\(\int x^m \log (d (b x+c x^2)^n) \, dx\) [59]

Optimal result

Integrand size = 18, antiderivative size = 66 \[ \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=-\frac {2 n x^{1+m}}{(1+m)^2}+\frac {n x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {c x}{b}\right )}{(1+m)^2}+\frac {x^{1+m} \log \left (d \left (b x+c x^2\right )^n\right )}{1+m} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.73 \[ \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\frac {x^{1+m} \left (-2 n+n \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {c x}{b}\right )+(1+m) \log \left (d (x (b+c x))^n\right )\right )}{(1+m)^2} \] Input:

Integrate[x^m*Log[d*(b*x + c*x^2)^n],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3005, 90, 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.

\(\displaystyle \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx\)

\(\Big \downarrow \) 3005

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 74

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

Input:

Int[x^m*Log[d*(b*x + c*x^2)^n],x]
 

Output:

-((n*((2*x^(1 + m))/(1 + m) - (x^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m 
, -((c*x)/b)])/(1 + m)))/(1 + m)) + (x^(1 + m)*Log[d*(b*x + c*x^2)^n])/(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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

rule 3005

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_. 
), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))) 
, x] - Simp[b*n*(p/(e*(m + 1)))   Int[SimplifyIntegrand[(d + e*x)^(m + 1)*( 
a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, 
d, e, m, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || 
 IntegerQ[m]) && NeQ[m, -1]
 

Maple [F]

\[\int x^{m} \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )d x\]

Input:

int(x^m*ln(d*(c*x^2+b*x)^n),x)
 

Output:

int(x^m*ln(d*(c*x^2+b*x)^n),x)
 

Fricas [F]

\[ \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\int { x^{m} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) \,d x } \] Input:

integrate(x^m*log(d*(c*x^2+b*x)^n),x, algorithm="fricas")
 

Output:

integral(x^m*log((c*x^2 + b*x)^n*d), x)
 

Sympy [F]

\[ \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\int x^{m} \log {\left (d \left (b x + c x^{2}\right )^{n} \right )}\, dx \] Input:

integrate(x**m*ln(d*(c*x**2+b*x)**n),x)
 

Output:

Integral(x**m*log(d*(b*x + c*x**2)**n), x)
 

Maxima [F]

\[ \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\int { x^{m} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) \,d x } \] Input:

integrate(x^m*log(d*(c*x^2+b*x)^n),x, algorithm="maxima")
 

Output:

(x*x^m*log((c*x + b)^n) + x*x^m*log(x^n))/(m + 1) + integrate((((m + 1)*lo 
g(d) - 2*n)*c*x + ((m + 1)*log(d) - n)*b)*x^m/(c*(m + 1)*x + b*(m + 1)), x 
)
 

Giac [F]

\[ \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\int { x^{m} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ) \,d x } \] Input:

integrate(x^m*log(d*(c*x^2+b*x)^n),x, algorithm="giac")
 

Output:

integrate(x^m*log((c*x^2 + b*x)^n*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx=\int x^m\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right ) \,d x \] Input:

int(x^m*log(d*(b*x + c*x^2)^n),x)
 

Output:

int(x^m*log(d*(b*x + c*x^2)^n), x)
 

Reduce [F]

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

int(x^m*log(d*(c*x^2+b*x)^n),x)
 

Output:

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