\(\int (d+e x)^m \log (c (a+\frac {b}{x})^p) \, dx\) [209]

Optimal result

Integrand size = 20, antiderivative size = 135 \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\frac {a p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {a (d+e x)}{a d-b e}\right )}{e (a d-b e) (1+m) (2+m)}-\frac {p (d+e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {e x}{d}\right )}{d e \left (2+3 m+m^2\right )}+\frac {(d+e x)^{1+m} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.91 \[ \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx=\frac {(d+e x)^{1+m} \left (-a d p (d+e x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,\frac {a (d+e x)}{a d-b e}\right )+(a d-b e) \left (p (d+e x) \operatorname {Hypergeometric2F1}\left (1,2+m,3+m,1+\frac {e x}{d}\right )-d (2+m) \log \left (c \left (a+\frac {b}{x}\right )^p\right )\right )\right )}{d e (-a d+b e) (1+m) (2+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2913, 1016, 97, 75, 78}

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.

\(\displaystyle \int (d+e x)^m \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx\)

\(\Big \downarrow \) 2913

\(\displaystyle \frac {b p \int \frac {(d+e x)^{m+1}}{\left (a+\frac {b}{x}\right ) x^2}dx}{e (m+1)}+\frac {(d+e x)^{m+1} \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e (m+1)}\)

\(\Big \downarrow \) 1016

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

\(\Big \downarrow \) 97

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

\(\Big \downarrow \) 75

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

\(\Big \downarrow \) 78

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

Input:

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

Output:

(b*p*((a*(d + e*x)^(2 + m)*Hypergeometric2F1[1, 2 + m, 3 + m, (a*(d + e*x) 
)/(a*d - b*e)])/(b*(a*d - b*e)*(2 + m)) - ((d + e*x)^(2 + m)*Hypergeometri 
c2F1[1, 2 + m, 3 + m, 1 + (e*x)/d])/(b*d*(2 + m))))/(e*(1 + m)) + ((d + e* 
x)^(1 + m)*Log[c*(a + b/x)^p])/(e*(1 + m))
 

Defintions of rubi rules used

rule 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 1016

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^( 
p_.), x_Symbol] :> Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ 
[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] ||  !I 
ntegerQ[p])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((d + e*x)**m*log(c*(a + b/x)**p), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(log(c*(a + b/x)^p)*(d + e*x)^m,x)
 

Output:

int(log(c*(a + b/x)^p)*(d + e*x)^m, x)
 

Reduce [F]

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

int((e*x+d)^m*log(c*(a+b/x)^p),x)
 

Output:

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