Integrand size = 11, antiderivative size = 59 \[ \int \sqrt {\sqrt [4]{x}+x} \, dx=\frac {1}{3} \sqrt [4]{x} \sqrt {\sqrt [4]{x}+x}+\frac {2}{3} x \sqrt {\sqrt [4]{x}+x}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {\sqrt [4]{x}+x}}\right ) \] Output:
1/3*x^(1/4)*(x^(1/4)+x)^(1/2)+2/3*x*(x^(1/4)+x)^(1/2)-1/3*arctanh(x^(1/2)/ (x^(1/4)+x)^(1/2))
Time = 0.43 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.80 \[ \int \sqrt {\sqrt [4]{x}+x} \, dx=\frac {1}{3} \sqrt {\sqrt [4]{x}+x} \left (\sqrt [4]{x}+2 x\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {\sqrt [4]{x}+x}}\right ) \] Input:
Integrate[Sqrt[x^(1/4) + x],x]
Output:
(Sqrt[x^(1/4) + x]*(x^(1/4) + 2*x))/3 - ArcTanh[Sqrt[x]/Sqrt[x^(1/4) + x]] /3
Time = 0.35 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1910, 1924, 1930, 1935, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {x+\sqrt [4]{x}} \, dx\) |
\(\Big \downarrow \) 1910 |
\(\displaystyle \frac {1}{4} \int \frac {\sqrt [4]{x}}{\sqrt {x+\sqrt [4]{x}}}dx+\frac {2}{3} \sqrt {x+\sqrt [4]{x}} x\) |
\(\Big \downarrow \) 1924 |
\(\displaystyle \int \frac {x}{\sqrt {x+\sqrt [4]{x}}}d\sqrt [4]{x}+\frac {2}{3} \sqrt {x+\sqrt [4]{x}} x\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle -\frac {1}{2} \int \frac {\sqrt [4]{x}}{\sqrt {x+\sqrt [4]{x}}}d\sqrt [4]{x}+\frac {2}{3} \sqrt {x+\sqrt [4]{x}} x+\frac {1}{3} \sqrt {x+\sqrt [4]{x}} \sqrt [4]{x}\) |
\(\Big \downarrow \) 1935 |
\(\displaystyle -\frac {1}{3} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt {x}}{\sqrt {x+\sqrt [4]{x}}}+\frac {2}{3} \sqrt {x+\sqrt [4]{x}} x+\frac {1}{3} \sqrt {x+\sqrt [4]{x}} \sqrt [4]{x}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {1}{3} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt {x+\sqrt [4]{x}}}\right )+\frac {2}{3} \sqrt {x+\sqrt [4]{x}} x+\frac {1}{3} \sqrt {x+\sqrt [4]{x}} \sqrt [4]{x}\) |
Input:
Int[Sqrt[x^(1/4) + x],x]
Output:
(x^(1/4)*Sqrt[x^(1/4) + x])/3 + (2*x*Sqrt[x^(1/4) + x])/3 - ArcTanh[Sqrt[x ]/Sqrt[x^(1/4) + x]]/3
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[x*((a*x^j + b*x^n)^p/(n*p + 1)), x] + Simp[a*(n - j)*(p/(n*p + 1)) Int[x^j*(a*x^j + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && !IntegerQ[p] && LtQ[0, j, n] && GtQ[p, 0] && NeQ[n*p + 1, 0]
Int[(x_)^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp [1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x)^p, x ], x, x^n], x] /; FreeQ[{a, b, j, m, n, p}, x] && !IntegerQ[p] && NeQ[n, j ] && IntegerQ[Simplify[j/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1 ]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a*x^j + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))) I nt[(c*x)^(m - (n - j))*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, p}, x] && !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && Gt Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
Int[(x_)^(m_.)/Sqrt[(a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> Simp [-2/(n - j) Subst[Int[1/(1 - a*x^2), x], x, x^(j/2)/Sqrt[a*x^j + b*x^n]], x] /; FreeQ[{a, b, j, n}, x] && EqQ[m, j/2 - 1] && NeQ[n, j]
Time = 0.59 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.64
method | result | size |
meijerg | \(-\frac {2 \left (-\frac {\sqrt {\pi }\, x^{\frac {3}{8}} \left (6 x^{\frac {3}{4}}+3\right ) \sqrt {x^{\frac {3}{4}}+1}}{6}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (x^{\frac {3}{8}}\right )}{2}\right )}{3 \sqrt {\pi }}\) | \(38\) |
derivativedivides | \(-\frac {1}{6 {\left (\frac {\sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}+1\right )}^{2}}+\frac {1}{\frac {6 \sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}+6}-\frac {\ln \left (\frac {\sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}+1\right )}{6}+\frac {1}{6 {\left (\frac {\sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}-1\right )}^{2}}+\frac {1}{\frac {6 \sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}-6}+\frac {\ln \left (\frac {\sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}-1\right )}{6}\) | \(102\) |
default | \(-\frac {1}{6 {\left (\frac {\sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}+1\right )}^{2}}+\frac {1}{\frac {6 \sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}+6}-\frac {\ln \left (\frac {\sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}+1\right )}{6}+\frac {1}{6 {\left (\frac {\sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}-1\right )}^{2}}+\frac {1}{\frac {6 \sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}-6}+\frac {\ln \left (\frac {\sqrt {x^{\frac {1}{4}}+x}}{\sqrt {x}}-1\right )}{6}\) | \(102\) |
Input:
int((x^(1/4)+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/3/Pi^(1/2)*(-1/6*Pi^(1/2)*x^(3/8)*(6*x^(3/4)+3)*(x^(3/4)+1)^(1/2)+1/2*P i^(1/2)*arcsinh(x^(3/8)))
Timed out. \[ \int \sqrt {\sqrt [4]{x}+x} \, dx=\text {Timed out} \] Input:
integrate((x^(1/4)+x)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \sqrt {\sqrt [4]{x}+x} \, dx=\int \sqrt {\sqrt [4]{x} + x}\, dx \] Input:
integrate((x**(1/4)+x)**(1/2),x)
Output:
Integral(sqrt(x**(1/4) + x), x)
\[ \int \sqrt {\sqrt [4]{x}+x} \, dx=\int { \sqrt {x + x^{\frac {1}{4}}} \,d x } \] Input:
integrate((x^(1/4)+x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(x + x^(1/4)), x)
Time = 0.58 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.76 \[ \int \sqrt {\sqrt [4]{x}+x} \, dx=\frac {1}{3} \, \sqrt {x + x^{\frac {1}{4}}} x^{\frac {1}{4}} {\left (2 \, x^{\frac {3}{4}} + 1\right )} - \frac {1}{6} \, \log \left (\sqrt {\frac {1}{x^{\frac {3}{4}}} + 1} + 1\right ) + \frac {1}{6} \, \log \left ({\left | \sqrt {\frac {1}{x^{\frac {3}{4}}} + 1} - 1 \right |}\right ) \] Input:
integrate((x^(1/4)+x)^(1/2),x, algorithm="giac")
Output:
1/3*sqrt(x + x^(1/4))*x^(1/4)*(2*x^(3/4) + 1) - 1/6*log(sqrt(1/x^(3/4) + 1 ) + 1) + 1/6*log(abs(sqrt(1/x^(3/4) + 1) - 1))
Time = 22.69 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.46 \[ \int \sqrt {\sqrt [4]{x}+x} \, dx=\frac {8\,x\,\sqrt {x+x^{1/4}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},\frac {3}{2};\ \frac {5}{2};\ -x^{3/4}\right )}{9\,\sqrt {x^{3/4}+1}} \] Input:
int((x + x^(1/4))^(1/2),x)
Output:
(8*x*(x + x^(1/4))^(1/2)*hypergeom([-1/2, 3/2], 5/2, -x^(3/4)))/(9*(x^(3/4 ) + 1)^(1/2))
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.86 \[ \int \sqrt {\sqrt [4]{x}+x} \, dx=\frac {x^{\frac {3}{8}} \sqrt {x^{\frac {3}{4}}+1}}{3}+\frac {2 x^{\frac {9}{8}} \sqrt {x^{\frac {3}{4}}+1}}{3}-\frac {\mathrm {log}\left (\sqrt {x^{\frac {3}{4}}+1}+x^{\frac {3}{8}}\right )}{6}+\frac {\mathrm {log}\left (\sqrt {x^{\frac {3}{4}}+1}-x^{\frac {3}{8}}\right )}{6} \] Input:
int((x^(1/4)+x)^(1/2),x)
Output:
(2*x**(3/8)*sqrt(x**(3/4) + 1) + 4*x**(1/8)*sqrt(x**(3/4) + 1)*x - log(sqr t(x**(3/4) + 1) + x**(3/8)) + log(sqrt(x**(3/4) + 1) - x**(3/8)))/6