\(\int \frac {x^2 \log (c (a+b x)^p)}{d+e x} \, dx\) [220]

Optimal result

Integrand size = 21, antiderivative size = 159 \[ \int \frac {x^2 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {d p x}{e^2}+\frac {a p x}{2 b e}-\frac {p x^2}{4 e}-\frac {a^2 p \log (a+b x)}{2 b^2 e}+\frac {x^2 \log \left (c (a+b x)^p\right )}{2 e}-\frac {d (a+b x) \log \left (c (a+b x)^p\right )}{b e^2}+\frac {d^2 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^3}+\frac {d^2 p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^3} \] Output:

d*p*x/e^2+1/2*a*p*x/b/e-1/4*p*x^2/e-1/2*a^2*p*ln(b*x+a)/b^2/e+1/2*x^2*ln(c 
*(b*x+a)^p)/e-d*(b*x+a)*ln(c*(b*x+a)^p)/b/e^2+d^2*ln(c*(b*x+a)^p)*ln(b*(e* 
x+d)/(-a*e+b*d))/e^3+d^2*p*polylog(2,-e*(b*x+a)/(-a*e+b*d))/e^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.80 \[ \int \frac {x^2 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {b e p x (4 b d+2 a e-b e x)-2 a^2 e^2 p \log (a+b x)+b \log \left (c (a+b x)^p\right ) \left (-4 a d e+2 b e x (-2 d+e x)+4 b d^2 \log \left (\frac {b (d+e x)}{b d-a e}\right )\right )+4 b^2 d^2 p \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )}{4 b^2 e^3} \] Input:

Integrate[(x^2*Log[c*(a + b*x)^p])/(d + e*x),x]
 

Output:

(b*e*p*x*(4*b*d + 2*a*e - b*e*x) - 2*a^2*e^2*p*Log[a + b*x] + b*Log[c*(a + 
 b*x)^p]*(-4*a*d*e + 2*b*e*x*(-2*d + e*x) + 4*b*d^2*Log[(b*(d + e*x))/(b*d 
 - a*e)]) + 4*b^2*d^2*p*PolyLog[2, (e*(a + b*x))/(-(b*d) + a*e)])/(4*b^2*e 
^3)
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2863, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {a^2 p \log (a+b x)}{2 b^2 e}+\frac {d^2 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^3}-\frac {d (a+b x) \log \left (c (a+b x)^p\right )}{b e^2}+\frac {x^2 \log \left (c (a+b x)^p\right )}{2 e}+\frac {d^2 p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^3}+\frac {a p x}{2 b e}+\frac {d p x}{e^2}-\frac {p x^2}{4 e}\)

Input:

Int[(x^2*Log[c*(a + b*x)^p])/(d + e*x),x]
 

Output:

(d*p*x)/e^2 + (a*p*x)/(2*b*e) - (p*x^2)/(4*e) - (a^2*p*Log[a + b*x])/(2*b^ 
2*e) + (x^2*Log[c*(a + b*x)^p])/(2*e) - (d*(a + b*x)*Log[c*(a + b*x)^p])/( 
b*e^2) + (d^2*Log[c*(a + b*x)^p]*Log[(b*(d + e*x))/(b*d - a*e)])/e^3 + (d^ 
2*p*PolyLog[2, -((e*(a + b*x))/(b*d - a*e))])/e^3
 

Defintions of rubi rules used

rule 2009

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

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

Maple [A] (verified)

Time = 2.67 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.34

method	result	size
parts	\(\frac {x^{2} \ln \left (c \left (b x +a \right )^{p}\right )}{2 e}-\frac {d x \ln \left (c \left (b x +a \right )^{p}\right )}{e^{2}}+\frac {\ln \left (c \left (b x +a \right )^{p}\right ) d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {p b \left (\frac {d^{2} \left (\frac {\operatorname {dilog}\left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{b}+\frac {\ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{b}\right )}{e^{2}}+\frac {-\frac {\left (e x +d \right ) a e +3 d \left (e x +d \right ) b -\frac {\left (e x +d \right )^{2} b}{2}}{b^{2}}+\frac {e a \left (e a +2 b d \right ) \ln \left (\left (e x +d \right ) b +e a -b d \right )}{b^{3}}}{2 e^{2}}\right )}{e}\)	\(213\)
risch	\(\frac {\ln \left (\left (b x +a \right )^{p}\right ) x^{2}}{2 e}-\frac {\ln \left (\left (b x +a \right )^{p}\right ) d x}{e^{2}}+\frac {\ln \left (\left (b x +a \right )^{p}\right ) d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {p \,d^{2} \operatorname {dilog}\left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{e^{3}}-\frac {p \,d^{2} \ln \left (e x +d \right ) \ln \left (\frac {\left (e x +d \right ) b +e a -b d}{e a -b d}\right )}{e^{3}}-\frac {p \,x^{2}}{4 e}+\frac {d p x}{e^{2}}+\frac {5 p \,d^{2}}{4 e^{3}}+\frac {a p x}{2 b e}+\frac {p d a}{2 b \,e^{2}}-\frac {p \,a^{2} \ln \left (\left (e x +d \right ) b +e a -b d \right )}{2 b^{2} e}-\frac {p a \ln \left (\left (e x +d \right ) b +e a -b d \right ) d}{b \,e^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+\frac {i \pi \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\frac {1}{2} e \,x^{2}-d x}{e^{2}}+\frac {d^{2} \ln \left (e x +d \right )}{e^{3}}\right )\)	\(369\)

Input:

int(x^2*ln(c*(b*x+a)^p)/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/2*x^2*ln(c*(b*x+a)^p)/e-d*x*ln(c*(b*x+a)^p)/e^2+ln(c*(b*x+a)^p)*d^2/e^3* 
ln(e*x+d)-p*b/e*(d^2/e^2*(dilog(((e*x+d)*b+e*a-b*d)/(a*e-b*d))/b+ln(e*x+d) 
*ln(((e*x+d)*b+e*a-b*d)/(a*e-b*d))/b)+1/2/e^2*(-1/b^2*((e*x+d)*a*e+3*d*(e* 
x+d)*b-1/2*(e*x+d)^2*b)+e*a*(a*e+2*b*d)/b^3*ln((e*x+d)*b+e*a-b*d)))
 

Fricas [F]

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

integrate(x^2*log(c*(b*x+a)^p)/(e*x+d),x, algorithm="fricas")
 

Output:

integral(x^2*log((b*x + a)^p*c)/(e*x + d), x)
 

Sympy [F]

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

integrate(x**2*ln(c*(b*x+a)**p)/(e*x+d),x)
 

Output:

Integral(x**2*log(c*(a + b*x)**p)/(d + e*x), x)
 

Maxima [F]

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

integrate(x^2*log(c*(b*x+a)^p)/(e*x+d),x, algorithm="maxima")
 

Output:

integrate(x^2*log((b*x + a)^p*c)/(e*x + d), x)
 

Giac [F]

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

integrate(x^2*log(c*(b*x+a)^p)/(e*x+d),x, algorithm="giac")
 

Output:

integrate(x^2*log((b*x + a)^p*c)/(e*x + d), x)
 

Mupad [F(-1)]

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

int((x^2*log(c*(a + b*x)^p))/(d + e*x),x)
 

Output:

int((x^2*log(c*(a + b*x)^p))/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {x^2 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (\left (b x +a \right )^{p} c \right )}{b e \,x^{2}+a e x +b d x +a d}d x \right ) a \,b^{2} d^{2} e p -4 \left (\int \frac {\mathrm {log}\left (\left (b x +a \right )^{p} c \right )}{b e \,x^{2}+a e x +b d x +a d}d x \right ) b^{3} d^{3} p +2 \mathrm {log}\left (\left (b x +a \right )^{p} c \right )^{2} b^{2} d^{2}-2 \,\mathrm {log}\left (\left (b x +a \right )^{p} c \right ) a^{2} e^{2} p -4 \,\mathrm {log}\left (\left (b x +a \right )^{p} c \right ) a b d e p -4 \,\mathrm {log}\left (\left (b x +a \right )^{p} c \right ) b^{2} d e p x +2 \,\mathrm {log}\left (\left (b x +a \right )^{p} c \right ) b^{2} e^{2} p \,x^{2}+2 a b \,e^{2} p^{2} x +4 b^{2} d e \,p^{2} x -b^{2} e^{2} p^{2} x^{2}}{4 b^{2} e^{3} p} \] Input:

int(x^2*log(c*(b*x+a)^p)/(e*x+d),x)
 

Output:

(4*int(log((a + b*x)**p*c)/(a*d + a*e*x + b*d*x + b*e*x**2),x)*a*b**2*d**2 
*e*p - 4*int(log((a + b*x)**p*c)/(a*d + a*e*x + b*d*x + b*e*x**2),x)*b**3* 
d**3*p + 2*log((a + b*x)**p*c)**2*b**2*d**2 - 2*log((a + b*x)**p*c)*a**2*e 
**2*p - 4*log((a + b*x)**p*c)*a*b*d*e*p - 4*log((a + b*x)**p*c)*b**2*d*e*p 
*x + 2*log((a + b*x)**p*c)*b**2*e**2*p*x**2 + 2*a*b*e**2*p**2*x + 4*b**2*d 
*e*p**2*x - b**2*e**2*p**2*x**2)/(4*b**2*e**3*p)