Optimal. Leaf size=130 \[ -\frac {925}{864} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5}{24} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {1}{9} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {6553 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2592}+\frac {2}{81} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {103, 159, 163,
56, 222, 95, 210} \begin {gather*} \frac {6553 \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2592}+\frac {2}{81} \sqrt {7} \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )+\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{5/2}-\frac {5}{24} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {925}{864} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 95
Rule 103
Rule 159
Rule 163
Rule 210
Rule 222
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{2+3 x} \, dx &=\frac {1}{9} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {1}{9} \int \frac {\left (-8-\frac {45 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {5}{24} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {1}{9} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {1}{108} \int \frac {\sqrt {3+5 x} \left (\frac {981}{2}+\frac {2775 x}{4}\right )}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {925}{864} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5}{24} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {1}{9} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {1}{648} \int \frac {-\frac {32541}{4}-\frac {98295 x}{8}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {925}{864} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5}{24} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {1}{9} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {7}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {32765 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{5184}\\ &=-\frac {925}{864} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5}{24} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {1}{9} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {14}{81} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {\left (6553 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2592}\\ &=-\frac {925}{864} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {5}{24} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {1}{9} \sqrt {1-2 x} (3+5 x)^{5/2}+\frac {6553 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2592}+\frac {2}{81} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 114, normalized size = 0.88 \begin {gather*} \frac {6 \sqrt {1-2 x} \left (-1803+2935 x+17100 x^2+12000 x^3\right )-6553 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )+128 \sqrt {7} \sqrt {3+5 x} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{5184 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 115, normalized size = 0.88
method | result | size |
default | \(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (28800 x^{2} \sqrt {-10 x^{2}-x +3}+6553 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-128 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+23760 x \sqrt {-10 x^{2}-x +3}-7212 \sqrt {-10 x^{2}-x +3}\right )}{10368 \sqrt {-10 x^{2}-x +3}}\) | \(115\) |
risch | \(-\frac {\left (2400 x^{2}+1980 x -601\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{864 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {6553 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{10368}+\frac {\sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right )}{81}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(131\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 83, normalized size = 0.64 \begin {gather*} -\frac {5}{18} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {145}{72} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {6553}{10368} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{81} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {119}{864} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.92, size = 113, normalized size = 0.87 \begin {gather*} \frac {1}{864} \, {\left (2400 \, x^{2} + 1980 \, x - 601\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {6553}{10368} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + \frac {1}{81} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {5}{2}}}{3 x + 2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 186 vs.
\(2 (92) = 184\).
time = 1.77, size = 186, normalized size = 1.43 \begin {gather*} -\frac {1}{810} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{4320} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 15 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 925 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {6553}{10368} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{3\,x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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