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

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {45, 2463, 2436, 2332, 2442, 2441, 2440, 2438} \[ \int \frac {x^2 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=-\frac {a^2 p \log (a+b x)}{2 b^2 e}+\frac {d^2 \log \left (c (a+b x)^p\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{e^3}-\frac {d (a+b x) \log \left (c (a+b x)^p\right )}{b e^2}+\frac {x^2 \log \left (c (a+b x)^p\right )}{2 e}+\frac {d^2 p \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{e^3}+\frac {a p x}{2 b e}+\frac {d p x}{e^2}-\frac {p x^2}{4 e} \]

[In]

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

[Out]

(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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

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

Rule 2442

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

Rule 2463

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]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {d \log \left (c (a+b x)^p\right )}{e^2}+\frac {x \log \left (c (a+b x)^p\right )}{e}+\frac {d^2 \log \left (c (a+b x)^p\right )}{e^2 (d+e x)}\right ) \, dx \\ & = -\frac {d \int \log \left (c (a+b x)^p\right ) \, dx}{e^2}+\frac {d^2 \int \frac {\log \left (c (a+b x)^p\right )}{d+e x} \, dx}{e^2}+\frac {\int x \log \left (c (a+b x)^p\right ) \, dx}{e} \\ & = \frac {x^2 \log \left (c (a+b x)^p\right )}{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 \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+b x\right )}{b e^2}-\frac {\left (b d^2 p\right ) \int \frac {\log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{e^3}-\frac {(b p) \int \frac {x^2}{a+b x} \, dx}{2 e} \\ & = \frac {d p x}{e^2}+\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 {\left (d^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{e^3}-\frac {(b p) \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{2 e} \\ & = \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 \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{e^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.80 \[ \int \frac {x^2 \log \left (c (a+b x)^p\right )}{d+e x} \, dx=\frac {b e p x (4 b d+2 a e-b e x)-2 a^2 e^2 p \log (a+b x)+b \log \left (c (a+b x)^p\right ) \left (-4 a d e+2 b e x (-2 d+e x)+4 b d^2 \log \left (\frac {b (d+e x)}{b d-a e}\right )\right )+4 b^2 d^2 p \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )}{4 b^2 e^3} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.34

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

[In]

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

[Out]

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+a*e-b*d)/(a*e-b*d))/b+ln(e*x+d)*ln(((e*x+d)*b+a*e-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)+a*e*(a*e+2*b*d)/b^3*ln((e*x+d)*b+a*e-b*d)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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