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

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

[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

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Rubi [A]  time = 0.476373, antiderivative size = 355, normalized size of antiderivative = 0.74, 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 = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ \frac{1}{90} b n \left (\frac{360 d^5 \left (d+e \sqrt{x}\right )}{e^6}-\frac{450 d^4 \left (d+e \sqrt{x}\right )^2}{e^6}+\frac{400 d^3 \left (d+e \sqrt{x}\right )^3}{e^6}-\frac{225 d^2 \left (d+e \sqrt{x}\right )^4}{e^6}-\frac{60 d^6 \log \left (d+e \sqrt{x}\right )}{e^6}+\frac{72 d \left (d+e \sqrt{x}\right )^5}{e^6}-\frac{10 \left (d+e \sqrt{x}\right )^6}{e^6}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2-\frac{4 b^2 d^5 n^2 \sqrt{x}}{e^5}+\frac{5 b^2 d^4 n^2 \left (d+e \sqrt{x}\right )^2}{2 e^6}-\frac{40 b^2 d^3 n^2 \left (d+e \sqrt{x}\right )^3}{27 e^6}+\frac{5 b^2 d^2 n^2 \left (d+e \sqrt{x}\right )^4}{8 e^6}+\frac{b^2 d^6 n^2 \log ^2\left (d+e \sqrt{x}\right )}{3 e^6}-\frac{4 b^2 d n^2 \left (d+e \sqrt{x}\right )^5}{25 e^6}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^6}{54 e^6} \]

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) + (b*n*((360*d^5*(d + e*Sqrt[x]))/e^6 - (
450*d^4*(d + e*Sqrt[x])^2)/e^6 + (400*d^3*(d + e*Sqrt[x])^3)/e^6 - (225*d^2*(d + e*Sqrt[x])^4)/e^6 + (72*d*(d
+ e*Sqrt[x])^5)/e^6 - (10*(d + e*Sqrt[x])^6)/e^6 - (60*d^6*Log[d + e*Sqrt[x]])/e^6)*(a + b*Log[c*(d + e*Sqrt[x
])^n]))/90 + (x^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/3

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 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 43

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 2334

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]}, Simp[u*(a + b*Log[c*x^n]), 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 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 2301

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]

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 \operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 0.316471, size = 295, normalized size = 0.61 \[ \frac{e \sqrt{x} \left (1800 a^2 e^5 x^{5/2}+60 a b n \left (-15 d^2 e^3 x^{3/2}+20 d^3 e^2 x-30 d^4 e \sqrt{x}+60 d^5+12 d e^4 x^2-10 e^5 x^{5/2}\right )+b^2 n^2 \left (555 d^2 e^3 x^{3/2}-1140 d^3 e^2 x+2610 d^4 e \sqrt{x}-8820 d^5-264 d e^4 x^2+100 e^5 x^{5/2}\right )\right )-60 b \left (60 a \left (d^6-e^6 x^3\right )+b n \left (-20 d^3 e^3 x^{3/2}+15 d^2 e^4 x^2+30 d^4 e^2 x-60 d^5 e \sqrt{x}-147 d^6-12 d e^5 x^{5/2}+10 e^6 x^3\right )\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )-1800 b^2 \left (d^6-e^6 x^3\right ) \log ^2\left (c \left (d+e \sqrt{x}\right )^n\right )}{5400 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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) + 1
2*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*(d^6 - e^6*x^3) + b*n*(-147*d^6 - 60*d^5*e*Sqrt[x] + 30*d^4*
e^2*x - 20*d^3*e^3*x^(3/2) + 15*d^2*e^4*x^2 - 12*d*e^5*x^(5/2) + 10*e^6*x^3))*Log[c*(d + e*Sqrt[x])^n] - 1800*
b^2*(d^6 - e^6*x^3)*Log[c*(d + e*Sqrt[x])^n]^2)/(5400*e^6)

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Maple [F]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

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 = 1.07361, size = 437, normalized size = 0.91 \begin{align*} \frac{1}{3} \, b^{2} x^{3} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} + \frac{2}{3} \, a b x^{3} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + \frac{1}{3} \, a^{2} x^{3} - \frac{1}{90} \, a b e n{\left (\frac{60 \, d^{6} \log \left (e \sqrt{x} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac{5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac{3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt{x}}{e^{6}}\right )} - \frac{1}{5400} \,{\left (60 \, e n{\left (\frac{60 \, d^{6} \log \left (e \sqrt{x} + d\right )}{e^{7}} + \frac{10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac{5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac{3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt{x}}{e^{6}}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) - \frac{{\left (100 \, e^{6} x^{3} - 264 \, d e^{5} x^{\frac{5}{2}} + 555 \, d^{2} e^{4} x^{2} + 1800 \, d^{6} \log \left (e \sqrt{x} + d\right )^{2} - 1140 \, d^{3} e^{3} x^{\frac{3}{2}} + 2610 \, d^{4} e^{2} x + 8820 \, d^{6} \log \left (e \sqrt{x} + d\right ) - 8820 \, d^{5} e \sqrt{x}\right )} n^{2}}{e^{6}}\right )} b^{2} \end{align*}

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

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Fricas [A]  time = 2.25006, size = 1081, normalized size = 2.25 \begin{align*} \frac{1800 \, b^{2} e^{6} x^{3} \log \left (c\right )^{2} + 100 \,{\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n + 18 \, a^{2} e^{6}\right )} x^{3} + 15 \,{\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x^{2} + 1800 \,{\left (b^{2} e^{6} n^{2} x^{3} - b^{2} d^{6} n^{2}\right )} \log \left (e \sqrt{x} + d\right )^{2} + 90 \,{\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x - 60 \,{\left (15 \, b^{2} d^{2} e^{4} n^{2} x^{2} + 30 \, b^{2} d^{4} e^{2} n^{2} x - 147 \, b^{2} d^{6} n^{2} + 60 \, a b d^{6} n + 10 \,{\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n\right )} x^{3} - 60 \,{\left (b^{2} e^{6} n x^{3} - b^{2} d^{6} n\right )} \log \left (c\right ) - 4 \,{\left (3 \, b^{2} d e^{5} n^{2} x^{2} + 5 \, b^{2} d^{3} e^{3} n^{2} x + 15 \, b^{2} d^{5} e n^{2}\right )} \sqrt{x}\right )} \log \left (e \sqrt{x} + d\right ) - 300 \,{\left (3 \, b^{2} d^{2} e^{4} n x^{2} + 6 \, b^{2} d^{4} e^{2} n x + 2 \,{\left (b^{2} e^{6} n - 6 \, a b e^{6}\right )} x^{3}\right )} \log \left (c\right ) - 12 \,{\left (735 \, b^{2} d^{5} e n^{2} - 300 \, a b d^{5} e n + 2 \,{\left (11 \, b^{2} d e^{5} n^{2} - 30 \, a b d e^{5} n\right )} x^{2} + 5 \,{\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x - 20 \,{\left (3 \, b^{2} d e^{5} n x^{2} + 5 \, b^{2} d^{3} e^{3} n x + 15 \, b^{2} d^{5} e n\right )} \log \left (c\right )\right )} \sqrt{x}}{5400 \, e^{6}} \end{align*}

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*e^6*x^3*log(c)^2 + 100*(b^2*e^6*n^2 - 6*a*b*e^6*n + 18*a^2*e^6)*x^3 + 15*(37*b^2*d^2*e^4*n^2
- 60*a*b*d^2*e^4*n)*x^2 + 1800*(b^2*e^6*n^2*x^3 - b^2*d^6*n^2)*log(e*sqrt(x) + d)^2 + 90*(29*b^2*d^4*e^2*n^2 -
20*a*b*d^4*e^2*n)*x - 60*(15*b^2*d^2*e^4*n^2*x^2 + 30*b^2*d^4*e^2*n^2*x - 147*b^2*d^6*n^2 + 60*a*b*d^6*n + 10
*(b^2*e^6*n^2 - 6*a*b*e^6*n)*x^3 - 60*(b^2*e^6*n*x^3 - b^2*d^6*n)*log(c) - 4*(3*b^2*d*e^5*n^2*x^2 + 5*b^2*d^3*
e^3*n^2*x + 15*b^2*d^5*e*n^2)*sqrt(x))*log(e*sqrt(x) + d) - 300*(3*b^2*d^2*e^4*n*x^2 + 6*b^2*d^4*e^2*n*x + 2*(
b^2*e^6*n - 6*a*b*e^6)*x^3)*log(c) - 12*(735*b^2*d^5*e*n^2 - 300*a*b*d^5*e*n + 2*(11*b^2*d*e^5*n^2 - 30*a*b*d*
e^5*n)*x^2 + 5*(19*b^2*d^3*e^3*n^2 - 20*a*b*d^3*e^3*n)*x - 20*(3*b^2*d*e^5*n*x^2 + 5*b^2*d^3*e^3*n*x + 15*b^2*
d^5*e*n)*log(c))*sqrt(x))/e^6

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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Giac [B]  time = 1.36998, size = 1619, normalized size = 3.37 \begin{align*} \text{result too large to display} \end{align*}

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*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d)^2 - 10800*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x)*e +
d)^2 + 27000*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d)^2 - 36000*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt(
x)*e + d)^2 + 27000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d)^2 - 10800*(sqrt(x)*e + d)*d^5*e^(-4)*log(s
qrt(x)*e + d)^2 - 600*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d) + 4320*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x
)*e + d) - 13500*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d) + 24000*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt
(x)*e + d) - 27000*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d) + 21600*(sqrt(x)*e + d)*d^5*e^(-4)*log(sqrt
(x)*e + d) + 100*(sqrt(x)*e + d)^6*e^(-4) - 864*(sqrt(x)*e + d)^5*d*e^(-4) + 3375*(sqrt(x)*e + d)^4*d^2*e^(-4)
- 8000*(sqrt(x)*e + d)^3*d^3*e^(-4) + 13500*(sqrt(x)*e + d)^2*d^4*e^(-4) - 21600*(sqrt(x)*e + d)*d^5*e^(-4))*
b^2*n^2*e^(-1) + 60*(60*(sqrt(x)*e + d)^6*e^(-4)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(
x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt(x)*e + d) - 1200*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt(x
)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)*d^5*e^(-4)*log(sqrt(x)*e
+ d) - 10*(sqrt(x)*e + d)^6*e^(-4) + 72*(sqrt(x)*e + d)^5*d*e^(-4) - 225*(sqrt(x)*e + d)^4*d^2*e^(-4) + 400*(s
qrt(x)*e + d)^3*d^3*e^(-4) - 450*(sqrt(x)*e + d)^2*d^4*e^(-4) + 360*(sqrt(x)*e + d)*d^5*e^(-4))*b^2*n*e^(-1)*l
og(c) + 1800*((sqrt(x)*e + d)^6 - 6*(sqrt(x)*e + d)^5*d + 15*(sqrt(x)*e + d)^4*d^2 - 20*(sqrt(x)*e + d)^3*d^3
+ 15*(sqrt(x)*e + d)^2*d^4 - 6*(sqrt(x)*e + d)*d^5)*b^2*e^(-5)*log(c)^2 + 60*(60*(sqrt(x)*e + d)^6*e^(-4)*log(
sqrt(x)*e + d) - 360*(sqrt(x)*e + d)^5*d*e^(-4)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-4)*log(sqrt
(x)*e + d) - 1200*(sqrt(x)*e + d)^3*d^3*e^(-4)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-4)*log(sqrt(
x)*e + d) - 360*(sqrt(x)*e + d)*d^5*e^(-4)*log(sqrt(x)*e + d) - 10*(sqrt(x)*e + d)^6*e^(-4) + 72*(sqrt(x)*e +
d)^5*d*e^(-4) - 225*(sqrt(x)*e + d)^4*d^2*e^(-4) + 400*(sqrt(x)*e + d)^3*d^3*e^(-4) - 450*(sqrt(x)*e + d)^2*d^
4*e^(-4) + 360*(sqrt(x)*e + d)*d^5*e^(-4))*a*b*n*e^(-1) + 3600*((sqrt(x)*e + d)^6 - 6*(sqrt(x)*e + d)^5*d + 15
*(sqrt(x)*e + d)^4*d^2 - 20*(sqrt(x)*e + d)^3*d^3 + 15*(sqrt(x)*e + d)^2*d^4 - 6*(sqrt(x)*e + d)*d^5)*a*b*e^(-
5)*log(c) + 1800*((sqrt(x)*e + d)^6 - 6*(sqrt(x)*e + d)^5*d + 15*(sqrt(x)*e + d)^4*d^2 - 20*(sqrt(x)*e + d)^3*
d^3 + 15*(sqrt(x)*e + d)^2*d^4 - 6*(sqrt(x)*e + d)*d^5)*a^2*e^(-5))*e^(-1)