Optimal. Leaf size=104 \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}+\frac{5}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{5}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0452857, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {323, 330, 52} \[ \frac{1}{3} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{5/2}+\frac{5}{12} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} x^{3/2}+\frac{5}{8} \sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 323
Rule 330
Rule 52
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx &=\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{6} \int \frac{x^{3/2}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=\frac{5}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{8} \int \frac{\sqrt{x}}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}}} \, dx\\ &=\frac{5}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{5}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{16} \int \frac{1}{\sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}} \, dx\\ &=\frac{5}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{5}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} \sqrt{1+x}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{8} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} \sqrt{x}+\frac{5}{12} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}+\frac{1}{3} \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{5/2}+\frac{5}{8} \cosh ^{-1}\left (\sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0349079, size = 67, normalized size = 0.64 \[ \frac{1}{24} \left (\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1} \sqrt{x} \left (8 x^2+10 x+15\right )+30 \tanh ^{-1}\left (\sqrt{\frac{\sqrt{x}-1}{\sqrt{x}+1}}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 65, normalized size = 0.6 \begin{align*}{\frac{1}{24}\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}} \left ( 8\,{x}^{5/2}\sqrt{-1+x}+10\,{x}^{3/2}\sqrt{-1+x}+15\,\sqrt{x}\sqrt{-1+x}+15\,\ln \left ( \sqrt{x}+\sqrt{-1+x} \right ) \right ){\frac{1}{\sqrt{-1+x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.936321, size = 63, normalized size = 0.61 \begin{align*} \frac{1}{3} \, \sqrt{x - 1} x^{\frac{5}{2}} + \frac{5}{12} \, \sqrt{x - 1} x^{\frac{3}{2}} + \frac{5}{8} \, \sqrt{x - 1} \sqrt{x} + \frac{5}{8} \, \log \left (2 \, \sqrt{x - 1} + 2 \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.947352, size = 186, normalized size = 1.79 \begin{align*} \frac{1}{24} \,{\left (8 \, x^{2} + 10 \, x + 15\right )} \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - \frac{5}{16} \, \log \left (2 \, \sqrt{x} \sqrt{\sqrt{x} + 1} \sqrt{\sqrt{x} - 1} - 2 \, x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{5}{2}}}{\sqrt{\sqrt{x} - 1} \sqrt{\sqrt{x} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]