Optimal. Leaf size=138 \[ \frac {838101 \sqrt {1-2 x} \sqrt {3+5 x}}{2048}+\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {9219111 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2048 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {91, 81, 52, 56,
222} \begin {gather*} -\frac {9219111 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2048 \sqrt {10}}+\frac {9}{80} \sqrt {1-2 x} (5 x+3)^{7/2}+\frac {49 (5 x+3)^{7/2}}{22 \sqrt {1-2 x}}+\frac {25397 \sqrt {1-2 x} (5 x+3)^{5/2}}{3520}+\frac {25397}{512} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {838101 \sqrt {1-2 x} \sqrt {5 x+3}}{2048} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 81
Rule 91
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}-\frac {1}{22} \int \frac {(3+5 x)^{5/2} \left (\frac {1833}{2}+99 x\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {76191 \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx}{1760}\\ &=\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {25397}{128} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {838101 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{1024}\\ &=\frac {838101 \sqrt {1-2 x} \sqrt {3+5 x}}{2048}+\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {9219111 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{4096}\\ &=\frac {838101 \sqrt {1-2 x} \sqrt {3+5 x}}{2048}+\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {9219111 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2048 \sqrt {5}}\\ &=\frac {838101 \sqrt {1-2 x} \sqrt {3+5 x}}{2048}+\frac {25397}{512} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {25397 \sqrt {1-2 x} (3+5 x)^{5/2}}{3520}+\frac {49 (3+5 x)^{7/2}}{22 \sqrt {1-2 x}}+\frac {9}{80} \sqrt {1-2 x} (3+5 x)^{7/2}-\frac {9219111 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2048 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 78, normalized size = 0.57 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (-1405233+966014 x+517096 x^2+243520 x^3+57600 x^4\right )+9219111 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{20480 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 140, normalized size = 1.01
method | result | size |
default | \(-\frac {\left (-1152000 x^{4} \sqrt {-10 x^{2}-x +3}-4870400 x^{3} \sqrt {-10 x^{2}-x +3}+18438222 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -10341920 x^{2} \sqrt {-10 x^{2}-x +3}-9219111 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-19320280 x \sqrt {-10 x^{2}-x +3}+28104660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{40960 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 109, normalized size = 0.79 \begin {gather*} -\frac {1125 \, x^{5}}{8 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {21725 \, x^{4}}{32 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {414505 \, x^{3}}{256 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3190679 \, x^{2}}{1024 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {9219111}{40960} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {4128123 \, x}{2048 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {4215699}{2048 \, \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 = 0.95, size = 91, normalized size = 0.66 \begin {gather*} \frac {9219111 \, \sqrt {10} {\left (2 \, x - 1\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 (57600 \, x^{4} + 243520 \, x^{3} + 517096 \, x^{2} + 966014 \, x - 1405233\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{40960 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{2} \left (5 x + 3\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 97, normalized size = 0.70 \begin {gather*} -\frac {9219111}{20480} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} + 329 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 25397 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1396835 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 46095555 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{256000 \, {\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 (3\,x+2\right )}^2\,{\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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