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

Optimal. Leaf size=342 \[ -\frac{b d^4 n \log \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{e^4}+\frac{4 b d^3 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{e^4}-\frac{3 b d^2 n \left (d+e \sqrt{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{e^4}+\frac{4 b d n \left (d+e \sqrt{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{3 e^4}-\frac{b n \left (d+e \sqrt{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{4 e^4}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2-\frac{4 b^2 d^3 n^2 \sqrt{x}}{e^3}+\frac{3 b^2 d^2 n^2 \left (d+e \sqrt{x}\right )^2}{2 e^4}+\frac{b^2 d^4 n^2 \log ^2\left (d+e \sqrt{x}\right )}{2 e^4}-\frac{4 b^2 d n^2 \left (d+e \sqrt{x}\right )^3}{9 e^4}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^4}{16 e^4} \]

[Out]

(3*b^2*d^2*n^2*(d + e*Sqrt[x])^2)/(2*e^4) - (4*b^2*d*n^2*(d + e*Sqrt[x])^3)/(9*e^4) + (b^2*n^2*(d + e*Sqrt[x])
^4)/(16*e^4) - (4*b^2*d^3*n^2*Sqrt[x])/e^3 + (b^2*d^4*n^2*Log[d + e*Sqrt[x]]^2)/(2*e^4) + (4*b*d^3*n*(d + e*Sq
rt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^4 - (3*b*d^2*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/
e^4 + (4*b*d*n*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*e^4) - (b*n*(d + e*Sqrt[x])^4*(a + b*Log
[c*(d + e*Sqrt[x])^n]))/(4*e^4) - (b*d^4*n*Log[d + e*Sqrt[x]]*(a + b*Log[c*(d + e*Sqrt[x])^n]))/e^4 + (x^2*(a
+ b*Log[c*(d + e*Sqrt[x])^n])^2)/2

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Rubi [A]  time = 0.361032, antiderivative size = 263, normalized size of antiderivative = 0.77, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2454, 2398, 2411, 43, 2334, 12, 14, 2301} \[ \frac{1}{12} b n \left (\frac{48 d^3 \left (d+e \sqrt{x}\right )}{e^4}-\frac{36 d^2 \left (d+e \sqrt{x}\right )^2}{e^4}-\frac{12 d^4 \log \left (d+e \sqrt{x}\right )}{e^4}+\frac{16 d \left (d+e \sqrt{x}\right )^3}{e^4}-\frac{3 \left (d+e \sqrt{x}\right )^4}{e^4}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2-\frac{4 b^2 d^3 n^2 \sqrt{x}}{e^3}+\frac{3 b^2 d^2 n^2 \left (d+e \sqrt{x}\right )^2}{2 e^4}+\frac{b^2 d^4 n^2 \log ^2\left (d+e \sqrt{x}\right )}{2 e^4}-\frac{4 b^2 d n^2 \left (d+e \sqrt{x}\right )^3}{9 e^4}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^4}{16 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b^2*d^2*n^2*(d + e*Sqrt[x])^2)/(2*e^4) - (4*b^2*d*n^2*(d + e*Sqrt[x])^3)/(9*e^4) + (b^2*n^2*(d + e*Sqrt[x])
^4)/(16*e^4) - (4*b^2*d^3*n^2*Sqrt[x])/e^3 + (b^2*d^4*n^2*Log[d + e*Sqrt[x]]^2)/(2*e^4) + (b*n*((48*d^3*(d + e
*Sqrt[x]))/e^4 - (36*d^2*(d + e*Sqrt[x])^2)/e^4 + (16*d*(d + e*Sqrt[x])^3)/e^4 - (3*(d + e*Sqrt[x])^4)/e^4 - (
12*d^4*Log[d + e*Sqrt[x]])/e^4)*(a + b*Log[c*(d + e*Sqrt[x])^n]))/12 + (x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2
)/2

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 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2-(b e n) \operatorname{Subst}\left (\int \frac{x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx,x,\sqrt{x}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2-(b n) \operatorname{Subst}\left (\int \frac{\left (-\frac{d}{e}+\frac{x}{e}\right )^4 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e \sqrt{x}\right )\\ &=\frac{1}{12} b n \left (\frac{48 d^3 \left (d+e \sqrt{x}\right )}{e^4}-\frac{36 d^2 \left (d+e \sqrt{x}\right )^2}{e^4}+\frac{16 d \left (d+e \sqrt{x}\right )^3}{e^4}-\frac{3 \left (d+e \sqrt{x}\right )^4}{e^4}-\frac{12 d^4 \log \left (d+e \sqrt{x}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2+\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3\right )+12 d^4 \log (x)}{12 e^4 x} \, dx,x,d+e \sqrt{x}\right )\\ &=\frac{1}{12} b n \left (\frac{48 d^3 \left (d+e \sqrt{x}\right )}{e^4}-\frac{36 d^2 \left (d+e \sqrt{x}\right )^2}{e^4}+\frac{16 d \left (d+e \sqrt{x}\right )^3}{e^4}-\frac{3 \left (d+e \sqrt{x}\right )^4}{e^4}-\frac{12 d^4 \log \left (d+e \sqrt{x}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{1}{2} x^2 \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 (-48 d^3+36 d^2 x-16 d x^2+3 x^3\right )+12 d^4 \log (x)}{x} \, dx,x,d+e \sqrt{x}\right )}{12 e^4}\\ &=\frac{1}{12} b n \left (\frac{48 d^3 \left (d+e \sqrt{x}\right )}{e^4}-\frac{36 d^2 \left (d+e \sqrt{x}\right )^2}{e^4}+\frac{16 d \left (d+e \sqrt{x}\right )^3}{e^4}-\frac{3 \left (d+e \sqrt{x}\right )^4}{e^4}-\frac{12 d^4 \log \left (d+e \sqrt{x}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{1}{2} x^2 \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 (-48 d^3+36 d^2 x-16 d x^2+3 x^3+\frac{12 d^4 \log (x)}{x}\right ) \, dx,x,d+e \sqrt{x}\right )}{12 e^4}\\ &=\frac{3 b^2 d^2 n^2 \left (d+e \sqrt{x}\right )^2}{2 e^4}-\frac{4 b^2 d n^2 \left (d+e \sqrt{x}\right )^3}{9 e^4}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^4}{16 e^4}-\frac{4 b^2 d^3 n^2 \sqrt{x}}{e^3}+\frac{1}{12} b n \left (\frac{48 d^3 \left (d+e \sqrt{x}\right )}{e^4}-\frac{36 d^2 \left (d+e \sqrt{x}\right )^2}{e^4}+\frac{16 d \left (d+e \sqrt{x}\right )^3}{e^4}-\frac{3 \left (d+e \sqrt{x}\right )^4}{e^4}-\frac{12 d^4 \log \left (d+e \sqrt{x}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2+\frac{\left (b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,d+e \sqrt{x}\right )}{e^4}\\ &=\frac{3 b^2 d^2 n^2 \left (d+e \sqrt{x}\right )^2}{2 e^4}-\frac{4 b^2 d n^2 \left (d+e \sqrt{x}\right )^3}{9 e^4}+\frac{b^2 n^2 \left (d+e \sqrt{x}\right )^4}{16 e^4}-\frac{4 b^2 d^3 n^2 \sqrt{x}}{e^3}+\frac{b^2 d^4 n^2 \log ^2\left (d+e \sqrt{x}\right )}{2 e^4}+\frac{1}{12} b n \left (\frac{48 d^3 \left (d+e \sqrt{x}\right )}{e^4}-\frac{36 d^2 \left (d+e \sqrt{x}\right )^2}{e^4}+\frac{16 d \left (d+e \sqrt{x}\right )^3}{e^4}-\frac{3 \left (d+e \sqrt{x}\right )^4}{e^4}-\frac{12 d^4 \log \left (d+e \sqrt{x}\right )}{e^4}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2\\ \end{align*}

Mathematica [A]  time = 0.192595, size = 223, normalized size = 0.65 \[ \frac{e \sqrt{x} \left (72 a^2 e^3 x^{3/2}+12 a b n \left (-6 d^2 e \sqrt{x}+12 d^3+4 d e^2 x-3 e^3 x^{3/2}\right )+b^2 n^2 \left (78 d^2 e \sqrt{x}-300 d^3-28 d e^2 x+9 e^3 x^{3/2}\right )\right )-12 b \left (12 a \left (d^4-e^4 x^2\right )+b n \left (6 d^2 e^2 x-12 d^3 e \sqrt{x}-25 d^4-4 d e^3 x^{3/2}+3 e^4 x^2\right )\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )-72 b^2 \left (d^4-e^4 x^2\right ) \log ^2\left (c \left (d+e \sqrt{x}\right )^n\right )}{144 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[x]*(72*a^2*e^3*x^(3/2) + 12*a*b*n*(12*d^3 - 6*d^2*e*Sqrt[x] + 4*d*e^2*x - 3*e^3*x^(3/2)) + b^2*n^2*(-3
00*d^3 + 78*d^2*e*Sqrt[x] - 28*d*e^2*x + 9*e^3*x^(3/2))) - 12*b*(12*a*(d^4 - e^4*x^2) + b*n*(-25*d^4 - 12*d^3*
e*Sqrt[x] + 6*d^2*e^2*x - 4*d*e^3*x^(3/2) + 3*e^4*x^2))*Log[c*(d + e*Sqrt[x])^n] - 72*b^2*(d^4 - e^4*x^2)*Log[
c*(d + e*Sqrt[x])^n]^2)/(144*e^4)

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int x \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*(a+b*ln(c*(d+e*x^(1/2))^n))^2,x)

[Out]

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

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Maxima [A]  time = 1.06406, size = 347, normalized size = 1.01 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} - \frac{1}{12} \, a b e n{\left (\frac{12 \, d^{4} \log \left (e \sqrt{x} + d\right )}{e^{5}} + \frac{3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac{3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt{x}}{e^{4}}\right )} + a b x^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + \frac{1}{2} \, a^{2} x^{2} - \frac{1}{144} \,{\left (12 \, e n{\left (\frac{12 \, d^{4} \log \left (e \sqrt{x} + d\right )}{e^{5}} + \frac{3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac{3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt{x}}{e^{4}}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) - \frac{{\left (9 \, e^{4} x^{2} + 72 \, d^{4} \log \left (e \sqrt{x} + d\right )^{2} - 28 \, d e^{3} x^{\frac{3}{2}} + 78 \, d^{2} e^{2} x + 300 \, d^{4} \log \left (e \sqrt{x} + d\right ) - 300 \, d^{3} e \sqrt{x}\right )} n^{2}}{e^{4}}\right )} b^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2*log((e*sqrt(x) + d)^n*c)^2 - 1/12*a*b*e*n*(12*d^4*log(e*sqrt(x) + d)/e^5 + (3*e^3*x^2 - 4*d*e^2*x^
(3/2) + 6*d^2*e*x - 12*d^3*sqrt(x))/e^4) + a*b*x^2*log((e*sqrt(x) + d)^n*c) + 1/2*a^2*x^2 - 1/144*(12*e*n*(12*
d^4*log(e*sqrt(x) + d)/e^5 + (3*e^3*x^2 - 4*d*e^2*x^(3/2) + 6*d^2*e*x - 12*d^3*sqrt(x))/e^4)*log((e*sqrt(x) +
d)^n*c) - (9*e^4*x^2 + 72*d^4*log(e*sqrt(x) + d)^2 - 28*d*e^3*x^(3/2) + 78*d^2*e^2*x + 300*d^4*log(e*sqrt(x) +
d) - 300*d^3*e*sqrt(x))*n^2/e^4)*b^2

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Fricas [A]  time = 2.24195, size = 788, normalized size = 2.3 \begin{align*} \frac{72 \, b^{2} e^{4} x^{2} \log \left (c\right )^{2} + 9 \,{\left (b^{2} e^{4} n^{2} - 4 \, a b e^{4} n + 8 \, a^{2} e^{4}\right )} x^{2} + 72 \,{\left (b^{2} e^{4} n^{2} x^{2} - b^{2} d^{4} n^{2}\right )} \log \left (e \sqrt{x} + d\right )^{2} + 6 \,{\left (13 \, b^{2} d^{2} e^{2} n^{2} - 12 \, a b d^{2} e^{2} n\right )} x - 12 \,{\left (6 \, b^{2} d^{2} e^{2} n^{2} x - 25 \, b^{2} d^{4} n^{2} + 12 \, a b d^{4} n + 3 \,{\left (b^{2} e^{4} n^{2} - 4 \, a b e^{4} n\right )} x^{2} - 12 \,{\left (b^{2} e^{4} n x^{2} - b^{2} d^{4} n\right )} \log \left (c\right ) - 4 \,{\left (b^{2} d e^{3} n^{2} x + 3 \, b^{2} d^{3} e n^{2}\right )} \sqrt{x}\right )} \log \left (e \sqrt{x} + d\right ) - 36 \,{\left (2 \, b^{2} d^{2} e^{2} n x +{\left (b^{2} e^{4} n - 4 \, a b e^{4}\right )} x^{2}\right )} \log \left (c\right ) - 4 \,{\left (75 \, b^{2} d^{3} e n^{2} - 36 \, a b d^{3} e n +{\left (7 \, b^{2} d e^{3} n^{2} - 12 \, a b d e^{3} n\right )} x - 12 \,{\left (b^{2} d e^{3} n x + 3 \, b^{2} d^{3} e n\right )} \log \left (c\right )\right )} \sqrt{x}}{144 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(72*b^2*e^4*x^2*log(c)^2 + 9*(b^2*e^4*n^2 - 4*a*b*e^4*n + 8*a^2*e^4)*x^2 + 72*(b^2*e^4*n^2*x^2 - b^2*d^4
*n^2)*log(e*sqrt(x) + d)^2 + 6*(13*b^2*d^2*e^2*n^2 - 12*a*b*d^2*e^2*n)*x - 12*(6*b^2*d^2*e^2*n^2*x - 25*b^2*d^
4*n^2 + 12*a*b*d^4*n + 3*(b^2*e^4*n^2 - 4*a*b*e^4*n)*x^2 - 12*(b^2*e^4*n*x^2 - b^2*d^4*n)*log(c) - 4*(b^2*d*e^
3*n^2*x + 3*b^2*d^3*e*n^2)*sqrt(x))*log(e*sqrt(x) + d) - 36*(2*b^2*d^2*e^2*n*x + (b^2*e^4*n - 4*a*b*e^4)*x^2)*
log(c) - 4*(75*b^2*d^3*e*n^2 - 36*a*b*d^3*e*n + (7*b^2*d*e^3*n^2 - 12*a*b*d*e^3*n)*x - 12*(b^2*d*e^3*n*x + 3*b
^2*d^3*e*n)*log(c))*sqrt(x))/e^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \log{\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]

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

[Out]

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

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Giac [B]  time = 1.2789, size = 1073, normalized size = 3.14 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*((72*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d)^2
+ 432*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d)^2
- 36*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d) + 192*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d) - 432*(s
qrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d) + 576*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d) + 9*(sqrt(x)
*e + d)^4*e^(-2) - 64*(sqrt(x)*e + d)^3*d*e^(-2) + 216*(sqrt(x)*e + d)^2*d^2*e^(-2) - 576*(sqrt(x)*e + d)*d^3*
e^(-2))*b^2*n^2*e^(-1) + 12*(12*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)^3*d*e^(-2)*lo
g(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt
(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-2) + 16*(sqrt(x)*e + d)^3*d*e^(-2) - 36*(sqrt(x)*e + d)^2*d^2*e^(-2) + 48
*(sqrt(x)*e + d)*d^3*e^(-2))*b^2*n*e^(-1)*log(c) + 72*((sqrt(x)*e + d)^4 - 4*(sqrt(x)*e + d)^3*d + 6*(sqrt(x)*
e + d)^2*d^2 - 4*(sqrt(x)*e + d)*d^3)*b^2*e^(-3)*log(c)^2 + 12*(12*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d)
- 48*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d) - 48*
(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-2) + 16*(sqrt(x)*e + d)^3*d*e^(-2) - 3
6*(sqrt(x)*e + d)^2*d^2*e^(-2) + 48*(sqrt(x)*e + d)*d^3*e^(-2))*a*b*n*e^(-1) + 144*((sqrt(x)*e + d)^4 - 4*(sqr
t(x)*e + d)^3*d + 6*(sqrt(x)*e + d)^2*d^2 - 4*(sqrt(x)*e + d)*d^3)*a*b*e^(-3)*log(c) + 72*((sqrt(x)*e + d)^4 -
4*(sqrt(x)*e + d)^3*d + 6*(sqrt(x)*e + d)^2*d^2 - 4*(sqrt(x)*e + d)*d^3)*a^2*e^(-3))*e^(-1)