Optimal. Leaf size=144 \[ -\frac {608185 \sqrt {1-2 x}}{504 \sqrt {5 x+3}}+\frac {13409 \sqrt {1-2 x}}{168 (3 x+2) \sqrt {5 x+3}}+\frac {77 \sqrt {1-2 x}}{12 (3 x+2)^2 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}+\frac {463881 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}} \]
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Rubi [A] time = 0.05, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 151, 152, 12, 93, 204} \[ -\frac {608185 \sqrt {1-2 x}}{504 \sqrt {5 x+3}}+\frac {13409 \sqrt {1-2 x}}{168 (3 x+2) \sqrt {5 x+3}}+\frac {77 \sqrt {1-2 x}}{12 (3 x+2)^2 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x}}{9 (3 x+2)^3 \sqrt {5 x+3}}+\frac {463881 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {1}{9} \int \frac {\frac {275}{2}-198 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {1}{126} \int \frac {\frac {50743}{4}-16170 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}+\frac {1}{882} \int \frac {\frac {5986981}{8}-\frac {1407945 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {321469533}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{4851}\\ &=-\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}-\frac {463881}{112} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}-\frac {463881}{56} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {608185 \sqrt {1-2 x}}{504 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x}}{9 (2+3 x)^3 \sqrt {3+5 x}}+\frac {77 \sqrt {1-2 x}}{12 (2+3 x)^2 \sqrt {3+5 x}}+\frac {13409 \sqrt {1-2 x}}{168 (2+3 x) \sqrt {3+5 x}}+\frac {463881 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 79, normalized size = 0.55 \[ \frac {1}{392} \left (463881 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\frac {7 \sqrt {1-2 x} \left (1824555 x^3+3608883 x^2+2378026 x+521968\right )}{(3 x+2)^3 \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 116, normalized size = 0.81 \[ \frac {463881 \, \sqrt {7} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1824555 \, x^{3} + 3608883 \, x^{2} + 2378026 \, x + 521968\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{784 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.68, size = 377, normalized size = 2.62 \[ -\frac {463881}{7840} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {55}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {11 \, {\left (33989 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 15023680 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 1769566400 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{28 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 250, normalized size = 1.74 \[ -\frac {\left (62623935 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+162822231 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+25543770 \sqrt {-10 x^{2}-x +3}\, x^{3}+158647302 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+50524362 \sqrt {-10 x^{2}-x +3}\, x^{2}+68654388 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+33292364 \sqrt {-10 x^{2}-x +3}\, x +11133144 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+7307552 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{784 \left (3 x +2\right )^{3} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 211, normalized size = 1.47 \[ -\frac {463881}{784} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {608185 \, x}{252 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {635003}{504 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {49}{27 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {1561}{108 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {4367}{24 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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