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

Optimal. Leaf size=297 \[ \frac {b^3 p \log (a x+b)}{3 a^3 e}+\frac {b^2 d p \log (a x+b)}{2 a^2 e^2}-\frac {b^2 p x}{3 a^2 e}-\frac {d^3 \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {d^3 p \text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{e^4}+\frac {b d^2 p \log (a x+b)}{a e^3}-\frac {b d p x}{2 a e^2}+\frac {b p x^2}{6 a e}-\frac {d^3 p \text {Li}_2\left (\frac {e x}{d}+1\right )}{e^4}-\frac {d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4} \]

[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

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Rubi [A]  time = 0.32, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2466, 2448, 263, 31, 2455, 193, 43, 2462, 260, 2416, 2394, 2315, 2393, 2391} \[ \frac {d^3 p \text {PolyLog}\left (2,\frac {a (d+e x)}{a d-b e}\right )}{e^4}-\frac {d^3 p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^4}+\frac {b^2 d p \log (a x+b)}{2 a^2 e^2}-\frac {b^2 p x}{3 a^2 e}+\frac {b^3 p \log (a x+b)}{3 a^3 e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^3}-\frac {d^3 \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{e^4}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e}+\frac {b d^2 p \log (a x+b)}{a e^3}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e (a x+b)}{a d-b e}\right )}{e^4}-\frac {b d p x}{2 a e^2}+\frac {b p x^2}{6 a e}-\frac {d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*d*p*x)/(2*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 43

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 193

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 260

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 263

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 2315

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

Rule 2391

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 2393

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 2394

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 2416

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 2448

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 2455

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 2462

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 2466

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

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Mathematica [A]  time = 0.23, size = 251, normalized size = 0.85 \[ \frac {\frac {b e^3 p \left (2 b^2 \log \left (a+\frac {b}{x}\right )+a x (a x-2 b)+2 b^2 \log (x)\right )}{a^3}+\frac {3 b d e^2 p (b \log (a x+b)-a x)}{a^2}-6 d^3 \log (d+e x) \log \left (c \left (a+\frac {b}{x}\right )^p\right )+6 d^2 e x \log \left (c \left (a+\frac {b}{x}\right )^p\right )-3 d e^2 x^2 \log \left (c \left (a+\frac {b}{x}\right )^p\right )+2 e^3 x^3 \log \left (c \left (a+\frac {b}{x}\right )^p\right )-6 d^3 p \left (-\text {Li}_2\left (\frac {a (d+e x)}{a d-b e}\right )+\log (d+e x) \left (\log \left (-\frac {e x}{d}\right )-\log \left (\frac {e (a x+b)}{b e-a d}\right )\right )+\text {Li}_2\left (\frac {e x}{d}+1\right )\right )+\frac {6 b d^2 e p \left (\log \left (a+\frac {b}{x}\right )+\log (x)\right )}{a}}{6 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \log \left (c \left (\frac {a x + b}{x}\right )^{p}\right )}{e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{e x +d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{e x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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