\(\int (a+b \log (c (d+e \sqrt {x})^n))^3 \, dx\) [417]

Optimal result

Integrand size = 20, antiderivative size = 284 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^2}-\frac {12 a b^2 d n^2 \sqrt {x}}{e}+\frac {12 b^3 d n^3 \sqrt {x}}{e}-\frac {12 b^3 d n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^2}+\frac {6 b d n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}-\frac {3 b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2} \]

[Out]

-12*a*b^2*d*n^2*x^(1/2)/e+12*b^3*d*n^3*x^(1/2)/e-12*b^3*d*n^2*ln(c*(d+e*x^(1/2))^n)*(d+e*x^(1/2))/e^2+6*b*d*n*
(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))/e^2-2*d*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))/e^2-3/4*b^3*n^
3*(d+e*x^(1/2))^2/e^2+3/2*b^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^2/e^2-3/2*b*n*(a+b*ln(c*(d+e*x^(1/
2))^n))^2*(d+e*x^(1/2))^2/e^2+(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^2/e^2

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2501, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^2}-\frac {12 a b^2 d n^2 \sqrt {x}}{e}-\frac {3 b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^2}+\frac {6 b d n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}-\frac {12 b^3 d n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^2}+\frac {12 b^3 d n^3 \sqrt {x}}{e} \]

[In]

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

[Out]

(-3*b^3*n^3*(d + e*Sqrt[x])^2)/(4*e^2) - (12*a*b^2*d*n^2*Sqrt[x])/e + (12*b^3*d*n^3*Sqrt[x])/e - (12*b^3*d*n^2
*(d + e*Sqrt[x])*Log[c*(d + e*Sqrt[x])^n])/e^2 + (3*b^2*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])
)/(2*e^2) + (6*b*d*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^2 - (3*b*n*(d + e*Sqrt[x])^2*(a + b
*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^2) - (2*d*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^2 + ((d + e
*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^2

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

Rule 2436

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

Rule 2437

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

Rule 2448

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

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]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e}-\frac {(2 d) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e} \\ & = \frac {2 \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^2}-\frac {(2 d) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^2} \\ & = -\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}-\frac {(3 b n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^2}+\frac {(6 b d n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^2} \\ & = \frac {6 b d n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}-\frac {3 b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}+\frac {\left (3 b^2 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^2}-\frac {\left (12 b^2 d n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^2} \\ & = -\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^2}-\frac {12 a b^2 d n^2 \sqrt {x}}{e}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^2}+\frac {6 b d n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}-\frac {3 b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}-\frac {\left (12 b^3 d n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt {x}\right )}{e^2} \\ & = -\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^2}-\frac {12 a b^2 d n^2 \sqrt {x}}{e}+\frac {12 b^3 d n^3 \sqrt {x}}{e}-\frac {12 b^3 d n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^2}+\frac {6 b d n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}-\frac {3 b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.85 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {-8 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3+4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3+24 b d n \left (\left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-2 b n \left (e (a-b n) \sqrt {x}+b \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )\right )-3 b n \left (2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+b n \left (b e n \left (2 d \sqrt {x}+e x\right )-2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )\right )\right )}{4 e^2} \]

[In]

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

[Out]

(-8*d*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3 + 4*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]
)^3 + 24*b*d*n*((d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2 - 2*b*n*(e*(a - b*n)*Sqrt[x] + b*(d + e*Sqr
t[x])*Log[c*(d + e*Sqrt[x])^n])) - 3*b*n*(2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2 + b*n*(b*e*n*
(2*d*Sqrt[x] + e*x) - 2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))))/(4*e^2)

Maple [F]

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

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (248) = 496\).

Time = 0.37 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.86 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {4 \, b^{3} e^{2} x \log \left (c\right )^{3} + 4 \, {\left (b^{3} e^{2} n^{3} x - b^{3} d^{2} n^{3}\right )} \log \left (e \sqrt {x} + d\right )^{3} - 6 \, {\left (b^{3} e^{2} n - 2 \, a b^{2} e^{2}\right )} x \log \left (c\right )^{2} + 6 \, {\left (2 \, b^{3} d e n^{3} \sqrt {x} + 3 \, b^{3} d^{2} n^{3} - 2 \, a b^{2} d^{2} n^{2} - {\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2}\right )} x + 2 \, {\left (b^{3} e^{2} n^{2} x - b^{3} d^{2} n^{2}\right )} \log \left (c\right )\right )} \log \left (e \sqrt {x} + d\right )^{2} + 6 \, {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n + 2 \, a^{2} b e^{2}\right )} x \log \left (c\right ) - {\left (3 \, b^{3} e^{2} n^{3} - 6 \, a b^{2} e^{2} n^{2} + 6 \, a^{2} b e^{2} n - 4 \, a^{3} e^{2}\right )} x - 6 \, {\left (7 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2} + 2 \, a^{2} b d^{2} n - 2 \, {\left (b^{3} e^{2} n x - b^{3} d^{2} n\right )} \log \left (c\right )^{2} - {\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2} + 2 \, a^{2} b e^{2} n\right )} x - 2 \, {\left (3 \, b^{3} d^{2} n^{2} - 2 \, a b^{2} d^{2} n - {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n\right )} x\right )} \log \left (c\right ) + 2 \, {\left (3 \, b^{3} d e n^{3} - 2 \, b^{3} d e n^{2} \log \left (c\right ) - 2 \, a b^{2} d e n^{2}\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right ) + 6 \, {\left (7 \, b^{3} d e n^{3} + 2 \, b^{3} d e n \log \left (c\right )^{2} - 6 \, a b^{2} d e n^{2} + 2 \, a^{2} b d e n - 2 \, {\left (3 \, b^{3} d e n^{2} - 2 \, a b^{2} d e n\right )} \log \left (c\right )\right )} \sqrt {x}}{4 \, e^{2}} \]

[In]

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

[Out]

1/4*(4*b^3*e^2*x*log(c)^3 + 4*(b^3*e^2*n^3*x - b^3*d^2*n^3)*log(e*sqrt(x) + d)^3 - 6*(b^3*e^2*n - 2*a*b^2*e^2)
*x*log(c)^2 + 6*(2*b^3*d*e*n^3*sqrt(x) + 3*b^3*d^2*n^3 - 2*a*b^2*d^2*n^2 - (b^3*e^2*n^3 - 2*a*b^2*e^2*n^2)*x +
2*(b^3*e^2*n^2*x - b^3*d^2*n^2)*log(c))*log(e*sqrt(x) + d)^2 + 6*(b^3*e^2*n^2 - 2*a*b^2*e^2*n + 2*a^2*b*e^2)*
x*log(c) - (3*b^3*e^2*n^3 - 6*a*b^2*e^2*n^2 + 6*a^2*b*e^2*n - 4*a^3*e^2)*x - 6*(7*b^3*d^2*n^3 - 6*a*b^2*d^2*n^
2 + 2*a^2*b*d^2*n - 2*(b^3*e^2*n*x - b^3*d^2*n)*log(c)^2 - (b^3*e^2*n^3 - 2*a*b^2*e^2*n^2 + 2*a^2*b*e^2*n)*x -
2*(3*b^3*d^2*n^2 - 2*a*b^2*d^2*n - (b^3*e^2*n^2 - 2*a*b^2*e^2*n)*x)*log(c) + 2*(3*b^3*d*e*n^3 - 2*b^3*d*e*n^2
*log(c) - 2*a*b^2*d*e*n^2)*sqrt(x))*log(e*sqrt(x) + d) + 6*(7*b^3*d*e*n^3 + 2*b^3*d*e*n*log(c)^2 - 6*a*b^2*d*e
*n^2 + 2*a^2*b*d*e*n - 2*(3*b^3*d*e*n^2 - 2*a*b^2*d*e*n)*log(c))*sqrt(x))/e^2

Sympy [F]

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

[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 [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.34 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {3}{2} \, {\left (e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )\right )} a^{2} b - \frac {3}{2} \, {\left (2 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {{\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n^{2}}{e^{2}}\right )} a b^{2} - \frac {1}{4} \, {\left (6 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - 4 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{3} + e n {\left (\frac {{\left (4 \, d^{2} \log \left (e \sqrt {x} + d\right )^{3} + 18 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + 3 \, e^{2} x + 42 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 42 \, d e \sqrt {x}\right )} n^{2}}{e^{3}} - \frac {6 \, {\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{e^{3}}\right )}\right )} b^{3} + a^{3} x \]

[In]

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

[Out]

-3/2*(e*n*(2*d^2*log(e*sqrt(x) + d)/e^3 + (e*x - 2*d*sqrt(x))/e^2) - 2*x*log((e*sqrt(x) + d)^n*c))*a^2*b - 3/2
*(2*e*n*(2*d^2*log(e*sqrt(x) + d)/e^3 + (e*x - 2*d*sqrt(x))/e^2)*log((e*sqrt(x) + d)^n*c) - 2*x*log((e*sqrt(x)
+ d)^n*c)^2 - (2*d^2*log(e*sqrt(x) + d)^2 + e^2*x + 6*d^2*log(e*sqrt(x) + d) - 6*d*e*sqrt(x))*n^2/e^2)*a*b^2
- 1/4*(6*e*n*(2*d^2*log(e*sqrt(x) + d)/e^3 + (e*x - 2*d*sqrt(x))/e^2)*log((e*sqrt(x) + d)^n*c)^2 - 4*x*log((e*
sqrt(x) + d)^n*c)^3 + e*n*((4*d^2*log(e*sqrt(x) + d)^3 + 18*d^2*log(e*sqrt(x) + d)^2 + 3*e^2*x + 42*d^2*log(e*
sqrt(x) + d) - 42*d*e*sqrt(x))*n^2/e^3 - 6*(2*d^2*log(e*sqrt(x) + d)^2 + e^2*x + 6*d^2*log(e*sqrt(x) + d) - 6*
d*e*sqrt(x))*n*log((e*sqrt(x) + d)^n*c)/e^3))*b^3 + a^3*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (248) = 496\).

Time = 0.31 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.51 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {\frac {{\left (4 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right )^{3} - 8 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right )^{3} - 6 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right )^{2} + 24 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right )^{2} + 6 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) - 48 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) - 3 \, {\left (e \sqrt {x} + d\right )}^{2} + 48 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{3} n^{3}}{e} + \frac {6 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right )^{2} - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right )^{2} - 2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) + 8 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) + {\left (e \sqrt {x} + d\right )}^{2} - 8 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{3} n^{2} \log \left (c\right )}{e} + \frac {6 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) - {\left (e \sqrt {x} + d\right )}^{2} + 4 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{3} n \log \left (c\right )^{2}}{e} + \frac {4 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{3} \log \left (c\right )^{3}}{e} + \frac {6 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right )^{2} - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right )^{2} - 2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) + 8 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) + {\left (e \sqrt {x} + d\right )}^{2} - 8 \, {\left (e \sqrt {x} + d\right )} d\right )} a b^{2} n^{2}}{e} + \frac {12 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) - {\left (e \sqrt {x} + d\right )}^{2} + 4 \, {\left (e \sqrt {x} + d\right )} d\right )} a b^{2} n \log \left (c\right )}{e} + \frac {12 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} a b^{2} \log \left (c\right )^{2}}{e} + \frac {6 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) - {\left (e \sqrt {x} + d\right )}^{2} + 4 \, {\left (e \sqrt {x} + d\right )} d\right )} a^{2} b n}{e} + \frac {12 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} a^{2} b \log \left (c\right )}{e} + \frac {4 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} a^{3}}{e}}{4 \, e} \]

[In]

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

[Out]

1/4*((4*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d)^3 - 8*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d)^3 - 6*(e*sqrt(x) + d)^
2*log(e*sqrt(x) + d)^2 + 24*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d)^2 + 6*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d) -
48*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) - 3*(e*sqrt(x) + d)^2 + 48*(e*sqrt(x) + d)*d)*b^3*n^3/e + 6*(2*(e*sqrt
(x) + d)^2*log(e*sqrt(x) + d)^2 - 4*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)^2*log(e*sqrt(x)
+ d) + 8*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) + (e*sqrt(x) + d)^2 - 8*(e*sqrt(x) + d)*d)*b^3*n^2*log(c)/e + 6
*(2*(e*sqrt(x) + d)^2*log(e*sqrt(x) + d) - 4*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) - (e*sqrt(x) + d)^2 + 4*(e*s
qrt(x) + d)*d)*b^3*n*log(c)^2/e + 4*((e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)*d)*b^3*log(c)^3/e + 6*(2*(e*sqrt(x)
+ d)^2*log(e*sqrt(x) + d)^2 - 4*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)^2*log(e*sqrt(x) +
d) + 8*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) + (e*sqrt(x) + d)^2 - 8*(e*sqrt(x) + d)*d)*a*b^2*n^2/e + 12*(2*(e*
sqrt(x) + d)^2*log(e*sqrt(x) + d) - 4*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) - (e*sqrt(x) + d)^2 + 4*(e*sqrt(x)
+ d)*d)*a*b^2*n*log(c)/e + 12*((e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)*d)*a*b^2*log(c)^2/e + 6*(2*(e*sqrt(x) + d
)^2*log(e*sqrt(x) + d) - 4*(e*sqrt(x) + d)*d*log(e*sqrt(x) + d) - (e*sqrt(x) + d)^2 + 4*(e*sqrt(x) + d)*d)*a^2
*b*n/e + 12*((e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d)*d)*a^2*b*log(c)/e + 4*((e*sqrt(x) + d)^2 - 2*(e*sqrt(x) + d
)*d)*a^3/e)/e

Mupad [B] (verification not implemented)

Time = 1.77 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.23 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=x\,\left (a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{2}-\frac {3\,b^3\,n^3}{4}\right )-\sqrt {x}\,\left (\frac {d\,\left (2\,a^3-3\,a^2\,b\,n+3\,a\,b^2\,n^2-\frac {3\,b^3\,n^3}{2}\right )}{e}-\frac {d\,\left (2\,a^3-6\,a\,b^2\,n^2+9\,b^3\,n^3\right )}{e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3\,\left (b^3\,x-\frac {b^3\,d^2}{e^2}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\sqrt {x}\,\left (\frac {3\,b\,d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {6\,b\,d\,\left (a^2-b^2\,n^2\right )}{e}\right )-\frac {3\,b\,x\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2}\right )-{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (\sqrt {x}\,\left (\frac {3\,b^2\,d\,\left (2\,a-b\,n\right )}{e}-\frac {6\,a\,b^2\,d}{e}\right )+\frac {3\,d\,\left (2\,a\,b^2\,d-3\,b^3\,d\,n\right )}{2\,e^2}-\frac {3\,b^2\,x\,\left (2\,a-b\,n\right )}{2}\right )-\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (6\,a^2\,b\,d^2\,n-18\,a\,b^2\,d^2\,n^2+21\,b^3\,d^2\,n^3\right )}{2\,e^2} \]

[In]

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

[Out]

x*(a^3 - (3*b^3*n^3)/4 + (3*a*b^2*n^2)/2 - (3*a^2*b*n)/2) - x^(1/2)*((d*(2*a^3 - (3*b^3*n^3)/2 + 3*a*b^2*n^2 -
3*a^2*b*n))/e - (d*(2*a^3 + 9*b^3*n^3 - 6*a*b^2*n^2))/e) + log(c*(d + e*x^(1/2))^n)^3*(b^3*x - (b^3*d^2)/e^2)
- log(c*(d + e*x^(1/2))^n)*(x^(1/2)*((3*b*d*(2*a^2 + b^2*n^2 - 2*a*b*n))/e - (6*b*d*(a^2 - b^2*n^2))/e) - (3*
b*x*(2*a^2 + b^2*n^2 - 2*a*b*n))/2) - log(c*(d + e*x^(1/2))^n)^2*(x^(1/2)*((3*b^2*d*(2*a - b*n))/e - (6*a*b^2*
d)/e) + (3*d*(2*a*b^2*d - 3*b^3*d*n))/(2*e^2) - (3*b^2*x*(2*a - b*n))/2) - (log(d + e*x^(1/2))*(21*b^3*d^2*n^3
- 18*a*b^2*d^2*n^2 + 6*a^2*b*d^2*n))/(2*e^2)