\(\int (f x)^q (a+b \log (c (d+e x^m)^n)) \, dx\) [399]

Optimal result

Integrand size = 22, antiderivative size = 92 \[ \int (f x)^q \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right ) \, dx=-\frac {b e m n x^{1+m} (f x)^q \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+q}{m},\frac {1+2 m+q}{m},-\frac {e x^m}{d}\right )}{d (1+q) (1+m+q)}+\frac {(f x)^{1+q} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (1+q)} \] Output:

-b*e*m*n*x^(1+m)*(f*x)^q*hypergeom([1, (1+m+q)/m],[(1+2*m+q)/m],-e*x^m/d)/ 
d/(1+q)/(1+m+q)+(f*x)^(1+q)*(a+b*ln(c*(d+e*x^m)^n))/f/(1+q)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.89 \[ \int (f x)^q \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right ) \, dx=\frac {x (f x)^q \left (-b e m n x^m \operatorname {Hypergeometric2F1}\left (1,\frac {1+m+q}{m},\frac {1+2 m+q}{m},-\frac {e x^m}{d}\right )+d (1+m+q) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )\right )}{d (1+q) (1+m+q)} \] Input:

Integrate[(f*x)^q*(a + b*Log[c*(d + e*x^m)^n]),x]
 

Output:

(x*(f*x)^q*(-(b*e*m*n*x^m*Hypergeometric2F1[1, (1 + m + q)/m, (1 + 2*m + q 
)/m, -((e*x^m)/d)]) + d*(1 + m + q)*(a + b*Log[c*(d + e*x^m)^n])))/(d*(1 + 
 q)*(1 + m + q))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2905, 30, 888}

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.

\(\displaystyle \int (f x)^q \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2905

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

\(\Big \downarrow \) 30

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

\(\Big \downarrow \) 888

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

Input:

Int[(f*x)^q*(a + b*Log[c*(d + e*x^m)^n]),x]
 

Output:

-((b*e*m*n*x^(1 + m)*(f*x)^q*Hypergeometric2F1[1, (1 + m + q)/m, (1 + 2*m 
+ q)/m, -((e*x^m)/d)])/(d*(1 + q)*(1 + m + q))) + ((f*x)^(1 + q)*(a + b*Lo 
g[c*(d + e*x^m)^n]))/(f*(1 + q))
 

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 2905

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] - Simp[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]
 

Maple [F]

\[\int \left (f x \right )^{q} \left (a +b \ln \left (c \left (d +e \,x^{m}\right )^{n}\right )\right )d x\]

Input:

int((f*x)^q*(a+b*ln(c*(d+e*x^m)^n)),x)
 

Output:

int((f*x)^q*(a+b*ln(c*(d+e*x^m)^n)),x)
 

Fricas [F]

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

integrate((f*x)^q*(a+b*log(c*(d+e*x^m)^n)),x, algorithm="fricas")
 

Output:

integral((f*x)^q*b*log((e*x^m + d)^n*c) + (f*x)^q*a, x)
 

Sympy [F]

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

integrate((f*x)**q*(a+b*ln(c*(d+e*x**m)**n)),x)
 

Output:

Integral((f*x)**q*(a + b*log(c*(d + e*x**m)**n)), x)
 

Maxima [F]

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

integrate((f*x)^q*(a+b*log(c*(d+e*x^m)^n)),x, algorithm="maxima")
 

Output:

(d^2*f^q*m^2*n*integrate(x^q/((m*(q + 1) - q^2 - 2*q - 1)*e^2*x^(2*m) + 2* 
(m*(q + 1) - q^2 - 2*q - 1)*d*e*x^m + (m*(q + 1) - q^2 - 2*q - 1)*d^2), x) 
 - (((m*(q + 1) - q^2 - 2*q - 1)*e*f^q*x*x^m + (m*(q + 1) - q^2 - 2*q - 1) 
*d*f^q*x)*x^q*log((e*x^m + d)^n) + (((m*(q + 1) - q^2 - 2*q - 1)*e*f^q*log 
(c) - (m^2*n - m*n*(q + 1))*e*f^q)*x*x^m - (d*f^q*m^2*n - (m*(q + 1) - q^2 
 - 2*q - 1)*d*f^q*log(c))*x)*x^q)/((q^3 - (q^2 + 2*q + 1)*m + 3*q^2 + 3*q 
+ 1)*e*x^m + (q^3 - (q^2 + 2*q + 1)*m + 3*q^2 + 3*q + 1)*d))*b + (f*x)^(q 
+ 1)*a/(f*(q + 1))
 

Giac [F]

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

integrate((f*x)^q*(a+b*log(c*(d+e*x^m)^n)),x, algorithm="giac")
 

Output:

integrate((b*log((e*x^m + d)^n*c) + a)*(f*x)^q, x)
 

Mupad [F(-1)]

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

int((f*x)^q*(a + b*log(c*(d + e*x^m)^n)),x)
 

Output:

int((f*x)^q*(a + b*log(c*(d + e*x^m)^n)), x)
 

Reduce [F]

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

int((f*x)^q*(a+b*log(c*(d+e*x^m)^n)),x)
 

Output:

(f**q*(x**q*log((x**m*e + d)**n*c)*b*q*x + x**q*log((x**m*e + d)**n*c)*b*x 
 + x**q*a*q*x + x**q*a*x - x**q*b*m*n*x + int(x**q/(x**m*e*q + x**m*e + d* 
q + d),x)*b*d*m*n*q**2 + 2*int(x**q/(x**m*e*q + x**m*e + d*q + d),x)*b*d*m 
*n*q + int(x**q/(x**m*e*q + x**m*e + d*q + d),x)*b*d*m*n))/(q**2 + 2*q + 1 
)