Optimal. Leaf size=85 \[ \frac {1}{4} \left (2+\sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt {3-2 \sqrt {2}} \sqrt {x^3-x}}{x+1}\right )+\frac {1}{4} \left (\sqrt {2}-2\right ) \tan ^{-1}\left (\frac {\sqrt {3+2 \sqrt {2}} \sqrt {x^3-x}}{x+1}\right ) \]
________________________________________________________________________________________
Rubi [C] time = 0.73, antiderivative size = 257, normalized size of antiderivative = 3.02, number of steps used = 16, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {1593, 2056, 6728, 944, 329, 222, 933, 168, 537} \begin {gather*} \frac {\left (2+\sqrt {2}\right ) \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{2 \sqrt {x^3-x}}-\frac {\left (2-\sqrt {2}\right ) \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{2 \sqrt {x^3-x}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{2+\sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \sqrt {x^3-x}}-\frac {\left (1-\sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{2} \left (2+\sqrt {2}\right );\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \sqrt {x^3-x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 168
Rule 222
Rule 329
Rule 537
Rule 933
Rule 944
Rule 1593
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {-x+x^2}{\left (-1+2 x+x^2\right ) \sqrt {-x+x^3}} \, dx &=\int \frac {(-1+x) x}{\left (-1+2 x+x^2\right ) \sqrt {-x+x^3}} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {(-1+x) \sqrt {x}}{\sqrt {-1+x^2} \left (-1+2 x+x^2\right )} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \left (\frac {\left (1-\sqrt {2}\right ) \sqrt {x}}{\left (2-2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}}+\frac {\left (1+\sqrt {2}\right ) \sqrt {x}}{\left (2+2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}}\right ) \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {\sqrt {x}}{\left (2-2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {\sqrt {x}}{\left (2+2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^2}} \, dx}{2 \sqrt {-x+x^3}}+\frac {\left (\left (1-\sqrt {2}\right ) \left (-1+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (2-2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^2}} \, dx}{2 \sqrt {-x+x^3}}-\frac {\left (\left (1+\sqrt {2}\right )^2 \sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \left (2+2 \sqrt {2}+2 x\right ) \sqrt {-1+x^2}} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\left (1-\sqrt {2}\right ) \left (-1+\sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x} \left (2-2 \sqrt {2}+2 x\right )} \, dx}{\sqrt {-x+x^3}}-\frac {\left (\left (1+\sqrt {2}\right )^2 \sqrt {x} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x} \sqrt {1+x} \left (2+2 \sqrt {2}+2 x\right )} \, dx}{\sqrt {-x+x^3}}+\frac {\left (\left (1-\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}+\frac {\left (\left (1+\sqrt {2}\right ) \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=-\frac {\left (2-\sqrt {2}\right ) \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{2 \sqrt {-x+x^3}}+\frac {\left (2+\sqrt {2}\right ) \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{2 \sqrt {-x+x^3}}-\frac {\left (2 \left (1-\sqrt {2}\right ) \left (-1+\sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \left (2-\sqrt {2}\right )-2 x^2\right ) \sqrt {1-x^2} \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {-x+x^3}}+\frac {\left (2 \left (1+\sqrt {2}\right )^2 \sqrt {x} \sqrt {1-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (2 \left (2+\sqrt {2}\right )-2 x^2\right ) \sqrt {1-x^2} \sqrt {2-x^2}} \, dx,x,\sqrt {1-x}\right )}{\sqrt {-x+x^3}}\\ &=-\frac {\left (2-\sqrt {2}\right ) \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{2 \sqrt {-x+x^3}}+\frac {\left (2+\sqrt {2}\right ) \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{2 \sqrt {-x+x^3}}+\frac {\left (4+3 \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{2+\sqrt {2}};\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \left (2+\sqrt {2}\right ) \sqrt {-x+x^3}}-\frac {\left (4-3 \sqrt {2}\right ) \sqrt {x} \sqrt {1-x^2} \Pi \left (\frac {1}{2} \left (2+\sqrt {2}\right );\sin ^{-1}\left (\sqrt {1-x}\right )|\frac {1}{2}\right )}{2 \left (2-\sqrt {2}\right ) \sqrt {-x+x^3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.73, size = 88, normalized size = 1.04 \begin {gather*} \frac {\sqrt {x} \sqrt {1-x^2} \left (2 F\left (\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )-\left (1+\sqrt {2}\right ) \Pi \left (1-\sqrt {2};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )+\left (\sqrt {2}-1\right ) \Pi \left (1+\sqrt {2};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )\right )}{\sqrt {x \left (x^2-1\right )}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.39, size = 73, normalized size = 0.86 \begin {gather*} \frac {1}{4} \left (2+\sqrt {2}\right ) \tan ^{-1}\left (\frac {\left (-1+\sqrt {2}\right ) \sqrt {-x+x^3}}{1+x}\right )+\frac {1}{4} \left (-2+\sqrt {2}\right ) \tan ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \sqrt {-x+x^3}}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 87, normalized size = 1.02 \begin {gather*} -\frac {1}{4} \, \sqrt {2} \sqrt {2 \, \sqrt {2} + 3} \arctan \left (\frac {\sqrt {x^{3} - x} \sqrt {2 \, \sqrt {2} + 3}}{x^{2} - x}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {-2 \, \sqrt {2} + 3} \arctan \left (\frac {\sqrt {x^{3} - x} \sqrt {-2 \, \sqrt {2} + 3}}{x^{2} - x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - x}{\sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.61, size = 222, normalized size = 2.61
method | result | size |
default | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {3 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \sqrt {2}\, \EllipticPi \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}-\frac {3 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \sqrt {2}\, \EllipticPi \left (\sqrt {1+x}, -\frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\) | \(222\) |
elliptic | \(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {3 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \sqrt {2}\, \EllipticPi \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}-\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}-\frac {3 \sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \sqrt {2}\, \EllipticPi \left (\sqrt {1+x}, -\frac {\sqrt {2}}{2}, \frac {\sqrt {2}}{2}\right )}{4 \sqrt {x^{3}-x}}\) | \(222\) |
trager | \(4 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} \ln \left (\frac {-8868864 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x^{2}+14781440 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +5912576 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5}-3959200 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x^{2}+9431776 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x +2278872 \sqrt {x^{3}-x}\, \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}+5472576 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3}-268344 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x^{2}+1475892 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x +181465 \sqrt {x^{3}-x}+1207548 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )}{\left (8 x \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-16 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-3 x -8\right )^{2}}\right )+3 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {-8868864 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x^{2}+14781440 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +5912576 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5}-3959200 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x^{2}+9431776 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x +2278872 \sqrt {x^{3}-x}\, \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}+5472576 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3}-268344 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x^{2}+1475892 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x +181465 \sqrt {x^{3}-x}+1207548 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )}{\left (8 x \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-16 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-3 x -8\right )^{2}}\right )-\frac {\ln \left (-\frac {-27452160 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x^{2}+45753600 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5} x +18301440 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{5}-37781952 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x^{2}+54200480 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3} x +2278872 \sqrt {x^{3}-x}\, \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}+16418528 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{3}-12987008 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x^{2}+15422072 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right ) x +1527689 \sqrt {x^{3}-x}+2435064 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )}{\left (8 x \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}-16 \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )^{2}+9 x -4\right )^{2}}\right ) \RootOf \left (64 \textit {\_Z}^{4}+48 \textit {\_Z}^{2}+1\right )}{2}\) | \(773\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - x}{\sqrt {x^{3} - x} {\left (x^{2} + 2 \, x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.80, size = 159, normalized size = 1.87 \begin {gather*} \frac {\sqrt {2}\,\sqrt {-x}\,\left (3\,\sqrt {2}+4\right )\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\frac {1}{\sqrt {2}+1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2\,\sqrt {x^3-x}\,\left (\sqrt {2}+1\right )}-\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}-\frac {\sqrt {2}\,\sqrt {-x}\,\left (3\,\sqrt {2}-4\right )\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (-\frac {1}{\sqrt {2}-1};\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{2\,\sqrt {x^3-x}\,\left (\sqrt {2}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (x - 1\right )}{\sqrt {x \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 2 x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________