Optimal. Leaf size=186 \[ -\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}+\frac {13}{50} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3+\frac {111 (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^2}{5000}-\frac {(1-2 x)^{5/2} \sqrt {5 x+3} (1990620 x+2725981)}{8000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {5 x+3}}{32000000}+\frac {118054167 \sqrt {1-2 x} \sqrt {5 x+3}}{320000000}+\frac {1298595837 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320000000 \sqrt {10}} \]
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Rubi [A] time = 0.06, antiderivative size = 186, 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 = {97, 153, 147, 50, 54, 216} \[ -\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt {5 x+3}}+\frac {13}{50} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^3+\frac {111 (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^2}{5000}-\frac {(1-2 x)^{5/2} \sqrt {5 x+3} (1990620 x+2725981)}{8000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {5 x+3}}{32000000}+\frac {118054167 \sqrt {1-2 x} \sqrt {5 x+3}}{320000000}+\frac {1298595837 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320000000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 97
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {2}{5} \int \frac {(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {1}{150} \int \frac {(1-2 x)^{3/2} (2+3 x)^2 \left (-162+\frac {333 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {\int \frac {(1-2 x)^{3/2} (2+3 x) \left (\frac {34731}{2}+\frac {99531 x}{4}\right )}{\sqrt {3+5 x}} \, dx}{7500}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {3577399 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{3200000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {118054167 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{64000000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {118054167 \sqrt {1-2 x} \sqrt {3+5 x}}{320000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {1298595837 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{640000000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {118054167 \sqrt {1-2 x} \sqrt {3+5 x}}{320000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {1298595837 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{320000000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt {3+5 x}}+\frac {118054167 \sqrt {1-2 x} \sqrt {3+5 x}}{320000000}+\frac {3577399 (1-2 x)^{3/2} \sqrt {3+5 x}}{32000000}+\frac {111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt {3+5 x}}{5000}+\frac {13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} \sqrt {3+5 x} (2725981+1990620 x)}{8000000}+\frac {1298595837 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320000000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 98, normalized size = 0.53 \[ \frac {1298595837 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \left (6912000000 x^7+4631040000 x^6-9103968000 x^5-4815780800 x^4+5550785640 x^3+1793366630 x^2-1029299623 x-168414751\right )}{3200000000 \sqrt {1-2 x} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 101, normalized size = 0.54 \[ -\frac {1298595837 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (3456000000 \, x^{6} + 4043520000 \, x^{5} - 2530224000 \, x^{4} - 3673002400 \, x^{3} + 938891620 \, x^{2} + 1366129125 \, x + 168414751\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{6400000000 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.38, size = 163, normalized size = 0.88 \[ \frac {1}{8000000000} \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (20 \, \sqrt {5} {\left (5 \, x + 3\right )} - 243 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9263 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 2532859 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 3473645 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 533500275 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {1298595837}{3200000000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {121 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{781250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{390625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 167, normalized size = 0.90 \[ \frac {\left (69120000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+80870400000 \sqrt {-10 x^{2}-x +3}\, x^{5}-50604480000 \sqrt {-10 x^{2}-x +3}\, x^{4}-73460048000 \sqrt {-10 x^{2}-x +3}\, x^{3}+18777832400 \sqrt {-10 x^{2}-x +3}\, x^{2}+6492979185 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+27322582500 \sqrt {-10 x^{2}-x +3}\, x +3895787511 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3368295020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{6400000000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 143, normalized size = 0.77 \[ -\frac {108 \, x^{7}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1809 \, x^{6}}{125 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {284499 \, x^{5}}{10000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3009863 \, x^{4}}{200000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {138769641 \, x^{3}}{8000000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {179336663 \, x^{2}}{32000000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1298595837}{6400000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1029299623 \, x}{320000000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {168414751}{320000000 \, \sqrt {-10 \, x^{2} - x + 3}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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