\(\int (f+g x) \log (c (d+e x^n)^p) \, dx\) [214]

Optimal result

Integrand size = 18, antiderivative size = 132 \[ \int (f+g x) \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e f n p x^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+\frac {1}{n},2+\frac {1}{n},-\frac {e x^n}{d}\right )}{d (1+n)}-\frac {e g n p x^{2+n} \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{n},2 \left (1+\frac {1}{n}\right ),-\frac {e x^n}{d}\right )}{2 d (2+n)}-\frac {f^2 p \log \left (d+e x^n\right )}{2 g}+\frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98 \[ \int (f+g x) \log \left (c \left (d+e x^n\right )^p\right ) \, dx=-\frac {e f n p x^{1+n} \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{n},1+\frac {1+n}{n},-\frac {e x^n}{d}\right )}{d (1+n)}-\frac {e g n p x^{2+n} \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{n},1+\frac {2+n}{n},-\frac {e x^n}{d}\right )}{2 d (2+n)}+f x \log \left (c \left (d+e x^n\right )^p\right )+\frac {1}{2} g x^2 \log \left (c \left (d+e x^n\right )^p\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2913, 2383, 2009}

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.

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

\(\Big \downarrow \) 2913

\(\displaystyle \frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g}-\frac {e n p \int \frac {x^{n-1} (f+g x)^2}{e x^n+d}dx}{2 g}\)

\(\Big \downarrow \) 2383

\(\displaystyle \frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g}-\frac {e n p \int \left (\frac {f^2 x^{n-1}}{e x^n+d}+\frac {2 f g x^n}{e x^n+d}+\frac {g^2 x^{n+1}}{e x^n+d}\right )dx}{2 g}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2383

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I 
nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n 
, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]
 

rule 2913

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.74 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.62 \[ \int (f+g x) \log \left (c \left (d+e x^n\right )^p\right ) \, dx=- \frac {d^{-2 - \frac {2}{n}} d^{1 + \frac {2}{n}} e g p x^{n + 2} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{2 \Gamma \left (2 + \frac {2}{n}\right )} - \frac {d^{-2 - \frac {2}{n}} d^{1 + \frac {2}{n}} e g p x^{n + 2} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{n \Gamma \left (2 + \frac {2}{n}\right )} + f x \log {\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac {g x^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{2} + \frac {d^{- \frac {1}{n}} d^{1 + \frac {1}{n}} e e^{\frac {1}{n}} e^{-1 - \frac {1}{n}} f p x \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{d n \Gamma \left (1 + \frac {1}{n}\right )} \] Input:

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

Output:

-d**(-2 - 2/n)*d**(1 + 2/n)*e*g*p*x**(n + 2)*lerchphi(e*x**n*exp_polar(I*p 
i)/d, 1, 1 + 2/n)*gamma(1 + 2/n)/(2*gamma(2 + 2/n)) - d**(-2 - 2/n)*d**(1 
+ 2/n)*e*g*p*x**(n + 2)*lerchphi(e*x**n*exp_polar(I*pi)/d, 1, 1 + 2/n)*gam 
ma(1 + 2/n)/(n*gamma(2 + 2/n)) + f*x*log(c*(d + e*x**n)**p) + g*x**2*log(c 
*(d + e*x**n)**p)/2 + d**(1 + 1/n)*e*e**(1/n)*e**(-1 - 1/n)*f*p*x*lerchphi 
(d*exp_polar(I*pi)/(e*x**n), 1, exp_polar(I*pi)/n)*gamma(1/n)/(d*d**(1/n)* 
n*gamma(1 + 1/n))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (f+g x) \log \left (c \left (d+e x^n\right )^p\right ) \, dx=\frac {\left (\int \frac {x}{x^{n} e +d}d x \right ) d g n p}{2}+\left (\int \frac {1}{x^{n} e +d}d x \right ) d f n p +\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) f x +\frac {\mathrm {log}\left (\left (x^{n} e +d \right )^{p} c \right ) g \,x^{2}}{2}-f n p x -\frac {g n p \,x^{2}}{4} \] Input:

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

Output:

(2*int(x/(x**n*e + d),x)*d*g*n*p + 4*int(1/(x**n*e + d),x)*d*f*n*p + 4*log 
((x**n*e + d)**p*c)*f*x + 2*log((x**n*e + d)**p*c)*g*x**2 - 4*f*n*p*x - g* 
n*p*x**2)/4