3.5.19 \(\int \frac {(a+b \log (c (d+e \sqrt {x})^n))^3}{x^2} \, dx\) [419]

Optimal. Leaf size=263 \[ -\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {3 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (\frac {d}{d+e \sqrt {x}}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {Li}_3\left (\frac {d}{d+e \sqrt {x}}\right )}{d^2} \]

[Out]

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

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Rubi [A]
time = 0.33, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438} \begin {gather*} \frac {6 b^2 e^2 n^2 \text {PolyLog}\left (2,\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {PolyLog}\left (2,\frac {e \sqrt {x}}{d}+1\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {PolyLog}\left (3,\frac {d}{d+e \sqrt {x}}\right )}{d^2}+\frac {6 b^2 e^2 n^2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {3 b e^2 n \log \left (1-\frac {d}{d+e \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2}-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*e*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(d^2*Sqrt[x]) - (3*b*e^2*n*Log[1 - d/(d + e*Sqrt
[x])]*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/d^2 - (a + b*Log[c*(d + e*Sqrt[x])^n])^3/x + (6*b^2*e^2*n^2*(a + b*L
og[c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[x])/d)])/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*Sqrt[x])/d])/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 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*} \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x^2} \, dx &=2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+(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,\sqrt {x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+(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+e \sqrt {x}\right )\\ &=-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\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+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+e \sqrt {x}\right )}{d}\\ &=-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}-\frac {(3 b e n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e \sqrt {x}\right )}{d^2}+\frac {\left (3 b e^2 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+e \sqrt {x}\right )}{d^2}+\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+e \sqrt {x}\right )}{d^2}\\ &=-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}-\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}+\frac {\left (6 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+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+e \sqrt {x}\right )}{d^2}\\ &=-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}+\frac {e^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}-\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^2}-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {\left (6 b^3 e^2 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right )}{d^2}\\ &=-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}+\frac {e^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{d^2}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x}+\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}-\frac {3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log \left (-\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^2}-\frac {6 b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {Li}_3\left (1+\frac {e \sqrt {x}}{d}\right )}{d^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(536\) vs. \(2(263)=526\).
time = 0.54, size = 536, normalized size = 2.04 \begin {gather*} \frac {-3 b d e n \sqrt {x} \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-3 b d^2 n \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+3 b e^2 n x \log \left (d+e \sqrt {x}\right ) \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-d^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3-\frac {3}{2} b e^2 n x \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \log (x)+3 b^2 n^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (\left (d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right ) \left (-2 e \sqrt {x}+\left (-d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right )\right )-2 e^2 x \left (-1+\log \left (d+e \sqrt {x}\right )\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )-2 e^2 x \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )\right )+b^3 n^3 \left (\left (d+e \sqrt {x}\right ) \log ^2\left (d+e \sqrt {x}\right ) \left (-3 e \sqrt {x}+\left (-d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right )\right )-3 e^2 x \left (-2+\log \left (d+e \sqrt {x}\right )\right ) \log \left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )-6 e^2 x \left (-1+\log \left (d+e \sqrt {x}\right )\right ) \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )+6 e^2 x \text {Li}_3\left (1+\frac {e \sqrt {x}}{d}\right )\right )}{d^2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )^{3}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b^3*d^2*n^3*sqrt(x)*log(sqrt(x)*e + d)^3 - 3*(2*b^3*n*x^(3/2)*e^2*log(sqrt(x)*e + d) - 2*b^3*d*n*x*e -
(b^3*n*x*e^2*log(x) + 2*b^3*d^2*log(c) + 2*a*b^2*d^2)*sqrt(x))*n^2*log(sqrt(x)*e + d)^2)/(d^2*x^(3/2)) - inte
grate(-1/2*(2*(b^3*d^2*log(c)^3 + 3*a*b^2*d^2*log(c)^2 + 3*a^2*b*d^2*log(c) + a^3*d^2)*x^(3/2)*e - 3*(2*b^3*n^
2*x^(5/2)*e^3*log(sqrt(x)*e + d) - 2*b^3*d*n^2*x^2*e^2 - 2*(b^3*d^2*log(c)^2 + 2*a*b^2*d^2*log(c) + a^2*b*d^2)
*x^(3/2)*e - 2*(b^3*d^3*log(c)^2 + 2*a*b^2*d^3*log(c) + a^2*b*d^3)*x - (b^3*n^2*x^2*e^3*log(x) + 2*(b^3*d^2*n*
log(c) + a*b^2*d^2*n)*x*e)*sqrt(x))*n*log(sqrt(x)*e + d) + 2*(b^3*d^3*log(c)^3 + 3*a*b^2*d^3*log(c)^2 + 3*a^2*
b*d^3*log(c) + a^3*d^3)*x)/(d^2*x^(7/2)*e + d^3*x^3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^3}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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