Optimal. Leaf size=71 \[ \frac {15}{4} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {33}{4} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {33}{4} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {(5 x+3)^{3/2}}{\sqrt {1-2 x}}+\frac {15}{4} \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)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {15}{2} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {15}{4} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {165}{8} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {15}{4} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {1}{4} \left (33 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {15}{4} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}}-\frac {33}{4} \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 = 0.22, size = 75, normalized size = 1.06 \begin {gather*} \frac {\sqrt {1-2 x} \sqrt {3+5 x} (-27+10 x)+33 \sqrt {10} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{-4+8 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (3+5 x \right )^{\frac {3}{2}}}{\left (1-2 x \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 62, normalized size = 0.87 \begin {gather*} -\frac {33}{16} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {33 \, \sqrt {-10 \, x^{2} - x + 3}}{4 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.62, size = 82, normalized size = 1.15 \begin {gather*} \frac {33 \, \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 (10 \, x - 27\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{16 \, {\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 = 1.70, size = 143, normalized size = 2.01 \begin {gather*} \begin {cases} \frac {25 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{2 \sqrt {10 x - 5}} - \frac {165 i \sqrt {x + \frac {3}{5}}}{4 \sqrt {10 x - 5}} + \frac {33 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{8} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {33 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{8} - \frac {25 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{2 \sqrt {5 - 10 x}} + \frac {165 \sqrt {x + \frac {3}{5}}}{4 \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 = 0.67, size = 58, normalized size = 0.82 \begin {gather*} -\frac {33}{8} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{20 \, {\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 )}^{3/2}}{{\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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