Optimal. Leaf size=135 \[ -\frac {4246733 \sqrt {1-2 x} \sqrt {3+5 x}}{14080}-\frac {101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac {4246733 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1280 \sqrt {10}} \]
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Rubi [A]
time = 0.03, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {4246733 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1280 \sqrt {10}}+\frac {(5 x+3)^{3/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac {101 (5 x+3)^{3/2} (3 x+2)^2}{22 \sqrt {1-2 x}}-\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2} (28200 x+59719)}{3520}-\frac {4246733 \sqrt {1-2 x} \sqrt {5 x+3}}{14080} \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)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^2 \sqrt {3+5 x} \left (42+\frac {135 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {9969}{2}-\frac {31725 x}{4}\right ) (2+3 x) \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac {4246733 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{7040}\\ &=-\frac {4246733 \sqrt {1-2 x} \sqrt {3+5 x}}{14080}-\frac {101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac {4246733 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2560}\\ &=-\frac {4246733 \sqrt {1-2 x} \sqrt {3+5 x}}{14080}-\frac {101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac {4246733 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1280 \sqrt {5}}\\ &=-\frac {4246733 \sqrt {1-2 x} \sqrt {3+5 x}}{14080}-\frac {101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt {1-2 x}}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac {4246733 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1280 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 83, normalized size = 0.61 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (1925361-5349344 x+1544724 x^2+447120 x^3+86400 x^4\right )+12740199 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{38400 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 154, normalized size = 1.14
method | result | size |
default | \(\frac {\left (-1728000 x^{4} \sqrt {-10 x^{2}-x +3}+50960796 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-8942400 x^{3} \sqrt {-10 x^{2}-x +3}-50960796 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -30894480 x^{2} \sqrt {-10 x^{2}-x +3}+12740199 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+106986880 x \sqrt {-10 x^{2}-x +3}-38507220 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{76800 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.53, size = 211, normalized size = 1.56 \begin {gather*} \frac {428267}{2560} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {35937}{25600} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) + \frac {9}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {297}{64} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x + \frac {6237}{1280} \, \sqrt {10 \, x^{2} - 21 \, x + 8} - \frac {6237}{128} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{48 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {189 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (2 \, x - 1\right )}} + \frac {3773 \, \sqrt {-10 \, x^{2} - x + 3}}{96 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {3479 \, \sqrt {-10 \, x^{2} - x + 3}}{6 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.50, size = 101, normalized size = 0.75 \begin {gather*} -\frac {12740199 \, \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 (86400 \, x^{4} + 447120 \, x^{3} + 1544724 \, x^{2} - 5349344 \, x + 1925361\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{76800 \, {\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 = 0.84, size = 97, normalized size = 0.72 \begin {gather*} \frac {4246733}{12800} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (27 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 111 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 8579 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 8493466 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 140142189 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{480000 \, {\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 )}^{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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