\(\int (a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^2 \, dx\) [431]

Optimal result

Integrand size = 20, antiderivative size = 152 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\frac {2 b e n \left (d+\frac {e}{\sqrt {x}}\right ) \sqrt {x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+\frac {2 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b^2 e^2 n^2 \log (x)}{d^2}-\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {2501, 2504, 2445, 2458, 2389, 2379, 2438, 2351, 31} \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=\frac {2 b e^2 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt {x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+\frac {2 b e n \sqrt {x} \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2-\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt {x}}}\right )}{d^2}+\frac {b^2 e^2 n^2 \log (x)}{d^2} \]

[In]

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

[Out]

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

Rule 31

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

Rule 2351

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

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.13 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \, dx=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+\frac {b e n \left (2 a d \sqrt {x}+2 b e n \log \left (d+\frac {e}{\sqrt {x}}\right )+2 b d \sqrt {x} \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-2 e \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \log \left (e+d \sqrt {x}\right )+b e n \log (x)+b e n \left (\log \left (e+d \sqrt {x}\right ) \left (\log \left (e+d \sqrt {x}\right )-2 \log \left (-\frac {d \sqrt {x}}{e}\right )\right )-2 \operatorname {PolyLog}\left (2,1+\frac {d \sqrt {x}}{e}\right )\right )\right )}{d^2} \]

[In]

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

[Out]

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

Maple [F]

\[\int {\left (a +b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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