{-2 n} (f x)^{2 n} \left (2 d^2 p \log \left (d+e x^n\right ) \left (3 p-2 \log \left (c \left (d+e x^n\right )^p\right )\right )+e x^n \left (2 e x^n \log ^2\left (c \left (d+e x^n\right )^p\right )+2 p \left (2 d-e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )+p^2 \left (e x^n-6 d\right )\right )+2 d^2 p^2 \log ^2\left (d+e x^n\right )\right )}{4 e^2 f n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((f*x)^(2*n)*(2*d^2*p^2*Log[d + e*x^n]^2 + 2*d^2*p*Log[d + e*x^n]*(3*p - 2*Log[c*(d + e*x^n)^p]) + e*x^n*(p^2*
(-6*d + e*x^n) + 2*p*(2*d - e*x^n)*Log[c*(d + e*x^n)^p] + 2*e*x^n*Log[c*(d + e*x^n)^p]^2)))/(4*e^2*f*n*x^(2*n)
)

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Maple [F]  time = 2.066, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{-1+2\,n} \left ( \ln \left ( c \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}\, 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)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.09321, size = 459, normalized size = 1.8 \begin{align*} \frac{{\left (e^{2} p^{2} - 2 \, e^{2} p \log \left (c\right ) + 2 \, e^{2} \log \left (c\right )^{2}\right )} f^{2 \, n - 1} x^{2 \, n} - 2 \,{\left (3 \, d e p^{2} - 2 \, d e p \log \left (c\right )\right )} f^{2 \, n - 1} x^{n} + 2 \,{\left (e^{2} f^{2 \, n - 1} p^{2} x^{2 \, n} - d^{2} f^{2 \, n - 1} p^{2}\right )} \log \left (e x^{n} + d\right )^{2} + 2 \,{\left (2 \, d e f^{2 \, n - 1} p^{2} x^{n} -{\left (e^{2} p^{2} - 2 \, e^{2} p \log \left (c\right )\right )} f^{2 \, n - 1} x^{2 \, n} +{\left (3 \, d^{2} p^{2} - 2 \, d^{2} p \log \left (c\right )\right )} f^{2 \, n - 1}\right )} \log \left (e x^{n} + d\right )}{4 \, e^{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)^2,x, algorithm="fricas")

[Out]

1/4*((e^2*p^2 - 2*e^2*p*log(c) + 2*e^2*log(c)^2)*f^(2*n - 1)*x^(2*n) - 2*(3*d*e*p^2 - 2*d*e*p*log(c))*f^(2*n -
1)*x^n + 2*(e^2*f^(2*n - 1)*p^2*x^(2*n) - d^2*f^(2*n - 1)*p^2)*log(e*x^n + d)^2 + 2*(2*d*e*f^(2*n - 1)*p^2*x^
n - (e^2*p^2 - 2*e^2*p*log(c))*f^(2*n - 1)*x^(2*n) + (3*d^2*p^2 - 2*d^2*p*log(c))*f^(2*n - 1))*log(e*x^n + d))
/(e^2*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+2*n)*ln(c*(d+e*x**n)**p)**2,x)

[Out]

Timed out

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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 )^{2}\,{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)^2,x, algorithm="giac")

[Out]

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