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

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

[Out]

1/2*b^2*d^4*n^2*ln(d+e*x^(1/2))^2/e^4-b*d^4*n*ln(d+e*x^(1/2))*(a+b*ln(c*(d+e*x^(1/2))^n))/e^4+1/2*x^2*(a+b*ln(
c*(d+e*x^(1/2))^n))^2-4*b^2*d^3*n^2*x^(1/2)/e^3+4*b*d^3*n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))/e^4+3/2*b^
2*d^2*n^2*(d+e*x^(1/2))^2/e^4-3*b*d^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^2/e^4-4/9*b^2*d*n^2*(d+e*x^(
1/2))^3/e^4+4/3*b*d*n*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^3/e^4+1/16*b^2*n^2*(d+e*x^(1/2))^4/e^4-1/4*b*n
*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^4/e^4

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Rubi [A]  time = 0.36, 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 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 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 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]

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 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 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])

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*}

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Mathematica [A]  time = 0.21, size = 223, normalized size = 0.65 \[ \frac {e \sqrt {x} \left (72 a^2 e^3 x^{3/2}+12 a b n \left (12 d^3-6 d^2 e \sqrt {x}+4 d e^2 x-3 e^3 x^{3/2}\right )+b^2 n^2 \left (-300 d^3+78 d^2 e \sqrt {x}-28 d e^2 x+9 e^3 x^{3/2}\right )\right )-12 b \left (12 a \left (d^4-e^4 x^2\right )+b n \left (-25 d^4-12 d^3 e \sqrt {x}+6 d^2 e^2 x-4 d e^3 x^{3/2}+3 e^4 x^2\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-72 b^2 \left (d^4-e^4 x^2\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )}{144 e^4} \]

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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fricas [A]  time = 0.45, size = 357, normalized size = 1.04 \[ \frac {72 \, b^{2} e^{4} x^{2} \log \relax (c)^{2} + 9 \, {\left (b^{2} e^{4} n^{2} - 4 \, a b e^{4} n + 8 \, a^{2} e^{4}\right )} x^{2} + 72 \, {\left (b^{2} e^{4} n^{2} x^{2} - b^{2} d^{4} n^{2}\right )} \log \left (e \sqrt {x} + d\right )^{2} + 6 \, {\left (13 \, b^{2} d^{2} e^{2} n^{2} - 12 \, a b d^{2} e^{2} n\right )} x - 12 \, {\left (6 \, b^{2} d^{2} e^{2} n^{2} x - 25 \, b^{2} d^{4} n^{2} + 12 \, a b d^{4} n + 3 \, {\left (b^{2} e^{4} n^{2} - 4 \, a b e^{4} n\right )} x^{2} - 12 \, {\left (b^{2} e^{4} n x^{2} - b^{2} d^{4} n\right )} \log \relax (c) - 4 \, {\left (b^{2} d e^{3} n^{2} x + 3 \, b^{2} d^{3} e n^{2}\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right ) - 36 \, {\left (2 \, b^{2} d^{2} e^{2} n x + {\left (b^{2} e^{4} n - 4 \, a b e^{4}\right )} x^{2}\right )} \log \relax (c) - 4 \, {\left (75 \, b^{2} d^{3} e n^{2} - 36 \, a b d^{3} e n + {\left (7 \, b^{2} d e^{3} n^{2} - 12 \, a b d e^{3} n\right )} x - 12 \, {\left (b^{2} d e^{3} n x + 3 \, b^{2} d^{3} e n\right )} \log \relax (c)\right )} \sqrt {x}}{144 \, e^{4}} \]

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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giac [B]  time = 0.25, size = 642, normalized size = 1.88 \[ \frac {1}{144} \, {\left (72 \, b^{2} x^{2} e \log \relax (c)^{2} + 144 \, a b x^{2} e \log \relax (c) + {\left (72 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right )^{2} - 288 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right )^{2} + 432 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right )^{2} - 288 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right )^{2} - 36 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + 192 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 432 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + 576 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + 9 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} - 64 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} + 216 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} - 576 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )}\right )} b^{2} n^{2} + 72 \, a^{2} x^{2} e + 12 \, {\left (12 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 48 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + 72 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 48 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 3 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} + 16 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} - 36 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} + 48 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )}\right )} b^{2} n \log \relax (c) + 12 \, {\left (12 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 48 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) + 72 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 48 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )} \log \left (\sqrt {x} e + d\right ) - 3 \, {\left (\sqrt {x} e + d\right )}^{4} e^{\left (-3\right )} + 16 \, {\left (\sqrt {x} e + d\right )}^{3} d e^{\left (-3\right )} - 36 \, {\left (\sqrt {x} e + d\right )}^{2} d^{2} e^{\left (-3\right )} + 48 \, {\left (\sqrt {x} e + d\right )} d^{3} e^{\left (-3\right )}\right )} a b n\right )} e^{\left (-1\right )} \]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a \right )^{2} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.55, size = 257, normalized size = 0.75 \[ \frac {1}{2} \, b^{2} x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {1}{12} \, a b e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} + a b x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{2} x^{2} - \frac {1}{144} \, {\left (12 \, e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - \frac {{\left (9 \, e^{4} x^{2} + 72 \, d^{4} \log \left (e \sqrt {x} + d\right )^{2} - 28 \, d e^{3} x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x + 300 \, d^{4} \log \left (e \sqrt {x} + d\right ) - 300 \, d^{3} e \sqrt {x}\right )} n^{2}}{e^{4}}\right )} b^{2} \]

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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mupad [B]  time = 0.56, size = 420, normalized size = 1.23 \[ x\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{3\,e}\right )}{2\,e}+\frac {b^2\,d^2\,n^2}{4\,e^2}\right )-x^{3/2}\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{3\,e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{9\,e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (\frac {b^2\,x^2}{2}-\frac {b^2\,d^4}{2\,e^4}\right )+x^2\,\left (\frac {a^2}{2}-\frac {a\,b\,n}{4}+\frac {b^2\,n^2}{16}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (x^{3/2}\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{3\,e}-\frac {4\,a\,b\,d}{3\,e}\right )-\frac {b\,x^2\,\left (4\,a-b\,n\right )}{4}+\frac {d^2\,\sqrt {x}\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )}{e^2}-\frac {d\,x\,\left (\frac {b\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )}{2\,e}\right )-\sqrt {x}\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^2-a\,b\,n+\frac {b^2\,n^2}{4}\right )}{e}-\frac {d\,\left (6\,a^2-b^2\,n^2\right )}{3\,e}\right )}{e}+\frac {b^2\,d^2\,n^2}{2\,e^2}\right )}{e}+\frac {b^2\,d^3\,n^2}{e^3}\right )+\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (25\,b^2\,d^4\,n^2-12\,a\,b\,d^4\,n\right )}{12\,e^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*((d*((d*(2*a^2 + (b^2*n^2)/4 - a*b*n))/e - (d*(6*a^2 - b^2*n^2))/(3*e)))/(2*e) + (b^2*d^2*n^2)/(4*e^2)) - x^
(3/2)*((d*(2*a^2 + (b^2*n^2)/4 - a*b*n))/(3*e) - (d*(6*a^2 - b^2*n^2))/(9*e)) + log(c*(d + e*x^(1/2))^n)^2*((b
^2*x^2)/2 - (b^2*d^4)/(2*e^4)) + x^2*(a^2/2 + (b^2*n^2)/16 - (a*b*n)/4) - log(c*(d + e*x^(1/2))^n)*(x^(3/2)*((
b*d*(4*a - b*n))/(3*e) - (4*a*b*d)/(3*e)) - (b*x^2*(4*a - b*n))/4 + (d^2*x^(1/2)*((b*d*(4*a - b*n))/e - (4*a*b
*d)/e))/e^2 - (d*x*((b*d*(4*a - b*n))/e - (4*a*b*d)/e))/(2*e)) - x^(1/2)*((d*((d*((d*(2*a^2 + (b^2*n^2)/4 - a*
b*n))/e - (d*(6*a^2 - b^2*n^2))/(3*e)))/e + (b^2*d^2*n^2)/(2*e^2)))/e + (b^2*d^3*n^2)/e^3) + (log(d + e*x^(1/2
))*(25*b^2*d^4*n^2 - 12*a*b*d^4*n))/(12*e^4)

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

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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