Optimal. Leaf size=193 \[ -\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}-\frac {5 \sqrt [4]{-1+2 \sqrt {x}}}{2 \sqrt {x}}-\frac {5 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}\right )}{2 \sqrt {2}}+\frac {5 \tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}\right )}{2 \sqrt {2}}-\frac {5 \log \left (1-\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}+\sqrt {-1+2 \sqrt {x}}\right )}{4 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}+\sqrt {-1+2 \sqrt {x}}\right )}{4 \sqrt {2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {272, 43, 65,
217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {5 \text {ArcTan}\left (1-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}\right )}{2 \sqrt {2}}+\frac {5 \text {ArcTan}\left (\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{2 \sqrt {2}}-\frac {\left (2 \sqrt {x}-1\right )^{5/4}}{x}-\frac {5 \sqrt [4]{2 \sqrt {x}-1}}{2 \sqrt {x}}-\frac {5 \log \left (\sqrt {2 \sqrt {x}-1}-\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{4 \sqrt {2}}+\frac {5 \log \left (\sqrt {2 \sqrt {x}-1}+\sqrt {2} \sqrt [4]{2 \sqrt {x}-1}+1\right )}{4 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 210
Rule 217
Rule 272
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x^2} \, dx &=2 \text {Subst}\left (\int \frac {(-1+2 x)^{5/4}}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}+\frac {5}{2} \text {Subst}\left (\int \frac {\sqrt [4]{-1+2 x}}{x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}-\frac {5 \sqrt [4]{-1+2 \sqrt {x}}}{2 \sqrt {x}}+\frac {5}{4} \text {Subst}\left (\int \frac {1}{x (-1+2 x)^{3/4}} \, dx,x,\sqrt {x}\right )\\ &=-\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}-\frac {5 \sqrt [4]{-1+2 \sqrt {x}}}{2 \sqrt {x}}+\frac {5}{2} \text {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {x^4}{2}} \, dx,x,\sqrt [4]{-1+2 \sqrt {x}}\right )\\ &=-\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}-\frac {5 \sqrt [4]{-1+2 \sqrt {x}}}{2 \sqrt {x}}+\frac {5}{4} \text {Subst}\left (\int \frac {1-x^2}{\frac {1}{2}+\frac {x^4}{2}} \, dx,x,\sqrt [4]{-1+2 \sqrt {x}}\right )+\frac {5}{4} \text {Subst}\left (\int \frac {1+x^2}{\frac {1}{2}+\frac {x^4}{2}} \, dx,x,\sqrt [4]{-1+2 \sqrt {x}}\right )\\ &=-\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}-\frac {5 \sqrt [4]{-1+2 \sqrt {x}}}{2 \sqrt {x}}+\frac {5}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+2 \sqrt {x}}\right )+\frac {5}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+2 \sqrt {x}}\right )-\frac {5 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+2 \sqrt {x}}\right )}{4 \sqrt {2}}-\frac {5 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+2 \sqrt {x}}\right )}{4 \sqrt {2}}\\ &=-\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}-\frac {5 \sqrt [4]{-1+2 \sqrt {x}}}{2 \sqrt {x}}-\frac {5 \log \left (1-\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}+\sqrt {-1+2 \sqrt {x}}\right )}{4 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}+\sqrt {-1+2 \sqrt {x}}\right )}{4 \sqrt {2}}+\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}\right )}{2 \sqrt {2}}-\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}\right )}{2 \sqrt {2}}\\ &=-\frac {\left (-1+2 \sqrt {x}\right )^{5/4}}{x}-\frac {5 \sqrt [4]{-1+2 \sqrt {x}}}{2 \sqrt {x}}-\frac {5 \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}\right )}{2 \sqrt {2}}+\frac {5 \tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}\right )}{2 \sqrt {2}}-\frac {5 \log \left (1-\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}+\sqrt {-1+2 \sqrt {x}}\right )}{4 \sqrt {2}}+\frac {5 \log \left (1+\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}+\sqrt {-1+2 \sqrt {x}}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 120, normalized size = 0.62 \begin {gather*} \frac {2 \left (2-9 \sqrt {x}\right ) \sqrt [4]{-1+2 \sqrt {x}}+5 \sqrt {2} x \tan ^{-1}\left (\frac {-1+\sqrt {-1+2 \sqrt {x}}}{\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}}\right )+5 \sqrt {2} x \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-1+2 \sqrt {x}}}{1+\sqrt {-1+2 \sqrt {x}}}\right )}{4 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 125, normalized size = 0.65
method | result | size |
meijerg | \(\frac {5 \mathrm {signum}\left (-1+2 \sqrt {x}\right )^{\frac {5}{4}} \left (-\frac {2 \Gamma \left (\frac {3}{4}\right )}{5 x}+\frac {2 \Gamma \left (\frac {3}{4}\right )}{\sqrt {x}}+\frac {\left (-2 \ln \left (2\right )+\frac {\pi }{2}-\frac {3}{2}+\frac {\ln \left (x \right )}{2}+i \pi \right ) \Gamma \left (\frac {3}{4}\right )}{2}+\frac {\Gamma \left (\frac {3}{4}\right ) \sqrt {x}\, \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 4\right ], 2 \sqrt {x}\right )}{4}\right )}{2 \Gamma \left (\frac {3}{4}\right ) \left (-\mathrm {signum}\left (-1+2 \sqrt {x}\right )\right )^{\frac {5}{4}}}\) | \(85\) |
derivativedivides | \(\frac {-\frac {9 \left (-1+2 \sqrt {x}\right )^{\frac {5}{4}}}{4}-\frac {5 \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}}{4}}{x}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}+\sqrt {-1+2 \sqrt {x}}}{1-\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}+\sqrt {-1+2 \sqrt {x}}}\right )+2 \arctan \left (1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}\right )+2 \arctan \left (-1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}\right )\right )}{8}\) | \(125\) |
default | \(\frac {-\frac {9 \left (-1+2 \sqrt {x}\right )^{\frac {5}{4}}}{4}-\frac {5 \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}}{4}}{x}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}+\sqrt {-1+2 \sqrt {x}}}{1-\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}+\sqrt {-1+2 \sqrt {x}}}\right )+2 \arctan \left (1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}\right )+2 \arctan \left (-1+\sqrt {2}\, \left (-1+2 \sqrt {x}\right )^{\frac {1}{4}}\right )\right )}{8}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 5.67, size = 157, normalized size = 0.81 \begin {gather*} \frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1\right ) - \frac {5}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1\right ) - \frac {9 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {5}{4}} + 5 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, \sqrt {x} - 1\right )}^{2} + 4 \, \sqrt {x} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 202, normalized size = 1.05 \begin {gather*} -\frac {20 \, \sqrt {2} x \arctan \left (\sqrt {2} \sqrt {\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1} - \sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} - 1\right ) + 20 \, \sqrt {2} x \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-4 \, \sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {2 \, \sqrt {x} - 1} + 4} - \sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + 1\right ) - 5 \, \sqrt {2} x \log \left (4 \, \sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {2 \, \sqrt {x} - 1} + 4\right ) + 5 \, \sqrt {2} x \log \left (-4 \, \sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {2 \, \sqrt {x} - 1} + 4\right ) + 4 \, {\left (9 \, \sqrt {x} - 2\right )} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}}{8 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 3.16, size = 44, normalized size = 0.23 \begin {gather*} - \frac {4 \cdot \sqrt [4]{2} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{2 \sqrt {x}}} \right )}}{x^{\frac {3}{8}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 142, normalized size = 0.74 \begin {gather*} \frac {5}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {5}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1\right ) - \frac {5}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}} + \sqrt {2 \, \sqrt {x} - 1} + 1\right ) - \frac {9 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {5}{4}} + 5 \, {\left (2 \, \sqrt {x} - 1\right )}^{\frac {1}{4}}}{4 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.28, size = 77, normalized size = 0.40 \begin {gather*} -\frac {5\,{\left (2\,\sqrt {x}-1\right )}^{1/4}}{4\,x}-\frac {9\,{\left (2\,\sqrt {x}-1\right )}^{5/4}}{4\,x}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (2\,\sqrt {x}-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {5}{4}+\frac {5}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (2\,\sqrt {x}-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {5}{4}-\frac {5}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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