Optimal. Leaf size=96 \[ \frac {(5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {25 (5 x+3)^{3/2}}{6 \sqrt {1-2 x}}-\frac {125}{8} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {275}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {47, 50, 54, 216} \begin {gather*} \frac {(5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {25 (5 x+3)^{3/2}}{6 \sqrt {1-2 x}}-\frac {125}{8} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {275}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {25}{6} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {125}{4} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {125}{8} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1375}{16} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {125}{8} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {1}{8} \left (275 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {125}{8} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {25 (3+5 x)^{3/2}}{6 \sqrt {1-2 x}}+\frac {(3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {275}{8} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 39, normalized size = 0.41 \begin {gather*} \frac {121 \sqrt {\frac {11}{2}} \, _2F_1\left (-\frac {5}{2},-\frac {3}{2};-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{12 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 118, normalized size = 1.23 \begin {gather*} -\frac {5 \sqrt {5} \sqrt {11-2 (5 x+3)} \left (12 (5 x+3)^{5/2}-440 (5 x+3)^{3/2}+1815 \sqrt {5 x+3}\right )}{24 (2 (5 x+3)-11)^2}-\frac {275}{4} \sqrt {\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.31, size = 97, normalized size = 1.01 \begin {gather*} -\frac {825 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, x + 1\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 ) + 4 \, {\left (300 \, x^{2} - 1840 \, x + 603\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{96 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 71, normalized size = 0.74 \begin {gather*} \frac {275}{16} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, \sqrt {5} {\left (5 \, x + 3\right )} - 110 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1815 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{120 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (5 x +3\right )^{\frac {5}{2}}}{\left (-2 x +1\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 129, normalized size = 1.34 \begin {gather*} \frac {275}{32} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {55 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{24 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {605 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {1925 \, \sqrt {-10 \, x^{2} - x + 3}}{48 \, {\left (2 \, x - 1\right )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.55, size = 729, normalized size = 7.59 \begin {gather*} \begin {cases} \frac {16500 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{- 960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} + 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} - \frac {8250 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5}}{- 960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} + 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} - \frac {18150 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{- 960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} + 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} + \frac {9075 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}}{- 960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} + 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} + \frac {30000 i \left (x + \frac {3}{5}\right )^{15}}{- 960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} + 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} - \frac {220000 i \left (x + \frac {3}{5}\right )^{14}}{- 960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} + 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} + \frac {181500 i \left (x + \frac {3}{5}\right )^{13}}{- 960 \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \sqrt {10 x - 5} + 1056 \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \sqrt {10 x - 5}} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {8250 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} - \frac {9075 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} + \frac {15000 \left (x + \frac {3}{5}\right )^{15}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} - \frac {110000 \left (x + \frac {3}{5}\right )^{14}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} + \frac {90750 \left (x + \frac {3}{5}\right )^{13}}{480 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {27}{2}} - 528 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {25}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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