Optimal. Leaf size=128 \[ -\frac {3}{50} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac {3 \sqrt {1-2 x} (3900 x+7889) (5 x+3)^{5/2}}{16000}-\frac {917953 \sqrt {1-2 x} (5 x+3)^{3/2}}{128000}-\frac {30292449 \sqrt {1-2 x} \sqrt {5 x+3}}{512000}+\frac {333216939 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{512000 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \begin {gather*} -\frac {3}{50} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{5/2}-\frac {3 \sqrt {1-2 x} (3900 x+7889) (5 x+3)^{5/2}}{16000}-\frac {917953 \sqrt {1-2 x} (5 x+3)^{3/2}}{128000}-\frac {30292449 \sqrt {1-2 x} \sqrt {5 x+3}}{512000}+\frac {333216939 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{512000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {1}{50} \int \frac {\left (-311-\frac {975 x}{2}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {917953 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{32000}\\ &=-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {30292449 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{256000}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1024000}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{512000 \sqrt {5}}\\ &=-\frac {30292449 \sqrt {1-2 x} \sqrt {3+5 x}}{512000}-\frac {917953 \sqrt {1-2 x} (3+5 x)^{3/2}}{128000}-\frac {3}{50} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{5/2} (7889+3900 x)}{16000}+\frac {333216939 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{512000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 88, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (6912000 x^4+26870400 x^3+46785120 x^2+51453140 x+49229901\right )+333216939 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{5120000 \sqrt {2 x-1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 141, normalized size = 1.10 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {1721161875 (1-2 x)^4}{(5 x+3)^4}+\frac {3212835500 (1-2 x)^3}{(5 x+3)^3}+\frac {2349918720 (1-2 x)^2}{(5 x+3)^2}+\frac {828154320 (1-2 x)}{5 x+3}+131554256\right )}{512000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {333216939 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{512000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 77, normalized size = 0.60 \begin {gather*} -\frac {1}{512000} \, {\left (6912000 \, x^{4} + 26870400 \, x^{3} + 46785120 \, x^{2} + 51453140 \, x + 49229901\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {333216939}{10240000} \, \sqrt {10} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.30, size = 72, normalized size = 0.56 \begin {gather*} -\frac {1}{25600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (24 \, {\left (36 \, {\left (80 \, x + 167\right )} {\left (5 \, x + 3\right )} + 27809\right )} {\left (5 \, x + 3\right )} + 4589765\right )} {\left (5 \, x + 3\right )} + 151462245\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 1666084695 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 121, normalized size = 0.95 \begin {gather*} \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-138240000 \sqrt {-10 x^{2}-x +3}\, x^{4}-537408000 \sqrt {-10 x^{2}-x +3}\, x^{3}-935702400 \sqrt {-10 x^{2}-x +3}\, x^{2}-1029062800 \sqrt {-10 x^{2}-x +3}\, x +333216939 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-984598020 \sqrt {-10 x^{2}-x +3}\right )}{10240000 \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 92, normalized size = 0.72 \begin {gather*} -\frac {27}{2} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {8397}{160} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {292407}{3200} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {2572657}{25600} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {333216939}{10240000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {49229901}{512000} \, \sqrt {-10 \, x^{2} - x + 3} \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}}{\sqrt {1-2\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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