Optimal. Leaf size=157 \[ -\frac {9444023 \sqrt {1-2 x} \sqrt {3+5 x}}{4096}-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {103884253 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4096 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 155, 152,
52, 56, 222} \begin {gather*} \frac {103884253 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{4096 \sqrt {10}}+\frac {(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac {373 (5 x+3)^{5/2} (3 x+2)^2}{66 \sqrt {1-2 x}}-\frac {9444023 \sqrt {1-2 x} (5 x+3)^{3/2}}{33792}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2} (40164 x+81191)}{1408}-\frac {9444023 \sqrt {1-2 x} \sqrt {5 x+3}}{4096} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 99
Rule 152
Rule 155
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^2 (3+5 x)^{3/2} \left (52+\frac {165 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {15989}{2}-\frac {50205 x}{4}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {9444023 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{8448}\\ &=-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {9444023 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{2048}\\ &=-\frac {9444023 \sqrt {1-2 x} \sqrt {3+5 x}}{4096}-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {103884253 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{8192}\\ &=-\frac {9444023 \sqrt {1-2 x} \sqrt {3+5 x}}{4096}-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {103884253 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{4096 \sqrt {5}}\\ &=-\frac {9444023 \sqrt {1-2 x} \sqrt {3+5 x}}{4096}-\frac {9444023 \sqrt {1-2 x} (3+5 x)^{3/2}}{33792}-\frac {373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac {103884253 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4096 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 87, normalized size = 0.55 \begin {gather*} \frac {-\frac {5 \sqrt {3+5 x} \left (47216961-129940960 x+40614996 x^2+15301008 x^3+5477760 x^4+1036800 x^5\right )}{(1-2 x)^{3/2}}-311652759 \sqrt {10} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {11}-\sqrt {5-10 x}}\right )}{61440} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 171, normalized size = 1.09
method | result | size |
default | \(\frac {\left (-20736000 x^{5} \sqrt {-10 x^{2}-x +3}-109555200 x^{4} \sqrt {-10 x^{2}-x +3}+1246611036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-306020160 x^{3} \sqrt {-10 x^{2}-x +3}-1246611036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -812299920 x^{2} \sqrt {-10 x^{2}-x +3}+311652759 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2598819200 x \sqrt {-10 x^{2}-x +3}-944339220 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{245760 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(171\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.54, size = 325, normalized size = 2.07 \begin {gather*} \frac {2606989}{2048} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {395307}{81920} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) + \frac {495}{256} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {343 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac {63 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{64 \, {\left (2 \, x - 1\right )}} - \frac {16335}{1024} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x + \frac {68607}{4096} \, \sqrt {10 \, x^{2} - 21 \, x + 8} - \frac {114345}{512} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {18865 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{192 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {24255 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{128 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {3465 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{128 \, {\left (2 \, x - 1\right )}} + \frac {207515 \, \sqrt {-10 \, x^{2} - x + 3}}{384 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {3721795 \, \sqrt {-10 \, x^{2} - x + 3}}{768 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 106, normalized size = 0.68 \begin {gather*} -\frac {311652759 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, 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 (1036800 \, x^{5} + 5477760 \, x^{4} + 15301008 \, x^{3} + 40614996 \, x^{2} - 129940960 \, x + 47216961\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{245760 \, {\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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 110, normalized size = 0.70 \begin {gather*} \frac {103884253}{40960} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, {\left (36 \, {\left (8 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 137 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 13627 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 9444023 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1038842530 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 17140901745 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{7680000 \, {\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 (3\,x+2\right )}^3\,{\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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