\(\int (f x)^m \log (c (d+e x)^p) \, dx\) [57]

Optimal result

Integrand size = 16, antiderivative size = 69 \[ \int (f x)^m \log \left (c (d+e x)^p\right ) \, dx=-\frac {e p (f x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,-\frac {e x}{d}\right )}{d f^2 (1+m) (2+m)}+\frac {(f x)^{1+m} \log \left (c (d+e x)^p\right )}{f (1+m)} \] Output:

-e*p*(f*x)^(2+m)*hypergeom([1, 2+m],[3+m],-e*x/d)/d/f^2/(1+m)/(2+m)+(f*x)^ 
(1+m)*ln(c*(e*x+d)^p)/f/(1+m)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.81 \[ \int (f x)^m \log \left (c (d+e x)^p\right ) \, dx=\frac {x (f x)^m \left (-e p x \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,-\frac {e x}{d}\right )+d (2+m) \log \left (c (d+e x)^p\right )\right )}{d (1+m) (2+m)} \] Input:

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

Output:

(x*(f*x)^m*(-(e*p*x*Hypergeometric2F1[1, 2 + m, 3 + m, -((e*x)/d)]) + d*(2 
 + m)*Log[c*(d + e*x)^p]))/(d*(1 + m)*(2 + m))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2842, 74}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (f x)^m \log \left (c (d+e x)^p\right ) \, dx\)

\(\Big \downarrow \) 2842

\(\displaystyle \frac {(f x)^{m+1} \log \left (c (d+e x)^p\right )}{f (m+1)}-\frac {e p \int \frac {(f x)^{m+1}}{d+e x}dx}{f (m+1)}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {(f x)^{m+1} \log \left (c (d+e x)^p\right )}{f (m+1)}-\frac {e p (f x)^{m+2} \operatorname {Hypergeometric2F1}\left (1,m+2,m+3,-\frac {e x}{d}\right )}{d f^2 (m+1) (m+2)}\)

Input:

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

Output:

-((e*p*(f*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, -((e*x)/d)])/(d*f^ 
2*(1 + m)*(2 + m))) + ((f*x)^(1 + m)*Log[c*(d + e*x)^p])/(f*(1 + m))
 

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-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 2842

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((f*x)**m*log(c*(d + e*x)**p), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (f x)^m \log \left (c (d+e x)^p\right ) \, dx=\frac {f^{m} \left (x^{m} \mathrm {log}\left (\left (e x +d \right )^{p} c \right ) e \,m^{2} x +x^{m} \mathrm {log}\left (\left (e x +d \right )^{p} c \right ) e m x +x^{m} d m p +x^{m} d p -x^{m} e m p x -\left (\int \frac {x^{m}}{e m \,x^{2}+d m x +e \,x^{2}+d x}d x \right ) d^{2} m^{3} p -2 \left (\int \frac {x^{m}}{e m \,x^{2}+d m x +e \,x^{2}+d x}d x \right ) d^{2} m^{2} p -\left (\int \frac {x^{m}}{e m \,x^{2}+d m x +e \,x^{2}+d x}d x \right ) d^{2} m p \right )}{e m \left (m^{2}+2 m +1\right )} \] Input:

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

Output:

(f**m*(x**m*log((d + e*x)**p*c)*e*m**2*x + x**m*log((d + e*x)**p*c)*e*m*x 
+ x**m*d*m*p + x**m*d*p - x**m*e*m*p*x - int(x**m/(d*m*x + d*x + e*m*x**2 
+ e*x**2),x)*d**2*m**3*p - 2*int(x**m/(d*m*x + d*x + e*m*x**2 + e*x**2),x) 
*d**2*m**2*p - int(x**m/(d*m*x + d*x + e*m*x**2 + e*x**2),x)*d**2*m*p))/(e 
*m*(m**2 + 2*m + 1))