Optimal. Leaf size=115 \[ \frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}+\frac {74}{45} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {346}{135} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {175}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {98, 154, 157, 54, 216, 93, 204} \[ \frac {7 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 (3 x+2)}+\frac {74}{45} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {346}{135} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {175}{27} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 98
Rule 154
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^2 \sqrt {3+5 x}} \, dx &=\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{3} \int \frac {\sqrt {1-2 x} \left (\frac {157}{2}+74 x\right )}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1}{45} \int \frac {\frac {2503}{2}+346 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {346}{135} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx+\frac {1225}{54} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {1225}{27} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {692 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{135 \sqrt {5}}\\ &=\frac {74}{45} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 (2+3 x)}+\frac {346}{135} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {175}{27} \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 = 0.18, size = 125, normalized size = 1.09 \[ \frac {15 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} (12 x+253)-4375 (3 x+2) \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-346 \sqrt {10-20 x} (3 x+2) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{675 \sqrt {2 x-1} (3 x+2)} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.93, size = 127, normalized size = 1.10 \[ -\frac {346 \, \sqrt {5} \sqrt {2} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 4375 \, \sqrt {7} {\left (3 \, x + 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 30 \, {\left (12 \, x + 253\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1350 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.17, size = 279, normalized size = 2.43 \[ \frac {35}{108} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {173}{675} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {4}{225} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1078 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{9 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 146, normalized size = 1.27 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (1038 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+13125 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+360 \sqrt {-10 x^{2}-x +3}\, x +692 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+8750 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+7590 \sqrt {-10 x^{2}-x +3}\right )}{1350 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 75, normalized size = 0.65 \[ \frac {173}{675} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {175}{54} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {4}{45} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{9 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^2\,\sqrt {5\,x+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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