Optimal. Leaf size=166 \[ \frac {63678595 \sqrt {1-2 x}}{12936 \sqrt {5 x+3}}-\frac {638165 \sqrt {1-2 x}}{1176 (5 x+3)^{3/2}}+\frac {25441 \sqrt {1-2 x}}{392 (3 x+2) (5 x+3)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {13246251 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 151, 152, 12, 93, 204} \[ \frac {63678595 \sqrt {1-2 x}}{12936 \sqrt {5 x+3}}-\frac {638165 \sqrt {1-2 x}}{1176 (5 x+3)^{3/2}}+\frac {25441 \sqrt {1-2 x}}{392 (3 x+2) (5 x+3)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {\sqrt {1-2 x}}{3 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {13246251 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{392 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {1}{3} \int \frac {-\frac {51}{2}+40 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {1}{42} \int \frac {-\frac {12921}{4}+4695 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {25441 \sqrt {1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}-\frac {1}{294} \int \frac {-\frac {2380137}{8}+381615 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {638165 \sqrt {1-2 x}}{1176 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {25441 \sqrt {1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac {\int \frac {-\frac {268650723}{16}+\frac {63178335 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{4851}\\ &=-\frac {638165 \sqrt {1-2 x}}{1176 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {25441 \sqrt {1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac {63678595 \sqrt {1-2 x}}{12936 \sqrt {3+5 x}}-\frac {2 \int -\frac {14425167339}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{53361}\\ &=-\frac {638165 \sqrt {1-2 x}}{1176 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {25441 \sqrt {1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac {63678595 \sqrt {1-2 x}}{12936 \sqrt {3+5 x}}+\frac {13246251}{784} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {638165 \sqrt {1-2 x}}{1176 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {25441 \sqrt {1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac {63678595 \sqrt {1-2 x}}{12936 \sqrt {3+5 x}}+\frac {13246251}{392} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {638165 \sqrt {1-2 x}}{1176 (3+5 x)^{3/2}}+\frac {\sqrt {1-2 x}}{3 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {313 \sqrt {1-2 x}}{84 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {25441 \sqrt {1-2 x}}{392 (2+3 x) (3+5 x)^{3/2}}+\frac {63678595 \sqrt {1-2 x}}{12936 \sqrt {3+5 x}}-\frac {13246251 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{392 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 84, normalized size = 0.51 \[ \frac {\frac {7 \sqrt {1-2 x} \left (8596610325 x^4+22161651840 x^3+21406565457 x^2+9181937962 x+1475586688\right )}{(3 x+2)^3 (5 x+3)^{3/2}}-437126283 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{90552} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 131, normalized size = 0.79 \[ -\frac {437126283 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (8596610325 \, x^{4} + 22161651840 \, x^{3} + 21406565457 \, x^{2} + 9181937962 \, x + 1475586688\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{181104 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.75, size = 439, normalized size = 2.64 \[ \frac {1}{1811040} \, \sqrt {5} {\left (437126283 \, \sqrt {70} \sqrt {2} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - 85750 \, \sqrt {2} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {3168 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {12672 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {2744280 \, {\left (22317 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 10704960 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 1323627200 \, \sqrt {2} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 298, normalized size = 1.80 \[ \frac {\left (295060241025 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+944192771280 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+120352544550 \sqrt {-10 x^{2}-x +3}\, x^{4}+1207779919929 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+310263125760 \sqrt {-10 x^{2}-x +3}\, x^{3}+771965015778 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+299691916398 \sqrt {-10 x^{2}-x +3}\, x^{2}+246539223612 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+128547131468 \sqrt {-10 x^{2}-x +3}\, x +31473092376 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+20658213632 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{181104 \left (3 x +2\right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 240, normalized size = 1.45 \[ \frac {13246251}{5488} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {63678595 \, x}{6468 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {66486521}{12936 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {207835 \, x}{84 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {49}{27 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {77}{4 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {24617}{72 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {2020657}{1512 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {1-2\,x}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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