\(\int \frac {x^3 \log (c (a+\frac {b}{x})^p)}{d+e x} \, dx\) [240]

Optimal result

Integrand size = 23, antiderivative size = 297 \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=-\frac {b d p x}{2 a e^2}-\frac {b^2 p x}{3 a^2 e}+\frac {b p x^2}{6 a e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {b d^2 p \log (b+a x)}{a e^3}+\frac {b^2 d p \log (b+a x)}{2 a^2 e^2}+\frac {b^3 p \log (b+a x)}{3 a^3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^4}-\frac {d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e^4}-\frac {d^3 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^4} \]

[Out]

-1/2*b*d*p*x/a/e^2-1/3*b^2*p*x/a^2/e+1/6*b*p*x^2/a/e+d^2*x*ln(c*(a+b/x)^p)/e^3-1/2*d*x^2*ln(c*(a+b/x)^p)/e^2+1
/3*x^3*ln(c*(a+b/x)^p)/e+b*d^2*p*ln(a*x+b)/a/e^3+1/2*b^2*d*p*ln(a*x+b)/a^2/e^2+1/3*b^3*p*ln(a*x+b)/a^3/e-d^3*l
n(c*(a+b/x)^p)*ln(e*x+d)/e^4-d^3*p*ln(-e*x/d)*ln(e*x+d)/e^4+d^3*p*ln(-e*(a*x+b)/(a*d-b*e))*ln(e*x+d)/e^4+d^3*p
*polylog(2,a*(e*x+d)/(a*d-b*e))/e^4-d^3*p*polylog(2,1+e*x/d)/e^4

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2516, 2498, 269, 31, 2505, 199, 45, 2512, 266, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\frac {b^3 p \log (a x+b)}{3 a^3 e}+\frac {b^2 d p \log (a x+b)}{2 a^2 e^2}-\frac {b^2 p x}{3 a^2 e}-\frac {d^3 \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{e^4}+\frac {b d^2 p \log (a x+b)}{a e^3}-\frac {b d p x}{2 a e^2}+\frac {b p x^2}{6 a e}-\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^4}-\frac {d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4} \]

[In]

Int[(x^3*Log[c*(a + b/x)^p])/(d + e*x),x]

[Out]

-1/2*(b*d*p*x)/(a*e^2) - (b^2*p*x)/(3*a^2*e) + (b*p*x^2)/(6*a*e) + (d^2*x*Log[c*(a + b/x)^p])/e^3 - (d*x^2*Log
[c*(a + b/x)^p])/(2*e^2) + (x^3*Log[c*(a + b/x)^p])/(3*e) + (b*d^2*p*Log[b + a*x])/(a*e^3) + (b^2*d*p*Log[b +
a*x])/(2*a^2*e^2) + (b^3*p*Log[b + a*x])/(3*a^3*e) - (d^3*Log[c*(a + b/x)^p]*Log[d + e*x])/e^4 - (d^3*p*Log[-(
(e*x)/d)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/e^4 + (d^3*p*PolyLog[2, (
a*(d + e*x))/(a*d - b*e)])/e^4 - (d^3*p*PolyLog[2, 1 + (e*x)/d])/e^4

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

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 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]

Rule 2498

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

Rule 2505

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] - Dist[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]

Rule 2512

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

Rule 2516

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3 (d+e x)}\right ) \, dx \\ & = \frac {d^2 \int \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx}{e^3}-\frac {d^3 \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx}{e^3}-\frac {d \int x \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx}{e^2}+\frac {\int x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \, dx}{e} \\ & = \frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^4}-\frac {\left (b d^3 p\right ) \int \frac {\log (d+e x)}{\left (a+\frac {b}{x}\right ) x^2} \, dx}{e^4}+\frac {\left (b d^2 p\right ) \int \frac {1}{\left (a+\frac {b}{x}\right ) x} \, dx}{e^3}-\frac {(b d p) \int \frac {1}{a+\frac {b}{x}} \, dx}{2 e^2}+\frac {(b p) \int \frac {x}{a+\frac {b}{x}} \, dx}{3 e} \\ & = \frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^4}-\frac {\left (b d^3 p\right ) \int \left (\frac {\log (d+e x)}{b x}-\frac {a \log (d+e x)}{b (b+a x)}\right ) \, dx}{e^4}+\frac {\left (b d^2 p\right ) \int \frac {1}{b+a x} \, dx}{e^3}-\frac {(b d p) \int \frac {x}{b+a x} \, dx}{2 e^2}+\frac {(b p) \int \frac {x^2}{b+a x} \, dx}{3 e} \\ & = \frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {b d^2 p \log (b+a x)}{a e^3}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^4}-\frac {\left (d^3 p\right ) \int \frac {\log (d+e x)}{x} \, dx}{e^4}+\frac {\left (a d^3 p\right ) \int \frac {\log (d+e x)}{b+a x} \, dx}{e^4}-\frac {(b d p) \int \left (\frac {1}{a}-\frac {b}{a (b+a x)}\right ) \, dx}{2 e^2}+\frac {(b p) \int \left (-\frac {b}{a^2}+\frac {x}{a}+\frac {b^2}{a^2 (b+a x)}\right ) \, dx}{3 e} \\ & = -\frac {b d p x}{2 a e^2}-\frac {b^2 p x}{3 a^2 e}+\frac {b p x^2}{6 a e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {b d^2 p \log (b+a x)}{a e^3}+\frac {b^2 d p \log (b+a x)}{2 a^2 e^2}+\frac {b^3 p \log (b+a x)}{3 a^3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^4}-\frac {d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^4}+\frac {\left (d^3 p\right ) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{e^3}-\frac {\left (d^3 p\right ) \int \frac {\log \left (\frac {e (b+a x)}{-a d+b e}\right )}{d+e x} \, dx}{e^3} \\ & = -\frac {b d p x}{2 a e^2}-\frac {b^2 p x}{3 a^2 e}+\frac {b p x^2}{6 a e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {b d^2 p \log (b+a x)}{a e^3}+\frac {b^2 d p \log (b+a x)}{2 a^2 e^2}+\frac {b^3 p \log (b+a x)}{3 a^3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^4}-\frac {d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^4}-\frac {d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4}-\frac {\left (d^3 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {a x}{-a d+b e}\right )}{x} \, dx,x,d+e x\right )}{e^4} \\ & = -\frac {b d p x}{2 a e^2}-\frac {b^2 p x}{3 a^2 e}+\frac {b p x^2}{6 a e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {b d^2 p \log (b+a x)}{a e^3}+\frac {b^2 d p \log (b+a x)}{2 a^2 e^2}+\frac {b^3 p \log (b+a x)}{3 a^3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^4}-\frac {d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{e^4}-\frac {d^3 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.04 \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\frac {b d^2 p \log \left (a+\frac {b}{x}\right )}{a e^3}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {b d^2 p \log (x)}{a e^3}-\frac {b p \left (\frac {2 b x}{a^2}-\frac {x^2}{a}-\frac {2 b^2 \log \left (a+\frac {b}{x}\right )}{a^3}-\frac {2 b^2 \log (x)}{a^3}\right )}{6 e}-\frac {b d p \left (\frac {x}{a}-\frac {b \log (b+a x)}{a^2}\right )}{2 e^2}-\frac {d^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right ) \log (d+e x)}{e^4}-\frac {d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e (b+a x)}{a d-b e}\right ) \log (d+e x)}{e^4}-\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {d+e x}{d}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e^4} \]

[In]

Integrate[(x^3*Log[c*(a + b/x)^p])/(d + e*x),x]

[Out]

(b*d^2*p*Log[a + b/x])/(a*e^3) + (d^2*x*Log[c*(a + b/x)^p])/e^3 - (d*x^2*Log[c*(a + b/x)^p])/(2*e^2) + (x^3*Lo
g[c*(a + b/x)^p])/(3*e) + (b*d^2*p*Log[x])/(a*e^3) - (b*p*((2*b*x)/a^2 - x^2/a - (2*b^2*Log[a + b/x])/a^3 - (2
*b^2*Log[x])/a^3))/(6*e) - (b*d*p*(x/a - (b*Log[b + a*x])/a^2))/(2*e^2) - (d^3*Log[c*(a + b/x)^p]*Log[d + e*x]
)/e^4 - (d^3*p*Log[-((e*x)/d)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(b + a*x))/(a*d - b*e))]*Log[d + e*x])/e^4
- (d^3*p*PolyLog[2, (d + e*x)/d])/e^4 + (d^3*p*PolyLog[2, (a*(d + e*x))/(a*d - b*e)])/e^4

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.01

method	result	size
parts	\(\frac {x^{3} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{3 e}-\frac {d \,x^{2} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{2 e^{2}}+\frac {d^{2} x \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{e^{3}}-\frac {d^{3} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right ) \ln \left (e x +d \right )}{e^{4}}+p b e \left (-\frac {\frac {5 a d \left (e x +d \right )-a \left (e x +d \right )^{2}+2 \left (e x +d \right ) b e}{a^{2}}+\frac {\left (-6 a^{2} d^{2}-3 a d e b -2 e^{2} b^{2}\right ) \ln \left (a d -a \left (e x +d \right )-b e \right )}{a^{3}}}{6 e^{4}}+\frac {d^{3} \operatorname {dilog}\left (\frac {-a d +a \left (e x +d \right )+b e}{-a d +b e}\right )}{e^{5} b}+\frac {d^{3} \ln \left (e x +d \right ) \ln \left (\frac {-a d +a \left (e x +d \right )+b e}{-a d +b e}\right )}{e^{5} b}-\frac {d^{3} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{5} b}-\frac {d^{3} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{5} b}\right )\)	\(301\)

[In]

int(x^3*ln(c*(a+b/x)^p)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/3*x^3*ln(c*(a+b/x)^p)/e-1/2*d*x^2*ln(c*(a+b/x)^p)/e^2+d^2*x*ln(c*(a+b/x)^p)/e^3-d^3*ln(c*(a+b/x)^p)*ln(e*x+d
)/e^4+p*b*e*(-1/6/e^4*(1/a^2*(5*a*d*(e*x+d)-a*(e*x+d)^2+2*(e*x+d)*b*e)+(-6*a^2*d^2-3*a*b*d*e-2*b^2*e^2)/a^3*ln
(a*d-a*(e*x+d)-b*e))+1/e^5*d^3/b*dilog((-a*d+a*(e*x+d)+b*e)/(-a*d+b*e))+1/e^5*d^3/b*ln(e*x+d)*ln((-a*d+a*(e*x+
d)+b*e)/(-a*d+b*e))-1/e^5*d^3/b*ln(e*x+d)*ln(-e*x/d)-1/e^5*d^3/b*dilog(-e*x/d))

Fricas [F]

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

[In]

integrate(x^3*log(c*(a+b/x)^p)/(e*x+d),x, algorithm="fricas")

[Out]

integral(x^3*log(c*((a*x + b)/x)^p)/(e*x + d), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\text {Timed out} \]

[In]

integrate(x**3*ln(c*(a+b/x)**p)/(e*x+d),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate(x^3*log(c*(a+b/x)^p)/(e*x+d),x, algorithm="maxima")

[Out]

integrate(x^3*log((a + b/x)^p*c)/(e*x + d), x)

Giac [F]

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

[In]

integrate(x^3*log(c*(a+b/x)^p)/(e*x+d),x, algorithm="giac")

[Out]

integrate(x^3*log((a + b/x)^p*c)/(e*x + d), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{d+e x} \, dx=\int \frac {x^3\,\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{d+e\,x} \,d x \]

[In]

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

[Out]

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