3.5.99 \(\int (a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n))^2 \, dx\) [499]

Optimal. Leaf size=227 \[ \frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}-\frac {2 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {2 b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.30, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2501, 2504, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \begin {gather*} \frac {2 b^2 e^3 n^2 \text {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3}-\frac {2 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {2 b e^2 n \sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 2356

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

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*} \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx &=3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=-\left (3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{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^3 (d+e x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{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 )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )\\ &=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{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 )^3} \, dx,x,d+\frac {e}{\sqrt [3]{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 )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}\\ &=\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {(2 b e 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 [3]{x}}\right )}{d^2}-\frac {\left (2 b e^2 n\right ) \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 [3]{x}}\right )}{d^2}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}\\ &=-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {\left (2 b e^2 n\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 [3]{x}}\right )}{d^3}+\frac {\left (2 b e^3 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}\\ &=\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {\left (2 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}\\ &=\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+\frac {e^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {2 b e^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (-\frac {e}{d \sqrt [3]{x}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}-\frac {2 b^2 e^3 n^2 \text {Li}_2\left (1+\frac {e}{d \sqrt [3]{x}}\right )}{d^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.27, size = 336, normalized size = 1.48 \begin {gather*} x \left (a+b \left (-n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )\right )^2-6 b e^3 n \left (a+b \left (-n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+\log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )\right ) \left (\frac {\sqrt [3]{x}}{3 d^2 e}-\frac {x^{2/3}}{6 d e^2}-\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^3}-\frac {x \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{3 e^3}+\frac {\log \left (\frac {e}{\sqrt [3]{x}}\right )}{3 d^3}\right )-b^2 e^3 n^2 \left (-\frac {\sqrt [3]{x} \left (1-2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{d^2 e}-\frac {x^{2/3} \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d e^2}-\frac {x \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^3}+\frac {3 \left (-\log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )+\log \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{d^3}+\frac {-\log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (-2 \log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )+\log \left (d+\frac {e}{\sqrt [3]{x}}\right )\right )+2 \text {Li}_2\left (\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )}{d^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^(1/3))^n))^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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^(1/3))^n))^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^(1/3))^n))^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________