Optimal. Leaf size=93 \[ \frac {825}{32} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {25}{8} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {1815}{32} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 93, 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 = {49, 52, 56, 222}
\begin {gather*} -\frac {1815}{32} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {(5 x+3)^{5/2}}{\sqrt {1-2 x}}+\frac {25}{8} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {825}{32} \sqrt {1-2 x} \sqrt {5 x+3} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {25}{2} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {25}{8} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {825}{16} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {825}{32} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {25}{8} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {9075}{64} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {825}{32} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {25}{8} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {1}{32} \left (1815 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {825}{32} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {25}{8} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {1815}{32} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 68, normalized size = 0.73 \begin {gather*} \frac {-2 \sqrt {3+5 x} \left (-1413+790 x+200 x^2\right )+1815 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{64 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (3+5 x \right )^{\frac {5}{2}}}{\left (1-2 x \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 75, normalized size = 0.81 \begin {gather*} -\frac {125 \, x^{3}}{4 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {2275 \, x^{2}}{16 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1815}{128} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {4695 \, x}{32 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4239}{32 \, \sqrt {-10 \, x^{2} - x + 3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.91, size = 87, normalized size = 0.94 \begin {gather*} \frac {1815 \, \sqrt {5} \sqrt {2} {\left (2 \, 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 (200 \, x^{2} + 790 \, x - 1413\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{128 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.75, size = 185, normalized size = 1.99 \begin {gather*} \begin {cases} \frac {125 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{4 \sqrt {10 x - 5}} + \frac {1375 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{16 \sqrt {10 x - 5}} - \frac {9075 i \sqrt {x + \frac {3}{5}}}{32 \sqrt {10 x - 5}} + \frac {1815 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{64} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {1815 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{64} - \frac {125 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{4 \sqrt {5 - 10 x}} - \frac {1375 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{16 \sqrt {5 - 10 x}} + \frac {9075 \sqrt {x + \frac {3}{5}}}{32 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 71, normalized size = 0.76 \begin {gather*} -\frac {1815}{64} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 55 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1815 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{160 \, {\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 )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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