Optimal. Leaf size=180 \[ \frac {107245 \sqrt {1-2 x} \sqrt {5 x+3}}{153664 (3 x+2)}+\frac {835 \sqrt {1-2 x} \sqrt {5 x+3}}{10976 (3 x+2)^2}-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{392 (3 x+2)^3}-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)^4}+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {1244755 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {99, 151, 12, 93, 204} \[ \frac {107245 \sqrt {1-2 x} \sqrt {5 x+3}}{153664 (3 x+2)}+\frac {835 \sqrt {1-2 x} \sqrt {5 x+3}}{10976 (3 x+2)^2}-\frac {13 \sqrt {1-2 x} \sqrt {5 x+3}}{392 (3 x+2)^3}-\frac {27 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)^4}+\frac {2 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {1244755 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {2}{7} \int \frac {-\frac {71}{2}-60 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {27 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {1}{98} \int \frac {-\frac {989}{4}-405 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {27 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {13 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}-\frac {\int \frac {-\frac {13125}{8}-1365 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{2058}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {27 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {13 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}+\frac {835 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}-\frac {\int \frac {-\frac {516915}{16}+\frac {87675 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{28812}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {27 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {13 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}+\frac {835 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {107245 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}-\frac {\int -\frac {26139855}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{201684}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {27 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {13 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}+\frac {835 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {107245 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}+\frac {1244755 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{307328}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {27 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {13 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}+\frac {835 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {107245 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}+\frac {1244755 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{153664}\\ &=\frac {2 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {27 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {13 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}+\frac {835 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {107245 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}-\frac {1244755 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{153664 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 95, normalized size = 0.53 \[ \frac {-7 \sqrt {5 x+3} \left (5791230 x^4+8897265 x^3+2075184 x^2-2239092 x-917264\right )-1244755 \sqrt {7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1075648 \sqrt {1-2 x} (3 x+2)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 131, normalized size = 0.73 \[ -\frac {1244755 \, \sqrt {7} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (5791230 \, x^{4} + 8897265 \, x^{3} + 2075184 \, x^{2} - 2239092 \, x - 917264\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2151296 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.90, size = 394, normalized size = 2.19 \[ \frac {248951}{4302592} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {32 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{84035 \, {\left (2 \, x - 1\right )}} - \frac {33 \, \sqrt {10} {\left (264101 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 272107080 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 72200520000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {5707629760000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {22830519040000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{537824 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 305, normalized size = 1.69 \[ \frac {\left (201650310 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+436909005 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+81077220 \sqrt {-10 x^{2}-x +3}\, x^{4}+268867080 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+124561710 \sqrt {-10 x^{2}-x +3}\, x^{3}-29874120 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+29052576 \sqrt {-10 x^{2}-x +3}\, x^{2}-79664320 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-31347288 \sqrt {-10 x^{2}-x +3}\, x -19916080 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-12841696 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{2151296 \left (3 x +2\right )^{4} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.34, size = 296, normalized size = 1.64 \[ \frac {1244755}{2151296} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {536225 \, x}{230496 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {189585}{153664 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{84 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {227}{3528 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {599}{14112 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {12725}{65856 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^5} \,d x \]
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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