3.5.8 \(\int x^2 (a+b \log (c (d+e \sqrt {x})^n))^2 \, dx\) [408]

Optimal. Leaf size=480 \[ \frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6}-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right )}{3 e^6}+\frac {4 b d^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^6}-\frac {5 b d^4 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^6}+\frac {40 b d^3 n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}-\frac {5 b d^2 n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}+\frac {4 b d n \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}-\frac {2 b d^6 n \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \]

[Out]

1/3*b^2*d^6*n^2*ln(d+e*x^(1/2))^2/e^6-2/3*b*d^6*n*ln(d+e*x^(1/2))*(a+b*ln(c*(d+e*x^(1/2))^n))/e^6+1/3*x^3*(a+b
*ln(c*(d+e*x^(1/2))^n))^2-4*b^2*d^5*n^2*x^(1/2)/e^5+4*b*d^5*n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))/e^6+5/
2*b^2*d^4*n^2*(d+e*x^(1/2))^2/e^6-5*b*d^4*n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^2/e^6-40/27*b^2*d^3*n^2*
(d+e*x^(1/2))^3/e^6+40/9*b*d^3*n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^3/e^6+5/8*b^2*d^2*n^2*(d+e*x^(1/2))
^4/e^6-5/2*b*d^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^4/e^6-4/25*b^2*d*n^2*(d+e*x^(1/2))^5/e^6+4/5*b*d*
n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^5/e^6+1/54*b^2*n^2*(d+e*x^(1/2))^6/e^6-1/9*b*n*(a+b*ln(c*(d+e*x^(1
/2))^n))*(d+e*x^(1/2))^6/e^6

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Rubi [A]
time = 0.33, antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2445, 2458, 45, 2372, 12, 14, 2338} \begin {gather*} -\frac {2 b d^6 n \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^6}+\frac {4 b d^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^6}-\frac {5 b d^4 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^6}+\frac {40 b d^3 n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}-\frac {5 b d^2 n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}+\frac {4 b d n \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right )}{3 e^6}-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*b^2*d^4*n^2*(d + e*Sqrt[x])^2)/(2*e^6) - (40*b^2*d^3*n^2*(d + e*Sqrt[x])^3)/(27*e^6) + (5*b^2*d^2*n^2*(d +
e*Sqrt[x])^4)/(8*e^6) - (4*b^2*d*n^2*(d + e*Sqrt[x])^5)/(25*e^6) + (b^2*n^2*(d + e*Sqrt[x])^6)/(54*e^6) - (4*b
^2*d^5*n^2*Sqrt[x])/e^5 + (b^2*d^6*n^2*Log[d + e*Sqrt[x]]^2)/(3*e^6) + (4*b*d^5*n*(d + e*Sqrt[x])*(a + b*Log[c
*(d + e*Sqrt[x])^n]))/e^6 - (5*b*d^4*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^6 + (40*b*d^3*n*(
d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*e^6) - (5*b*d^2*n*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*
Sqrt[x])^n]))/(2*e^6) + (4*b*d*n*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(5*e^6) - (b*n*(d + e*Sqr
t[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*e^6) - (2*b*d^6*n*Log[d + e*Sqrt[x]]*(a + b*Log[c*(d + e*Sqrt[x])
^n]))/(3*e^6) + (x^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -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])

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx &=2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {x^6 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt {x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {1}{3} (2 b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^6 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt {x}\right )\\ &=\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {1}{3} \left (2 b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{60 e^6 x} \, dx,x,d+e \sqrt {x}\right )\\ &=\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5\right )+60 d^6 \log (x)}{x} \, dx,x,d+e \sqrt {x}\right )}{90 e^6}\\ &=\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (-360 d^5+450 d^4 x-400 d^3 x^2+225 d^2 x^3-72 d x^4+10 x^5+\frac {60 d^6 \log (x)}{x}\right ) \, dx,x,d+e \sqrt {x}\right )}{90 e^6}\\ &=\frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6}-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+\frac {\left (2 b^2 d^6 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e \sqrt {x}\right )}{3 e^6}\\ &=\frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6}-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right )}{3 e^6}+\frac {1}{90} b n \left (\frac {360 d^5 \left (d+e \sqrt {x}\right )}{e^6}-\frac {450 d^4 \left (d+e \sqrt {x}\right )^2}{e^6}+\frac {400 d^3 \left (d+e \sqrt {x}\right )^3}{e^6}-\frac {225 d^2 \left (d+e \sqrt {x}\right )^4}{e^6}+\frac {72 d \left (d+e \sqrt {x}\right )^5}{e^6}-\frac {10 \left (d+e \sqrt {x}\right )^6}{e^6}-\frac {60 d^6 \log \left (d+e \sqrt {x}\right )}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 341, normalized size = 0.71 \begin {gather*} \frac {1800 b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right )+180 b d^6 n \log \left (d+e \sqrt {x}\right ) \left (-20 a+49 b n-20 b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+e \sqrt {x} \left (1800 a^2 e^5 x^{5/2}+60 a b n \left (60 d^5-30 d^4 e \sqrt {x}+20 d^3 e^2 x-15 d^2 e^3 x^{3/2}+12 d e^4 x^2-10 e^5 x^{5/2}\right )+b^2 n^2 \left (-8820 d^5+2610 d^4 e \sqrt {x}-1140 d^3 e^2 x+555 d^2 e^3 x^{3/2}-264 d e^4 x^2+100 e^5 x^{5/2}\right )+60 b \left (60 a e^5 x^{5/2}+b n \left (60 d^5-30 d^4 e \sqrt {x}+20 d^3 e^2 x-15 d^2 e^3 x^{3/2}+12 d e^4 x^2-10 e^5 x^{5/2}\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+1800 b^2 e^5 x^{5/2} \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{5400 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1800*b^2*d^6*n^2*Log[d + e*Sqrt[x]]^2 + 180*b*d^6*n*Log[d + e*Sqrt[x]]*(-20*a + 49*b*n - 20*b*Log[c*(d + e*Sq
rt[x])^n]) + e*Sqrt[x]*(1800*a^2*e^5*x^(5/2) + 60*a*b*n*(60*d^5 - 30*d^4*e*Sqrt[x] + 20*d^3*e^2*x - 15*d^2*e^3
*x^(3/2) + 12*d*e^4*x^2 - 10*e^5*x^(5/2)) + b^2*n^2*(-8820*d^5 + 2610*d^4*e*Sqrt[x] - 1140*d^3*e^2*x + 555*d^2
*e^3*x^(3/2) - 264*d*e^4*x^2 + 100*e^5*x^(5/2)) + 60*b*(60*a*e^5*x^(5/2) + b*n*(60*d^5 - 30*d^4*e*Sqrt[x] + 20
*d^3*e^2*x - 15*d^2*e^3*x^(3/2) + 12*d*e^4*x^2 - 10*e^5*x^(5/2)))*Log[c*(d + e*Sqrt[x])^n] + 1800*b^2*e^5*x^(5
/2)*Log[c*(d + e*Sqrt[x])^n]^2))/(5400*e^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 317, normalized size = 0.66 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{2} + \frac {2}{3} \, a b x^{3} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{2} x^{3} - \frac {1}{90} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) + {\left (30 \, d^{4} x e - 60 \, d^{5} \sqrt {x} - 20 \, d^{3} x^{\frac {3}{2}} e^{2} + 15 \, d^{2} x^{2} e^{3} - 12 \, d x^{\frac {5}{2}} e^{4} + 10 \, x^{3} e^{5}\right )} e^{\left (-6\right )}\right )} a b n e + \frac {1}{5400} \, {\left ({\left (1800 \, d^{6} \log \left (\sqrt {x} e + d\right )^{2} + 8820 \, d^{6} \log \left (\sqrt {x} e + d\right ) - 8820 \, d^{5} \sqrt {x} e + 2610 \, d^{4} x e^{2} - 1140 \, d^{3} x^{\frac {3}{2}} e^{3} + 555 \, d^{2} x^{2} e^{4} - 264 \, d x^{\frac {5}{2}} e^{5} + 100 \, x^{3} e^{6}\right )} n^{2} e^{\left (-6\right )} - 60 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) + {\left (30 \, d^{4} x e - 60 \, d^{5} \sqrt {x} - 20 \, d^{3} x^{\frac {3}{2}} e^{2} + 15 \, d^{2} x^{2} e^{3} - 12 \, d x^{\frac {5}{2}} e^{4} + 10 \, x^{3} e^{5}\right )} e^{\left (-6\right )}\right )} n e \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )\right )} b^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*x^3*log((sqrt(x)*e + d)^n*c)^2 + 2/3*a*b*x^3*log((sqrt(x)*e + d)^n*c) + 1/3*a^2*x^3 - 1/90*(60*d^6*e^(
-7)*log(sqrt(x)*e + d) + (30*d^4*x*e - 60*d^5*sqrt(x) - 20*d^3*x^(3/2)*e^2 + 15*d^2*x^2*e^3 - 12*d*x^(5/2)*e^4
+ 10*x^3*e^5)*e^(-6))*a*b*n*e + 1/5400*((1800*d^6*log(sqrt(x)*e + d)^2 + 8820*d^6*log(sqrt(x)*e + d) - 8820*d
^5*sqrt(x)*e + 2610*d^4*x*e^2 - 1140*d^3*x^(3/2)*e^3 + 555*d^2*x^2*e^4 - 264*d*x^(5/2)*e^5 + 100*x^3*e^6)*n^2*
e^(-6) - 60*(60*d^6*e^(-7)*log(sqrt(x)*e + d) + (30*d^4*x*e - 60*d^5*sqrt(x) - 20*d^3*x^(3/2)*e^2 + 15*d^2*x^2
*e^3 - 12*d*x^(5/2)*e^4 + 10*x^3*e^5)*e^(-6))*n*e*log((sqrt(x)*e + d)^n*c))*b^2

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Fricas [A]
time = 0.38, size = 451, normalized size = 0.94 \begin {gather*} \frac {1}{5400} \, {\left (1800 \, b^{2} x^{3} e^{6} \log \left (c\right )^{2} + 100 \, {\left (b^{2} n^{2} - 6 \, a b n + 18 \, a^{2}\right )} x^{3} e^{6} + 15 \, {\left (37 \, b^{2} d^{2} n^{2} - 60 \, a b d^{2} n\right )} x^{2} e^{4} + 90 \, {\left (29 \, b^{2} d^{4} n^{2} - 20 \, a b d^{4} n\right )} x e^{2} - 1800 \, {\left (b^{2} d^{6} n^{2} - b^{2} n^{2} x^{3} e^{6}\right )} \log \left (\sqrt {x} e + d\right )^{2} + 60 \, {\left (147 \, b^{2} d^{6} n^{2} - 30 \, b^{2} d^{4} n^{2} x e^{2} - 60 \, a b d^{6} n - 15 \, b^{2} d^{2} n^{2} x^{2} e^{4} - 10 \, {\left (b^{2} n^{2} - 6 \, a b n\right )} x^{3} e^{6} - 60 \, {\left (b^{2} d^{6} n - b^{2} n x^{3} e^{6}\right )} \log \left (c\right ) + 4 \, {\left (15 \, b^{2} d^{5} n^{2} e + 5 \, b^{2} d^{3} n^{2} x e^{3} + 3 \, b^{2} d n^{2} x^{2} e^{5}\right )} \sqrt {x}\right )} \log \left (\sqrt {x} e + d\right ) - 300 \, {\left (6 \, b^{2} d^{4} n x e^{2} + 3 \, b^{2} d^{2} n x^{2} e^{4} + 2 \, {\left (b^{2} n - 6 \, a b\right )} x^{3} e^{6}\right )} \log \left (c\right ) - 12 \, {\left (2 \, {\left (11 \, b^{2} d n^{2} - 30 \, a b d n\right )} x^{2} e^{5} + 5 \, {\left (19 \, b^{2} d^{3} n^{2} - 20 \, a b d^{3} n\right )} x e^{3} + 15 \, {\left (49 \, b^{2} d^{5} n^{2} - 20 \, a b d^{5} n\right )} e - 20 \, {\left (15 \, b^{2} d^{5} n e + 5 \, b^{2} d^{3} n x e^{3} + 3 \, b^{2} d n x^{2} e^{5}\right )} \log \left (c\right )\right )} \sqrt {x}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5400*(1800*b^2*x^3*e^6*log(c)^2 + 100*(b^2*n^2 - 6*a*b*n + 18*a^2)*x^3*e^6 + 15*(37*b^2*d^2*n^2 - 60*a*b*d^2
*n)*x^2*e^4 + 90*(29*b^2*d^4*n^2 - 20*a*b*d^4*n)*x*e^2 - 1800*(b^2*d^6*n^2 - b^2*n^2*x^3*e^6)*log(sqrt(x)*e +
d)^2 + 60*(147*b^2*d^6*n^2 - 30*b^2*d^4*n^2*x*e^2 - 60*a*b*d^6*n - 15*b^2*d^2*n^2*x^2*e^4 - 10*(b^2*n^2 - 6*a*
b*n)*x^3*e^6 - 60*(b^2*d^6*n - b^2*n*x^3*e^6)*log(c) + 4*(15*b^2*d^5*n^2*e + 5*b^2*d^3*n^2*x*e^3 + 3*b^2*d*n^2
*x^2*e^5)*sqrt(x))*log(sqrt(x)*e + d) - 300*(6*b^2*d^4*n*x*e^2 + 3*b^2*d^2*n*x^2*e^4 + 2*(b^2*n - 6*a*b)*x^3*e
^6)*log(c) - 12*(2*(11*b^2*d*n^2 - 30*a*b*d*n)*x^2*e^5 + 5*(19*b^2*d^3*n^2 - 20*a*b*d^3*n)*x*e^3 + 15*(49*b^2*
d^5*n^2 - 20*a*b*d^5*n)*e - 20*(15*b^2*d^5*n*e + 5*b^2*d^3*n*x*e^3 + 3*b^2*d*n*x^2*e^5)*log(c))*sqrt(x))*e^(-6
)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 956 vs. \(2 (419) = 838\).
time = 4.05, size = 956, normalized size = 1.99 \begin {gather*} \frac {1}{5400} \, {\left (1800 \, b^{2} x^{3} e \log \left (c\right )^{2} + 3600 \, a b x^{3} e \log \left (c\right ) + 1800 \, a^{2} x^{3} e + {\left (1800 \, {\left (\sqrt {x} e + d\right )}^{6} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right )^{2} - 10800 \, {\left (\sqrt {x} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right )^{2} + 27000 \, {\left (\sqrt {x} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right )^{2} - 36000 \, {\left (\sqrt {x} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right )^{2} + 27000 \, {\left (\sqrt {x} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right )^{2} - 10800 \, {\left (\sqrt {x} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right )^{2} - 600 \, {\left (\sqrt {x} e + d\right )}^{6} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + 4320 \, {\left (\sqrt {x} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 13500 \, {\left (\sqrt {x} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + 24000 \, {\left (\sqrt {x} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 27000 \, {\left (\sqrt {x} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + 21600 \, {\left (\sqrt {x} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + 100 \, {\left (\sqrt {x} e + d\right )}^{6} e^{\left (-5\right )} - 864 \, {\left (\sqrt {x} e + d\right )}^{5} d e^{\left (-5\right )} + 3375 \, {\left (\sqrt {x} e + d\right )}^{4} d^{2} e^{\left (-5\right )} - 8000 \, {\left (\sqrt {x} e + d\right )}^{3} d^{3} e^{\left (-5\right )} + 13500 \, {\left (\sqrt {x} e + d\right )}^{2} d^{4} e^{\left (-5\right )} - 21600 \, {\left (\sqrt {x} e + d\right )} d^{5} e^{\left (-5\right )}\right )} b^{2} n^{2} + 60 \, {\left (60 \, {\left (\sqrt {x} e + d\right )}^{6} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 360 \, {\left (\sqrt {x} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + 900 \, {\left (\sqrt {x} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 1200 \, {\left (\sqrt {x} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + 900 \, {\left (\sqrt {x} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 360 \, {\left (\sqrt {x} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 10 \, {\left (\sqrt {x} e + d\right )}^{6} e^{\left (-5\right )} + 72 \, {\left (\sqrt {x} e + d\right )}^{5} d e^{\left (-5\right )} - 225 \, {\left (\sqrt {x} e + d\right )}^{4} d^{2} e^{\left (-5\right )} + 400 \, {\left (\sqrt {x} e + d\right )}^{3} d^{3} e^{\left (-5\right )} - 450 \, {\left (\sqrt {x} e + d\right )}^{2} d^{4} e^{\left (-5\right )} + 360 \, {\left (\sqrt {x} e + d\right )} d^{5} e^{\left (-5\right )}\right )} b^{2} n \log \left (c\right ) + 60 \, {\left (60 \, {\left (\sqrt {x} e + d\right )}^{6} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 360 \, {\left (\sqrt {x} e + d\right )}^{5} d e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + 900 \, {\left (\sqrt {x} e + d\right )}^{4} d^{2} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 1200 \, {\left (\sqrt {x} e + d\right )}^{3} d^{3} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) + 900 \, {\left (\sqrt {x} e + d\right )}^{2} d^{4} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 360 \, {\left (\sqrt {x} e + d\right )} d^{5} e^{\left (-5\right )} \log \left (\sqrt {x} e + d\right ) - 10 \, {\left (\sqrt {x} e + d\right )}^{6} e^{\left (-5\right )} + 72 \, {\left (\sqrt {x} e + d\right )}^{5} d e^{\left (-5\right )} - 225 \, {\left (\sqrt {x} e + d\right )}^{4} d^{2} e^{\left (-5\right )} + 400 \, {\left (\sqrt {x} e + d\right )}^{3} d^{3} e^{\left (-5\right )} - 450 \, {\left (\sqrt {x} e + d\right )}^{2} d^{4} e^{\left (-5\right )} + 360 \, {\left (\sqrt {x} e + d\right )} d^{5} e^{\left (-5\right )}\right )} a b n\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5400*(1800*b^2*x^3*e*log(c)^2 + 3600*a*b*x^3*e*log(c) + 1800*a^2*x^3*e + (1800*(sqrt(x)*e + d)^6*e^(-5)*log(
sqrt(x)*e + d)^2 - 10800*(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^4*d^2*e^(-5)*
log(sqrt(x)*e + d)^2 - 36000*(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^2*d^4*e
^(-5)*log(sqrt(x)*e + d)^2 - 10800*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d)^2 - 600*(sqrt(x)*e + d)^6*e^(
-5)*log(sqrt(x)*e + d) + 4320*(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d) - 13500*(sqrt(x)*e + d)^4*d^2*e^(-
5)*log(sqrt(x)*e + d) + 24000*(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d) - 27000*(sqrt(x)*e + d)^2*d^4*e^
(-5)*log(sqrt(x)*e + d) + 21600*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d) + 100*(sqrt(x)*e + d)^6*e^(-5) -
864*(sqrt(x)*e + d)^5*d*e^(-5) + 3375*(sqrt(x)*e + d)^4*d^2*e^(-5) - 8000*(sqrt(x)*e + d)^3*d^3*e^(-5) + 1350
0*(sqrt(x)*e + d)^2*d^4*e^(-5) - 21600*(sqrt(x)*e + d)*d^5*e^(-5))*b^2*n^2 + 60*(60*(sqrt(x)*e + d)^6*e^(-5)*l
og(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(s
qrt(x)*e + d) - 1200*(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sq
rt(x)*e + d) - 360*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d) - 10*(sqrt(x)*e + d)^6*e^(-5) + 72*(sqrt(x)*e
+ d)^5*d*e^(-5) - 225*(sqrt(x)*e + d)^4*d^2*e^(-5) + 400*(sqrt(x)*e + d)^3*d^3*e^(-5) - 450*(sqrt(x)*e + d)^2
*d^4*e^(-5) + 360*(sqrt(x)*e + d)*d^5*e^(-5))*b^2*n*log(c) + 60*(60*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d
) - 360*(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d) -
1200*(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d) - 3
60*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d) - 10*(sqrt(x)*e + d)^6*e^(-5) + 72*(sqrt(x)*e + d)^5*d*e^(-5)
- 225*(sqrt(x)*e + d)^4*d^2*e^(-5) + 400*(sqrt(x)*e + d)^3*d^3*e^(-5) - 450*(sqrt(x)*e + d)^2*d^4*e^(-5) + 36
0*(sqrt(x)*e + d)*d^5*e^(-5))*a*b*n)*e^(-1)

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Mupad [B]
time = 1.78, size = 434, normalized size = 0.90 \begin {gather*} \frac {a^2\,x^3}{3}+\frac {b^2\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{3}+\frac {b^2\,n^2\,x^3}{54}+\frac {2\,a\,b\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3}-\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{3\,e^6}-\frac {a\,b\,n\,x^3}{9}-\frac {b^2\,n\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{9}+\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+e\,\sqrt {x}\right )}{30\,e^6}+\frac {37\,b^2\,d^2\,n^2\,x^2}{360\,e^2}-\frac {19\,b^2\,d^3\,n^2\,x^{3/2}}{90\,e^3}-\frac {49\,b^2\,d^5\,n^2\,\sqrt {x}}{30\,e^5}-\frac {11\,b^2\,d\,n^2\,x^{5/2}}{225\,e}+\frac {29\,b^2\,d^4\,n^2\,x}{60\,e^4}-\frac {b^2\,d^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{6\,e^2}+\frac {2\,b^2\,d^3\,n\,x^{3/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{9\,e^3}+\frac {2\,b^2\,d^5\,n\,\sqrt {x}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,e^5}+\frac {2\,a\,b\,d\,n\,x^{5/2}}{15\,e}-\frac {a\,b\,d^4\,n\,x}{3\,e^4}-\frac {2\,a\,b\,d^6\,n\,\ln \left (d+e\,\sqrt {x}\right )}{3\,e^6}+\frac {2\,b^2\,d\,n\,x^{5/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{15\,e}-\frac {b^2\,d^4\,n\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,e^4}-\frac {a\,b\,d^2\,n\,x^2}{6\,e^2}+\frac {2\,a\,b\,d^3\,n\,x^{3/2}}{9\,e^3}+\frac {2\,a\,b\,d^5\,n\,\sqrt {x}}{3\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*x^3)/3 + (b^2*x^3*log(c*(d + e*x^(1/2))^n)^2)/3 + (b^2*n^2*x^3)/54 + (2*a*b*x^3*log(c*(d + e*x^(1/2))^n))
/3 - (b^2*d^6*log(c*(d + e*x^(1/2))^n)^2)/(3*e^6) - (a*b*n*x^3)/9 - (b^2*n*x^3*log(c*(d + e*x^(1/2))^n))/9 + (
49*b^2*d^6*n^2*log(d + e*x^(1/2)))/(30*e^6) + (37*b^2*d^2*n^2*x^2)/(360*e^2) - (19*b^2*d^3*n^2*x^(3/2))/(90*e^
3) - (49*b^2*d^5*n^2*x^(1/2))/(30*e^5) - (11*b^2*d*n^2*x^(5/2))/(225*e) + (29*b^2*d^4*n^2*x)/(60*e^4) - (b^2*d
^2*n*x^2*log(c*(d + e*x^(1/2))^n))/(6*e^2) + (2*b^2*d^3*n*x^(3/2)*log(c*(d + e*x^(1/2))^n))/(9*e^3) + (2*b^2*d
^5*n*x^(1/2)*log(c*(d + e*x^(1/2))^n))/(3*e^5) + (2*a*b*d*n*x^(5/2))/(15*e) - (a*b*d^4*n*x)/(3*e^4) - (2*a*b*d
^6*n*log(d + e*x^(1/2)))/(3*e^6) + (2*b^2*d*n*x^(5/2)*log(c*(d + e*x^(1/2))^n))/(15*e) - (b^2*d^4*n*x*log(c*(d
+ e*x^(1/2))^n))/(3*e^4) - (a*b*d^2*n*x^2)/(6*e^2) + (2*a*b*d^3*n*x^(3/2))/(9*e^3) + (2*a*b*d^5*n*x^(1/2))/(3
*e^5)

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