### 3.109 $$\int \sqrt{x} \log (-1+4 x+4 \sqrt{(-1+x) x}) \, dx$$

Optimal. Leaf size=158 $-\frac{2 x^{3/2}}{9}-\frac{11 \sqrt{x^2-x}}{12 \sqrt{x}}-\frac{2 \left (x^2-x\right )^{3/2}}{9 x^{3/2}}+\frac{2}{3} x^{3/2} \log \left (4 \sqrt{x^2-x}+4 x-1\right )+\frac{\sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{24 \sqrt{2} \sqrt{x-1} \sqrt{x}}+\frac{\sqrt{x}}{12}-\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{24 \sqrt{2}}$

[Out]

Sqrt[x]/12 - (2*x^(3/2))/9 - (11*Sqrt[-x + x^2])/(12*Sqrt[x]) - (2*(-x + x^2)^(3/2))/(9*x^(3/2)) + (Sqrt[-x +
x^2]*ArcTan[(2*Sqrt[2]*Sqrt[-1 + x])/3])/(24*Sqrt[2]*Sqrt[-1 + x]*Sqrt[x]) - ArcTan[2*Sqrt[2]*Sqrt[x]]/(24*Sqr
t[2]) + (2*x^(3/2)*Log[-1 + 4*x + 4*Sqrt[-x + x^2]])/3

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Rubi [A]  time = 0.431156, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 12, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.522, Rules used = {2537, 2535, 6733, 6742, 203, 1588, 2000, 1146, 444, 50, 63, 204} $-\frac{2 x^{3/2}}{9}-\frac{11 \sqrt{x^2-x}}{12 \sqrt{x}}-\frac{2 \left (x^2-x\right )^{3/2}}{9 x^{3/2}}+\frac{2}{3} x^{3/2} \log \left (4 \sqrt{x^2-x}+4 x-1\right )+\frac{\sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{24 \sqrt{2} \sqrt{x-1} \sqrt{x}}+\frac{\sqrt{x}}{12}-\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{24 \sqrt{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[x]*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]],x]

[Out]

Sqrt[x]/12 - (2*x^(3/2))/9 - (11*Sqrt[-x + x^2])/(12*Sqrt[x]) - (2*(-x + x^2)^(3/2))/(9*x^(3/2)) + (Sqrt[-x +
x^2]*ArcTan[(2*Sqrt[2]*Sqrt[-1 + x])/3])/(24*Sqrt[2]*Sqrt[-1 + x]*Sqrt[x]) - ArcTan[2*Sqrt[2]*Sqrt[x]]/(24*Sqr
t[2]) + (2*x^(3/2)*Log[-1 + 4*x + 4*Sqrt[-x + x^2]])/3

Rule 2537

Int[Log[(d_.) + (f_.)*Sqrt[u_] + (e_.)*(x_)]*(v_.), x_Symbol] :> Int[v*Log[d + e*x + f*Sqrt[ExpandToSum[u, x]]
], x] /; FreeQ[{d, e, f}, x] && QuadraticQ[u, x] &&  !QuadraticMatchQ[u, x] && (EqQ[v, 1] || MatchQ[v, ((g_.)*
x)^(m_.) /; FreeQ[{g, m}, x]])

Rule 2535

Int[Log[(d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]]*((g_.)*(x_))^(m_.), x_Symbol] :> S
imp[((g*x)^(m + 1)*Log[d + e*x + f*Sqrt[a + b*x + c*x^2]])/(g*(m + 1)), x] + Dist[(f^2*(b^2 - 4*a*c))/(2*g*(m
+ 1)), Int[(g*x)^(m + 1)/((2*d*e - b*f^2)*(a + b*x + c*x^2) - f*(b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x +
c*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[e^2 - c*f^2, 0] && NeQ[m, -1] && IntegerQ[2*m]

Rule 6733

Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rule 1146

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[(b*x^2 + c*x^4)^FracPart
[p]/(x^(2*FracPart[p])*(b + c*x^2)^FracPart[p]), Int[x^(2*p)*(d + e*x^2)^q*(b + c*x^2)^p, x], x] /; FreeQ[{b,
c, d, e, p, q}, x] &&  !IntegerQ[p]

Rule 444

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{x} \log \left (-1+4 x+4 \sqrt{(-1+x) x}\right ) \, dx &=\int \sqrt{x} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right ) \, dx\\ &=\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{16}{3} \int \frac{x^{3/2}}{-4 (1+2 x) \sqrt{-x+x^2}+8 \left (-x+x^2\right )} \, dx\\ &=\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{32}{3} \operatorname{Subst}\left (\int \frac{x^4}{-4 \left (1+2 x^2\right ) \sqrt{-x^2+x^4}+8 \left (-x^2+x^4\right )} \, dx,x,\sqrt{x}\right )\\ &=\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{32}{3} \operatorname{Subst}\left (\int \left (\frac{1}{128}-\frac{x^2}{16}-\frac{1}{128 \left (1+8 x^2\right )}-\frac{x^2}{12 \sqrt{-x^2+x^4}}-\frac{1}{16} \sqrt{-x^2+x^4}+\frac{\sqrt{-x^2+x^4}}{48 \left (-1-8 x^2\right )}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{12}-\frac{2 x^{3/2}}{9}+\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{1}{12} \operatorname{Subst}\left (\int \frac{1}{1+8 x^2} \, dx,x,\sqrt{x}\right )+\frac{2}{9} \operatorname{Subst}\left (\int \frac{\sqrt{-x^2+x^4}}{-1-8 x^2} \, dx,x,\sqrt{x}\right )-\frac{2}{3} \operatorname{Subst}\left (\int \sqrt{-x^2+x^4} \, dx,x,\sqrt{x}\right )-\frac{8}{9} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-x^2+x^4}} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{12}-\frac{2 x^{3/2}}{9}-\frac{8 \sqrt{-x+x^2}}{9 \sqrt{x}}-\frac{2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}-\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{24 \sqrt{2}}+\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{\left (2 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{x \sqrt{-1+x^2}}{-1-8 x^2} \, dx,x,\sqrt{x}\right )}{9 \sqrt{-1+x} \sqrt{x}}\\ &=\frac{\sqrt{x}}{12}-\frac{2 x^{3/2}}{9}-\frac{8 \sqrt{-x+x^2}}{9 \sqrt{x}}-\frac{2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}-\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{24 \sqrt{2}}+\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )+\frac{\sqrt{-x+x^2} \operatorname{Subst}\left (\int \frac{\sqrt{-1+x}}{-1-8 x} \, dx,x,x\right )}{9 \sqrt{-1+x} \sqrt{x}}\\ &=\frac{\sqrt{x}}{12}-\frac{2 x^{3/2}}{9}-\frac{11 \sqrt{-x+x^2}}{12 \sqrt{x}}-\frac{2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}-\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{24 \sqrt{2}}+\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{\sqrt{-x+x^2} \operatorname{Subst}\left (\int \frac{1}{(-1-8 x) \sqrt{-1+x}} \, dx,x,x\right )}{8 \sqrt{-1+x} \sqrt{x}}\\ &=\frac{\sqrt{x}}{12}-\frac{2 x^{3/2}}{9}-\frac{11 \sqrt{-x+x^2}}{12 \sqrt{x}}-\frac{2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}-\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{24 \sqrt{2}}+\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )-\frac{\sqrt{-x+x^2} \operatorname{Subst}\left (\int \frac{1}{-9-8 x^2} \, dx,x,\sqrt{-1+x}\right )}{4 \sqrt{-1+x} \sqrt{x}}\\ &=\frac{\sqrt{x}}{12}-\frac{2 x^{3/2}}{9}-\frac{11 \sqrt{-x+x^2}}{12 \sqrt{x}}-\frac{2 \left (-x+x^2\right )^{3/2}}{9 x^{3/2}}+\frac{\sqrt{-x+x^2} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{-1+x}\right )}{24 \sqrt{2} \sqrt{-1+x} \sqrt{x}}-\frac{\tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )}{24 \sqrt{2}}+\frac{2}{3} x^{3/2} \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )\\ \end{align*}

Mathematica [C]  time = 0.640654, size = 209, normalized size = 1.32 $\frac{1}{576} \left (-128 x^{3/2}+384 x^{3/2} \log \left (4 x+4 \sqrt{(x-1) x}-1\right )-128 \sqrt{(x-1) x} \sqrt{x}+48 \sqrt{x}-\frac{400 \sqrt{(x-1) x}}{\sqrt{x}}+6 i \sqrt{2} \log \left (4 (8 x+1)^2\right )-3 i \sqrt{2} \log \left ((8 x+1) \left (-10 x-6 \sqrt{(x-1) x}+1\right )\right )-3 i \sqrt{2} \log \left ((8 x+1) \left (-10 x+6 \sqrt{(x-1) x}+1\right )\right )-12 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )+12 \sqrt{2} \tan ^{-1}\left (\frac{2 \sqrt{2} \sqrt{(x-1) x}}{3 \sqrt{x}}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[x]*Log[-1 + 4*x + 4*Sqrt[(-1 + x)*x]],x]

[Out]

(48*Sqrt[x] - 128*x^(3/2) - (400*Sqrt[(-1 + x)*x])/Sqrt[x] - 128*Sqrt[x]*Sqrt[(-1 + x)*x] - 12*Sqrt[2]*ArcTan[
2*Sqrt[2]*Sqrt[x]] + 12*Sqrt[2]*ArcTan[(2*Sqrt[2]*Sqrt[(-1 + x)*x])/(3*Sqrt[x])] + (6*I)*Sqrt[2]*Log[4*(1 + 8*
x)^2] - (3*I)*Sqrt[2]*Log[(1 + 8*x)*(1 - 10*x - 6*Sqrt[(-1 + x)*x])] + 384*x^(3/2)*Log[-1 + 4*x + 4*Sqrt[(-1 +
x)*x]] - (3*I)*Sqrt[2]*Log[(1 + 8*x)*(1 - 10*x + 6*Sqrt[(-1 + x)*x])])/576

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Maple [F]  time = 0.007, size = 0, normalized size = 0. \begin{align*} \int \sqrt{x}\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ) \, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*ln(-1+4*x+4*((-1+x)*x)^(1/2)),x)

[Out]

int(x^(1/2)*ln(-1+4*x+4*((-1+x)*x)^(1/2)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} \log \left (4 \, \sqrt{x - 1} \sqrt{x} + 4 \, x - 1\right ) - \frac{4}{9} \, x^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{x} + \int \frac{2 \, x^{2} + x}{3 \,{\left (4 \, x^{\frac{5}{2}} + 4 \,{\left (x^{2} - x\right )} \sqrt{x - 1} - 5 \, x^{\frac{3}{2}} + \sqrt{x}\right )}}\,{d x} + \frac{1}{3} \, \log \left (\sqrt{x} + 1\right ) - \frac{1}{3} \, \log \left (\sqrt{x} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*log(-1+4*x+4*((-1+x)*x)^(1/2)),x, algorithm="maxima")

[Out]

2/3*x^(3/2)*log(4*sqrt(x - 1)*sqrt(x) + 4*x - 1) - 4/9*x^(3/2) - 2/3*sqrt(x) + integrate(1/3*(2*x^2 + x)/(4*x^
(5/2) + 4*(x^2 - x)*sqrt(x - 1) - 5*x^(3/2) + sqrt(x)), x) + 1/3*log(sqrt(x) + 1) - 1/3*log(sqrt(x) - 1)

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Fricas [A]  time = 2.34709, size = 286, normalized size = 1.81 \begin{align*} \frac{96 \, x^{\frac{5}{2}} \log \left (4 \, x + 4 \, \sqrt{x^{2} - x} - 1\right ) - 3 \, \sqrt{2} x \arctan \left (2 \, \sqrt{2} \sqrt{x}\right ) - 3 \, \sqrt{2} x \arctan \left (\frac{3 \, \sqrt{2} \sqrt{x}}{4 \, \sqrt{x^{2} - x}}\right ) - 4 \, \sqrt{x^{2} - x}{\left (8 \, x + 25\right )} \sqrt{x} - 4 \,{\left (8 \, x^{2} - 3 \, x\right )} \sqrt{x}}{144 \, x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*log(-1+4*x+4*((-1+x)*x)^(1/2)),x, algorithm="fricas")

[Out]

1/144*(96*x^(5/2)*log(4*x + 4*sqrt(x^2 - x) - 1) - 3*sqrt(2)*x*arctan(2*sqrt(2)*sqrt(x)) - 3*sqrt(2)*x*arctan(
3/4*sqrt(2)*sqrt(x)/sqrt(x^2 - x)) - 4*sqrt(x^2 - x)*(8*x + 25)*sqrt(x) - 4*(8*x^2 - 3*x)*sqrt(x))/x

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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**(1/2)*ln(-1+4*x+4*((-1+x)*x)**(1/2)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \log \left (4 \, x + 4 \, \sqrt{{\left (x - 1\right )} x} - 1\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)*log(-1+4*x+4*((-1+x)*x)^(1/2)),x, algorithm="giac")

[Out]

integrate(sqrt(x)*log(4*x + 4*sqrt((x - 1)*x) - 1), x)