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

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

[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 = 0.59, antiderivative size = 283, normalized size of antiderivative = 1.08, number of steps used = 13, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391} \[ -\frac {6 b^2 e^2 n^2 \text {PolyLog}\left (2,\frac {e \sqrt {x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {PolyLog}\left (2,\frac {e \sqrt {x}}{d}+1\right )}{d^2}+\frac {6 b^3 e^2 n^3 \text {PolyLog}\left (3,\frac {e \sqrt {x}}{d}+1\right )}{d^2}+\frac {6 b^2 e^2 n^2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d^2}-\frac {3 b e^2 n \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2}+\frac {e^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{d^2}-\frac {3 b e n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{d^2 \sqrt {x}}-\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2317

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 2318

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 2344

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

Rule 2347

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 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 2398

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 2411

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 2454

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 6589

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 \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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]  time = 0.75, size = 536, normalized size = 2.04 \[ \frac {3 b^2 n^2 \left (-2 e^2 x \text {Li}_2\left (\frac {\sqrt {x} e}{d}+1\right )-2 e^2 x \left (\log \left (d+e \sqrt {x}\right )-1\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+\left (d+e \sqrt {x}\right ) \log \left (d+e \sqrt {x}\right ) \left (\left (e \sqrt {x}-d\right ) \log \left (d+e \sqrt {x}\right )-2 e \sqrt {x}\right )\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )-b n \log \left (d+e \sqrt {x}\right )\right )-3 b d^2 n \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )-b n \log \left (d+e \sqrt {x}\right )\right )^2-d^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )-b n \log \left (d+e \sqrt {x}\right )\right )^3+3 b e^2 n x \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )-b n \log \left (d+e \sqrt {x}\right )\right )^2-\frac {3}{2} b e^2 n x \log (x) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )-b n \log \left (d+e \sqrt {x}\right )\right )^2-3 b d e n \sqrt {x} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )-b n \log \left (d+e \sqrt {x}\right )\right )^2+b^3 n^3 \left (6 e^2 x \text {Li}_3\left (\frac {\sqrt {x} e}{d}+1\right )-6 e^2 x \text {Li}_2\left (\frac {\sqrt {x} e}{d}+1\right ) \left (\log \left (d+e \sqrt {x}\right )-1\right )-3 e^2 x \left (\log \left (d+e \sqrt {x}\right )-2\right ) \log \left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+\left (d+e \sqrt {x}\right ) \log ^2\left (d+e \sqrt {x}\right ) \left (\left (e \sqrt {x}-d\right ) \log \left (d+e \sqrt {x}\right )-3 e \sqrt {x}\right )\right )}{d^2 x} \]

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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fricas [F]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{3} + 3 \, a b^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a^{3}}{x^{2}}, x\right ) \]

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

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

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((e*sqrt(x) + d)^n*c) + a)^3/x^2, x)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a \right )^{3}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, b^{3} d^{2} n^{3} \sqrt {x} \log \left (e \sqrt {x} + d\right )^{3} - 3 \, {\left (2 \, b^{3} e^{2} n x^{\frac {3}{2}} \log \left (e \sqrt {x} + d\right ) - 2 \, b^{3} d e n x - {\left (b^{3} e^{2} n x \log \relax (x) + 2 \, b^{3} d^{2} \log \relax (c) + 2 \, a b^{2} d^{2}\right )} \sqrt {x}\right )} n^{2} \log \left (e \sqrt {x} + d\right )^{2}}{2 \, d^{2} x^{\frac {3}{2}}} - \int \frac {3 \, {\left (2 \, b^{3} e^{3} n^{2} x^{\frac {5}{2}} \log \left (e \sqrt {x} + d\right ) - 2 \, b^{3} d e^{2} n^{2} x^{2} - 2 \, {\left (b^{3} d^{2} e \log \relax (c)^{2} + 2 \, a b^{2} d^{2} e \log \relax (c) + a^{2} b d^{2} e\right )} x^{\frac {3}{2}} - 2 \, {\left (b^{3} d^{3} \log \relax (c)^{2} + 2 \, a b^{2} d^{3} \log \relax (c) + a^{2} b d^{3}\right )} x - {\left (b^{3} e^{3} n^{2} x^{2} \log \relax (x) + 2 \, {\left (b^{3} d^{2} e n \log \relax (c) + a b^{2} d^{2} e n\right )} x\right )} \sqrt {x}\right )} n \log \left (e \sqrt {x} + d\right ) - 2 \, {\left (b^{3} d^{2} e \log \relax (c)^{3} + 3 \, a b^{2} d^{2} e \log \relax (c)^{2} + 3 \, a^{2} b d^{2} e \log \relax (c) + a^{3} d^{2} e\right )} x^{\frac {3}{2}} - 2 \, {\left (b^{3} d^{3} \log \relax (c)^{3} + 3 \, a b^{2} d^{3} \log \relax (c)^{2} + 3 \, a^{2} b d^{3} \log \relax (c) + a^{3} d^{3}\right )} x}{2 \, {\left (d^{2} e x^{\frac {7}{2}} + d^{3} x^{3}\right )}}\,{d x} \]

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

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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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