Optimal. Leaf size=74 \[ -\frac {5 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {5}{2} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {49, 56, 222}
\begin {gather*} \frac {5}{2} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {(5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {5 \sqrt {5 x+3}}{2 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {5}{2} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {5 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {25}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {5 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {1}{2} \left (5 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {5 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {(3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}+\frac {5}{2} \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 = 0.10, size = 68, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {3+5 x} (-9+40 x)+15 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{12 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (3+5 x \right )^{\frac {3}{2}}}{\left (1-2 x \right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 93, normalized size = 1.26 \begin {gather*} \frac {5}{8} \, \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}}}{6 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {35 \, \sqrt {-10 \, x^{2} - x + 3}}{12 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 92, normalized size = 1.24 \begin {gather*} -\frac {15 \, \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 (40 \, x - 9\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24 \, {\left (4 \, x^{2} - 4 \, 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 = 2.32, size = 634, normalized size = 8.57 \begin {gather*} \begin {cases} - \frac {300 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} + \frac {150 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} + \frac {330 \sqrt {10} i \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5} \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} - \frac {165 \sqrt {10} \pi \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} + \frac {4000 i \left (x + \frac {3}{5}\right )^{8}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} - \frac {3300 i \left (x + \frac {3}{5}\right )^{7}}{240 \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \sqrt {10 x - 5} - 264 \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \sqrt {10 x - 5}} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {150 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} - \frac {165 \sqrt {10} \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} - \frac {2000 \left (x + \frac {3}{5}\right )^{8}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} + \frac {1650 \left (x + \frac {3}{5}\right )^{7}}{120 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {15}{2}} - 132 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right )^{\frac {13}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.09, size = 58, normalized size = 0.78 \begin {gather*} \frac {5}{4} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 33 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{30 \, {\left (2 \, x - 1\right )}^{2}} \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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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