∫ x3∕2 __________________________________________________∘ −1 + √_x ∘____

Optimal. Leaf size=73 \[ \frac {3}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {3}{4} \cosh ^{-1}\left (\sqrt {x}\right ) \]

3/4*arccosh(x^(1/2))+1/2*x^(3/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2))^(1/2)+3/4*x^(1/2)*(-1+x^(1/2))^(1/2)*(1+x^(1/2

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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {329, 336, 54} \begin {gather*} \frac {1}{2} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} x^{3/2}+\frac {3}{4} \sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1} \sqrt {x}+\frac {3}{4} \cosh ^{-1}\left (\sqrt {x}\right ) \end {gather*}

(3*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*Sqrt[x])/4 + (Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*x^(3/2))/2 + (3*Arc

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[b*(x/a)]/b, x] /; FreeQ[{a,

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(2*

n - 1)*(c*x)^(m - 2*n + 1)*(a1 + b1*x^n)^(p + 1)*((a2 + b2*x^n)^(p + 1)/(b1*b2*(m + 2*n*p + 1))), x] - Dist[a1

*a2*c^(2*n)*((m - 2*n + 1)/(b1*b2*(m + 2*n*p + 1))), Int[(c*x)^(m - 2*n)*(a1 + b1*x^n)^p*(a2 + b2*x^n)^p, x],

x] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && GtQ[m, 2*n - 1] && NeQ[m +

Int[((c_.)*(x_))^(m_)*((a1_) + (b1_.)*(x_)^(n_))^(p_)*((a2_) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k =

Denominator[m]}, Dist[k/c, Subst[Int[x^(k*(m + 1) - 1)*(a1 + b1*(x^(k*n)/c^n))^p*(a2 + b2*(x^(k*n)/c^n))^p, x]

, x, (c*x)^(1/k)], x]] /; FreeQ[{a1, b1, a2, b2, c, p}, x] && EqQ[a2*b1 + a1*b2, 0] && IGtQ[2*n, 0] && Fractio

\begin {align*} \int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx &=\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {3}{4} \int \frac {\sqrt {x}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\\ &=\frac {3}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {3}{8} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}} \, dx\\ &=\frac {3}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,\sqrt {x}\right )\\ &=\frac {3}{4} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \sqrt {x}+\frac {1}{2} \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}+\frac {3}{4} \cosh ^{-1}\left (\sqrt {x}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(406\) vs. \(2(73)=146\).
time = 1.46, size = 406, normalized size = 5.56 \begin {gather*} \frac {-4 \sqrt {1+\sqrt {x}} \left (-29568+50496 \sqrt {x}+98112 x+21840 x^{3/2}-2264 x^2-3368 x^{5/2}-4752 x^3-1136 x^{7/2}\right )-4 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} \left (51216+120600 \sqrt {x}+56904 x-4016 x^{3/2}-6344 x^2-6467 x^{5/2}-3120 x^3-194 x^{7/2}\right )+\sqrt {3} \left (-4 \sqrt {-1+\sqrt {x}} \left (-29568-84416 \sqrt {x}-64000 x-7152 x^{3/2}+5624 x^2+5144 x^{5/2}+3408 x^3+656 x^{7/2}\right )-4 \left (17072-20632 \sqrt {x}-73312 x-36244 x^{3/2}-510 x^2+2452 x^{5/2}+3640 x^3+1800 x^{7/2}+112 x^4\right )\right )}{-12416+13312 \sqrt {x}+49408 x+24960 x^{3/2}+1552 x^2+\sqrt {3} \sqrt {1+\sqrt {x}} \left (7168-11264 \sqrt {x}-22016 x-5248 x^{3/2}\right )+\sqrt {-1+\sqrt {x}} \left (21504+60416 \sqrt {x}+47104 x+9088 x^{3/2}+\sqrt {3} \sqrt {1+\sqrt {x}} \left (-12416-28672 \sqrt {x}-14400 x-896 x^{3/2}\right )\right )}-3 \tanh ^{-1}\left (\frac {-1+\sqrt {-1+\sqrt {x}}}{\sqrt {3}-\sqrt {1+\sqrt {x}}}\right ) \end {gather*}

(-4*Sqrt[1 + Sqrt[x]]*(-29568 + 50496*Sqrt[x] + 98112*x + 21840*x^(3/2) - 2264*x^2 - 3368*x^(5/2) - 4752*x^3 -

1136*x^(7/2)) - 4*Sqrt[-1 + Sqrt[x]]*Sqrt[1 + Sqrt[x]]*(51216 + 120600*Sqrt[x] + 56904*x - 4016*x^(3/2) - 634

4*x^2 - 6467*x^(5/2) - 3120*x^3 - 194*x^(7/2)) + Sqrt[3]*(-4*Sqrt[-1 + Sqrt[x]]*(-29568 - 84416*Sqrt[x] - 6400

0*x - 7152*x^(3/2) + 5624*x^2 + 5144*x^(5/2) + 3408*x^3 + 656*x^(7/2)) - 4*(17072 - 20632*Sqrt[x] - 73312*x -

36244*x^(3/2) - 510*x^2 + 2452*x^(5/2) + 3640*x^3 + 1800*x^(7/2) + 112*x^4)))/(-12416 + 13312*Sqrt[x] + 49408*

x + 24960*x^(3/2) + 1552*x^2 + Sqrt[3]*Sqrt[1 + Sqrt[x]]*(7168 - 11264*Sqrt[x] - 22016*x - 5248*x^(3/2)) + Sqr

t[-1 + Sqrt[x]]*(21504 + 60416*Sqrt[x] + 47104*x + 9088*x^(3/2) + Sqrt[3]*Sqrt[1 + Sqrt[x]]*(-12416 - 28672*Sq

rt[x] - 14400*x - 896*x^(3/2)))) - 3*ArcTanh[(-1 + Sqrt[-1 + Sqrt[x]])/(Sqrt[3] - Sqrt[1 + Sqrt[x]])]

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method	result	size

derivativedivides	\(\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{4 \sqrt {x -1}}\)	\(55\)
default	\(\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {\sqrt {x}+1}\, \left (2 x^{\frac {3}{2}} \sqrt {x -1}+3 \sqrt {x}\, \sqrt {x -1}+3 \ln \left (\sqrt {x}+\sqrt {x -1}\right )\right )}{4 \sqrt {x -1}}\)	\(55\)

1/4*(-1+x^(1/2))^(1/2)*(x^(1/2)+1)^(1/2)*(2*x^(3/2)*(x-1)^(1/2)+3*x^(1/2)*(x-1)^(1/2)+3*ln(x^(1/2)+(x-1)^(1/2)

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Maxima [A]
time = 0.28, size = 37, normalized size = 0.51 \begin {gather*} \frac {1}{2} \, \sqrt {x - 1} x^{\frac {3}{2}} + \frac {3}{4} \, \sqrt {x - 1} \sqrt {x} + \frac {3}{4} \, \log \left (2 \, \sqrt {x - 1} + 2 \, \sqrt {x}\right ) \end {gather*}

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Fricas [A]
time = 1.10, size = 52, normalized size = 0.71 \begin {gather*} \frac {1}{4} \, {\left (2 \, x + 3\right )} \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {3}{8} \, \log \left (2 \, \sqrt {x} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - 2 \, x + 1\right ) \end {gather*}

1/4*(2*x + 3)*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) - 1) - 3/8*log(2*sqrt(x)*sqrt(sqrt(x) + 1)*sqrt(sqrt(x) -

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{\frac {3}{2}}}{\sqrt {\sqrt {x} - 1} \sqrt {\sqrt {x} + 1}}\, dx \end {gather*}

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Giac [A]
time = 1.77, size = 59, normalized size = 0.81 \begin {gather*} \frac {1}{4} \, {\left ({\left (2 \, {\left (\sqrt {x} + 1\right )} {\left (\sqrt {x} - 2\right )} + 9\right )} {\left (\sqrt {x} + 1\right )} - 5\right )} \sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} - \frac {3}{2} \, \log \left (\sqrt {\sqrt {x} + 1} - \sqrt {\sqrt {x} - 1}\right ) \end {gather*}

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

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Mupad [B]
time = 18.76, size = 429, normalized size = 5.88 \begin {gather*} 3\,\mathrm {atanh}\left (\frac {\sqrt {\sqrt {x}-1}-\mathrm {i}}{\sqrt {\sqrt {x}+1}-1}\right )+\frac {\frac {23\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^3}+\frac {333\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^5}+\frac {671\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^7}+\frac {671\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^9}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^9}+\frac {333\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{11}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{11}}+\frac {23\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{13}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{13}}-\frac {3\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{15}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{15}}-\frac {3\,\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}{\sqrt {\sqrt {x}+1}-1}}{1+\frac {28\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{14}}+\frac {{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {\sqrt {x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\sqrt {x}+1}-1\right )}^2}} \end {gather*}

3*atanh(((x^(1/2) - 1)^(1/2) - 1i)/((x^(1/2) + 1)^(1/2) - 1)) + ((23*((x^(1/2) - 1)^(1/2) - 1i)^3)/((x^(1/2) +

1)^(1/2) - 1)^3 + (333*((x^(1/2) - 1)^(1/2) - 1i)^5)/((x^(1/2) + 1)^(1/2) - 1)^5 + (671*((x^(1/2) - 1)^(1/2)

- 1i)^7)/((x^(1/2) + 1)^(1/2) - 1)^7 + (671*((x^(1/2) - 1)^(1/2) - 1i)^9)/((x^(1/2) + 1)^(1/2) - 1)^9 + (333*(

(x^(1/2) - 1)^(1/2) - 1i)^11)/((x^(1/2) + 1)^(1/2) - 1)^11 + (23*((x^(1/2) - 1)^(1/2) - 1i)^13)/((x^(1/2) + 1)

^(1/2) - 1)^13 - (3*((x^(1/2) - 1)^(1/2) - 1i)^15)/((x^(1/2) + 1)^(1/2) - 1)^15 - (3*((x^(1/2) - 1)^(1/2) - 1i

))/((x^(1/2) + 1)^(1/2) - 1))/((28*((x^(1/2) - 1)^(1/2) - 1i)^4)/((x^(1/2) + 1)^(1/2) - 1)^4 - (8*((x^(1/2) -

1)^(1/2) - 1i)^2)/((x^(1/2) + 1)^(1/2) - 1)^2 - (56*((x^(1/2) - 1)^(1/2) - 1i)^6)/((x^(1/2) + 1)^(1/2) - 1)^6

+ (70*((x^(1/2) - 1)^(1/2) - 1i)^8)/((x^(1/2) + 1)^(1/2) - 1)^8 - (56*((x^(1/2) - 1)^(1/2) - 1i)^10)/((x^(1/2)

+ 1)^(1/2) - 1)^10 + (28*((x^(1/2) - 1)^(1/2) - 1i)^12)/((x^(1/2) + 1)^(1/2) - 1)^12 - (8*((x^(1/2) - 1)^(1/2

) - 1i)^14)/((x^(1/2) + 1)^(1/2) - 1)^14 + ((x^(1/2) - 1)^(1/2) - 1i)^16/((x^(1/2) + 1)^(1/2) - 1)^16 + 1)

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3.11.12 \(\int \frac {x^{3/2}}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}} \, dx\) [1012]