Optimal. Leaf size=151 \[ \frac{4 \sqrt{x^2-x}}{3 x^{3/2}}-\frac{2 \log \left (4 \sqrt{x^2-x}+4 x-1\right )}{3 x^{3/2}}+\frac{32 \sqrt{2} \sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{3 \sqrt{x-1} \sqrt{x}}+\frac{44}{3} \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x^2-x}}\right )-\frac{16}{3 \sqrt{x}}-\frac{32}{3} \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.478361, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {2537, 2535, 6733, 6742, 203, 1588, 2020, 2008, 2021, 1146, 444, 50, 63} \[ \frac{4 \sqrt{x^2-x}}{3 x^{3/2}}-\frac{2 \log \left (4 \sqrt{x^2-x}+4 x-1\right )}{3 x^{3/2}}+\frac{32 \sqrt{2} \sqrt{x^2-x} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{x-1}\right )}{3 \sqrt{x-1} \sqrt{x}}+\frac{44}{3} \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x^2-x}}\right )-\frac{16}{3 \sqrt{x}}-\frac{32}{3} \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2537
Rule 2535
Rule 6733
Rule 6742
Rule 203
Rule 1588
Rule 2020
Rule 2008
Rule 2021
Rule 1146
Rule 444
Rule 50
Rule 63
Rubi steps
\begin{align*} \int \frac{\log \left (-1+4 x+4 \sqrt{(-1+x) x}\right )}{x^{5/2}} \, dx &=\int \frac{\log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{x^{5/2}} \, dx\\ &=-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}-\frac{16}{3} \int \frac{1}{x^{3/2} \left (-4 (1+2 x) \sqrt{-x+x^2}+8 \left (-x+x^2\right )\right )} \, dx\\ &=-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}-\frac{32}{3} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (-4 \left (1+2 x^2\right ) \sqrt{-x^2+x^4}+8 \left (-x^2+x^4\right )\right )} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}-\frac{32}{3} \operatorname{Subst}\left (\int \left (-\frac{1}{2 x^2}+\frac{4}{1+8 x^2}-\frac{x^2}{12 \sqrt{-x^2+x^4}}+\frac{\sqrt{-x^2+x^4}}{4 x^4}-\frac{5 \sqrt{-x^2+x^4}}{4 x^2}+\frac{32 \sqrt{-x^2+x^4}}{3 \left (1+8 x^2\right )}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{16}{3 \sqrt{x}}-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}+\frac{8}{9} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{-x^2+x^4}} \, dx,x,\sqrt{x}\right )-\frac{8}{3} \operatorname{Subst}\left (\int \frac{\sqrt{-x^2+x^4}}{x^4} \, dx,x,\sqrt{x}\right )+\frac{40}{3} \operatorname{Subst}\left (\int \frac{\sqrt{-x^2+x^4}}{x^2} \, dx,x,\sqrt{x}\right )-\frac{128}{3} \operatorname{Subst}\left (\int \frac{1}{1+8 x^2} \, dx,x,\sqrt{x}\right )-\frac{1024}{9} \operatorname{Subst}\left (\int \frac{\sqrt{-x^2+x^4}}{1+8 x^2} \, dx,x,\sqrt{x}\right )\\ &=-\frac{16}{3 \sqrt{x}}+\frac{4 \sqrt{-x+x^2}}{3 x^{3/2}}+\frac{128 \sqrt{-x+x^2}}{9 \sqrt{x}}-\frac{32}{3} \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}-\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-x^2+x^4}} \, dx,x,\sqrt{x}\right )-\frac{40}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-x^2+x^4}} \, dx,x,\sqrt{x}\right )-\frac{\left (1024 \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{16}{3 \sqrt{x}}+\frac{4 \sqrt{-x+x^2}}{3 x^{3/2}}+\frac{128 \sqrt{-x+x^2}}{9 \sqrt{x}}-\frac{32}{3} \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}+\frac{4}{3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )+\frac{40}{3} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )-\frac{\left (512 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{-1+x}}{1+8 x} \, dx,x,x\right )}{9 \sqrt{-1+x} \sqrt{x}}\\ &=-\frac{16}{3 \sqrt{x}}+\frac{4 \sqrt{-x+x^2}}{3 x^{3/2}}-\frac{32}{3} \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )+\frac{44}{3} \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}+\frac{\left (64 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} (1+8 x)} \, dx,x,x\right )}{\sqrt{-1+x} \sqrt{x}}\\ &=-\frac{16}{3 \sqrt{x}}+\frac{4 \sqrt{-x+x^2}}{3 x^{3/2}}-\frac{32}{3} \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )+\frac{44}{3} \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}+\frac{\left (128 \sqrt{-x+x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{9+8 x^2} \, dx,x,\sqrt{-1+x}\right )}{\sqrt{-1+x} \sqrt{x}}\\ &=-\frac{16}{3 \sqrt{x}}+\frac{4 \sqrt{-x+x^2}}{3 x^{3/2}}+\frac{32 \sqrt{2} \sqrt{-x+x^2} \tan ^{-1}\left (\frac{2}{3} \sqrt{2} \sqrt{-1+x}\right )}{3 \sqrt{-1+x} \sqrt{x}}-\frac{32}{3} \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )+\frac{44}{3} \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{-x+x^2}}\right )-\frac{2 \log \left (-1+4 x+4 \sqrt{-x+x^2}\right )}{3 x^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.604722, size = 204, normalized size = 1.35 \[ \frac{2}{3} \left (\frac{2 \sqrt{(x-1) x}}{x^{3/2}}-\frac{\log \left (4 x+4 \sqrt{(x-1) x}-1\right )}{x^{3/2}}-\frac{8}{\sqrt{x}}+8 i \sqrt{2} \log \left (4 (8 x+1)^2\right )-4 i \sqrt{2} \log \left ((8 x+1) \left (-10 x-6 \sqrt{(x-1) x}+1\right )\right )-4 i \sqrt{2} \log \left ((8 x+1) \left (-10 x+6 \sqrt{(x-1) x}+1\right )\right )-16 \sqrt{2} \tan ^{-1}\left (2 \sqrt{2} \sqrt{x}\right )-22 \tan ^{-1}\left (\frac{\sqrt{(x-1) x}}{\sqrt{x}}\right )+16 \sqrt{2} \tan ^{-1}\left (\frac{2 \sqrt{2} \sqrt{(x-1) x}}{3 \sqrt{x}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.006, size = 0, normalized size = 0. \begin{align*} \int{\ln \left ( -1+4\,x+4\,\sqrt{ \left ( -1+x \right ) x} \right ){x}^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2}{3 \, \sqrt{x}} - \frac{2 \, \log \left (4 \, \sqrt{x - 1} \sqrt{x} + 4 \, x - 1\right )}{3 \, x^{\frac{3}{2}}} - \frac{2}{9 \, x^{\frac{3}{2}}} - \int \frac{2 \, x^{2} + x}{3 \,{\left (4 \, x^{\frac{11}{2}} - 5 \, x^{\frac{9}{2}} + x^{\frac{7}{2}} + 4 \,{\left (x^{5} - x^{4}\right )} \sqrt{x - 1}\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]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.19043, size = 311, normalized size = 2.06 \begin{align*} -\frac{2 \,{\left (16 \, \sqrt{2} x^{2} \arctan \left (2 \, \sqrt{2} \sqrt{x}\right ) + 16 \, \sqrt{2} x^{2} \arctan \left (\frac{3 \, \sqrt{2} \sqrt{x}}{4 \, \sqrt{x^{2} - x}}\right ) - 22 \, x^{2} \arctan \left (\frac{\sqrt{x}}{\sqrt{x^{2} - x}}\right ) + 8 \, x^{\frac{3}{2}} + \sqrt{x} \log \left (4 \, x + 4 \, \sqrt{x^{2} - x} - 1\right ) - 2 \, \sqrt{x^{2} - x} \sqrt{x}\right )}}{3 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]