\(\int (f x)^m \log (c (d+\frac {e}{x})^p) \, dx\) [58]

Optimal result

Integrand size = 18, antiderivative size = 67 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\frac {e p (f x)^m \operatorname {Hypergeometric2F1}\left (1,-m,1-m,-\frac {e}{d x}\right )}{d m (1+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)} \]

[Out]

e*p*(f*x)^m*hypergeom([1, -m],[1-m],-e/d/x)/d/m/(1+m)+(f*x)^(1+m)*ln(c*(d+e/x)^p)/f/(1+m)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2505, 16, 346, 66} \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\frac {(f x)^{m+1} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (m+1)}+\frac {e p (f x)^m \operatorname {Hypergeometric2F1}\left (1,-m,1-m,-\frac {e}{d x}\right )}{d m (m+1)} \]

[In]

Int[(f*x)^m*Log[c*(d + e/x)^p],x]

[Out]

(e*p*(f*x)^m*Hypergeometric2F1[1, -m, 1 - m, -(e/(d*x))])/(d*m*(1 + m)) + ((f*x)^(1 + m)*Log[c*(d + e/x)^p])/(
f*(1 + m))

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 346

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(-c^(-1))*(c*x)^(m + 1)*(1/x)^(m + 1),
Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m
]

Rule 2505

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]

Rubi steps \begin{align*} \text {integral}& = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)}+\frac {(e p) \int \frac {(f x)^{1+m}}{\left (d+\frac {e}{x}\right ) x^2} \, dx}{f (1+m)} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)}+\frac {(e f p) \int \frac {(f x)^{-1+m}}{d+\frac {e}{x}} \, dx}{1+m} \\ & = \frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)}-\frac {\left (e p \left (\frac {1}{x}\right )^m (f x)^m\right ) \text {Subst}\left (\int \frac {x^{-1-m}}{d+e x} \, dx,x,\frac {1}{x}\right )}{1+m} \\ & = \frac {e p (f x)^m \, _2F_1\left (1,-m;1-m;-\frac {e}{d x}\right )}{d m (1+m)}+\frac {(f x)^{1+m} \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{f (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\frac {(f x)^m \left (e p \operatorname {Hypergeometric2F1}\left (1,-m,1-m,-\frac {e}{d x}\right )+d m x \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )}{d m (1+m)} \]

[In]

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

[Out]

((f*x)^m*(e*p*Hypergeometric2F1[1, -m, 1 - m, -(e/(d*x))] + d*m*x*Log[c*(d + e/x)^p]))/(d*m*(1 + m))

Maple [F]

\[\int \left (f x \right )^{m} \ln \left (c \left (d +\frac {e}{x}\right )^{p}\right )d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right ) \,d x } \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 9.21 (sec) , antiderivative size = 231, normalized size of antiderivative = 3.45 \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=e p \left (\begin {cases} \frac {0^{m} \log {\left (d x + e \right )}}{d} & \text {for}\: \left (f = 0 \wedge m \neq -1\right ) \vee f = 0 \\\frac {d^{m - 1} f^{m + 1} m x^{m} \Phi \left (\frac {e e^{i \pi }}{d x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{d^{m} f m \Gamma \left (1 - m\right ) + d^{m} f \Gamma \left (1 - m\right )} & \text {for}\: m > -\infty \wedge m < \infty \wedge m \neq -1 \\\frac {\begin {cases} - \frac {1}{d x} & \text {for}\: e = 0 \\\frac {\begin {cases} \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \wedge \left |{x}\right | < 1 \\\log {\left (d \right )} \log {\left (x \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\left (d \right )} \log {\left (\frac {1}{x} \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\left (d \right )} + \operatorname {Li}_{2}\left (\frac {e e^{i \pi }}{d x}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}}{f} - \frac {\left (\begin {cases} \frac {1}{d x} & \text {for}\: e = 0 \\\frac {\log {\left (d + \frac {e}{x} \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (f x \right )}}{f} & \text {otherwise} \end {cases}\right ) + \left (\begin {cases} 0^{m} x & \text {for}\: f = 0 \\\frac {\begin {cases} \frac {\left (f x\right )^{m + 1}}{m + 1} & \text {for}\: m \neq -1 \\\log {\left (f x \right )} & \text {otherwise} \end {cases}}{f} & \text {otherwise} \end {cases}\right ) \log {\left (c \left (d + \frac {e}{x}\right )^{p} \right )} \]

[In]

integrate((f*x)**m*ln(c*(d+e/x)**p),x)

[Out]

e*p*Piecewise((0**m*log(d*x + e)/d, Eq(f, 0) | (Eq(f, 0) & Ne(m, -1))), (d**(m - 1)*f**(m + 1)*m*x**m*lerchphi
(e*exp_polar(I*pi)/(d*x), 1, m*exp_polar(I*pi))*gamma(-m)/(d**m*f*m*gamma(1 - m) + d**m*f*gamma(1 - m)), (m >
-oo) & (m < oo) & Ne(m, -1)), (Piecewise((-1/(d*x), Eq(e, 0)), (Piecewise((polylog(2, e*exp_polar(I*pi)/(d*x))
, (Abs(x) < 1) & (1/Abs(x) < 1)), (log(d)*log(x) + polylog(2, e*exp_polar(I*pi)/(d*x)), Abs(x) < 1), (-log(d)*
log(1/x) + polylog(2, e*exp_polar(I*pi)/(d*x)), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d)
+ meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) + polylog(2, e*exp_polar(I*pi)/(d*x)), True))/e, True))/f - P
iecewise((1/(d*x), Eq(e, 0)), (log(d + e/x)/e, True))*log(f*x)/f, True)) + Piecewise((0**m*x, Eq(f, 0)), (Piec
ewise(((f*x)**(m + 1)/(m + 1), Ne(m, -1)), (log(f*x), True))/f, True))*log(c*(d + e/x)**p)

Maxima [F]

\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\int { \left (f x\right )^{m} \log \left (c {\left (d + \frac {e}{x}\right )}^{p}\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (f x)^m \log \left (c \left (d+\frac {e}{x}\right )^p\right ) \, dx=\int \ln \left (c\,{\left (d+\frac {e}{x}\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \]

[In]

int(log(c*(d + e/x)^p)*(f*x)^m,x)

[Out]

int(log(c*(d + e/x)^p)*(f*x)^m, x)