\(\int (a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^3 \, dx\) [437]

Optimal result

Integrand size = 20, antiderivative size = 260 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=\frac {3 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+\frac {3 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^2}-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2}-\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,1+\frac {e}{d \sqrt {x}}\right )}{d^2}-\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2501, 2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438} \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3 \, dx=-\frac {6 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}-\frac {6 b^2 e^2 n^2 \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+\frac {3 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+\frac {3 b e n \sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,\frac {e}{d \sqrt {x}}+1\right )}{d^2}-\frac {6 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2} \]

[In]

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

[Out]

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

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2355

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

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

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 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2501

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p,
q}, x] && FractionQ[n]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 6724

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

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^n\right )\right )^3 \, dx,x,\sqrt {x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^3} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^2 (d+e x)} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {(3 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d}+\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d} \\ & = \frac {3 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+\frac {3 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {\left (6 b^2 e n^2\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}-\frac {\left (6 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2} \\ & = \frac {3 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+\frac {3 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^2}-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2}+\frac {\left (6 b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2}+\frac {\left (6 b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d}{x}\right )}{x} \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{d^2} \\ & = \frac {3 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+\frac {3 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt {x}}\right )}{d^2}-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2}-\frac {6 b^3 e^2 n^3 \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )}{d^2}-\frac {6 b^3 e^2 n^3 \text {Li}_3\left (\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2} \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

\[\int {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{3}d x\]

[In]

int((a+b*ln(c*(d+e/x^(1/2))^n))^3,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/2))^n))^3,x)

Fricas [F]

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

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3,x, algorithm="fricas")

[Out]

integral(b^3*log(c*((d*x + e*sqrt(x))/x)^n)^3 + 3*a*b^2*log(c*((d*x + e*sqrt(x))/x)^n)^2 + 3*a^2*b*log(c*((d*x
+ e*sqrt(x))/x)^n) + a^3, x)

Sympy [F]

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

[In]

integrate((a+b*ln(c*(d+e/x**(1/2))**n))**3,x)

[Out]

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

Maxima [F]

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

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3,x, algorithm="maxima")

[Out]

b^3*x*log((d*sqrt(x) + e)^n)^3 - 3*(e*n*(e*log(d*sqrt(x) + e)/d^2 - sqrt(x)/d) - x*log(c*(d + e/sqrt(x))^n))*a
^2*b + a^3*x - integrate(1/2*(2*(b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n))^3 + 3*(b^3*d*n*x - 2*(b^3*d*log(c) +
a*b^2*d)*x + 2*(b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c) + a*b^2*e)*sqrt(x))*log((d*sqrt(x) +
e)^n)^2 - 6*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2*n))^2 - 2*(b^3*d*log(c
)^3 + 3*a*b^2*d*log(c)^2)*x - 6*((b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n))^2 + (b^3*d*log(c)^2 + 2*a*b^2*d*log(
c))*x - 2*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2*n)) + (b^3*e*log(c)^2 + 2
*a*b^2*e*log(c))*sqrt(x))*log((d*sqrt(x) + e)^n) + 6*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c))*x + (b^3*e*log(c)^2
+ 2*a*b^2*e*log(c))*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2)*sqrt(x))/(d*x + e*sqrt(x
)), x)

Giac [F]

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

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

int((a + b*log(c*(d + e/x^(1/2))^n))^3,x)

[Out]

int((a + b*log(c*(d + e/x^(1/2))^n))^3, x)