Optimal. Leaf size=183 \[ \frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {13}{8} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac {999}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac {295101237 \sqrt {1-2 x} (5 x+3)^{3/2}}{409600}+\frac {\sqrt {1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac {9738340821 \sqrt {1-2 x} \sqrt {5 x+3}}{1638400}-\frac {107121749031 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1638400 \sqrt {10}} \]
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Rubi [A] time = 0.06, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 153, 147, 50, 54, 216} \begin {gather*} \frac {(5 x+3)^{5/2} (3 x+2)^4}{\sqrt {1-2 x}}+\frac {13}{8} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^3+\frac {999}{160} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^2+\frac {295101237 \sqrt {1-2 x} (5 x+3)^{3/2}}{409600}+\frac {\sqrt {1-2 x} (5 x+3)^{5/2} (3765060 x+7611023)}{51200}+\frac {9738340821 \sqrt {1-2 x} \sqrt {5 x+3}}{1638400}-\frac {107121749031 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1638400 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 97
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^3 (3+5 x)^{3/2} \left (61+\frac {195 x}{2}\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {1}{60} \int \frac {\left (-11805-\frac {74925 x}{4}\right ) (2+3 x)^2 (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}-\frac {\int \frac {(2+3 x) (3+5 x)^{3/2} \left (\frac {7494225}{4}+\frac {23531625 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{3000}\\ &=\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {295101237 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{102400}\\ &=\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {9738340821 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{819200}\\ &=\frac {9738340821 \sqrt {1-2 x} \sqrt {3+5 x}}{1638400}+\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {107121749031 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3276800}\\ &=\frac {9738340821 \sqrt {1-2 x} \sqrt {3+5 x}}{1638400}+\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {107121749031 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1638400 \sqrt {5}}\\ &=\frac {9738340821 \sqrt {1-2 x} \sqrt {3+5 x}}{1638400}+\frac {295101237 \sqrt {1-2 x} (3+5 x)^{3/2}}{409600}+\frac {999}{160} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}+\frac {13}{8} \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (7611023+3765060 x)}{51200}-\frac {107121749031 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1638400 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 98, normalized size = 0.54 \begin {gather*} \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (276480000 x^6+1479168000 x^5+3687379200 x^4+5945485120 x^3+7755469800 x^2+11734056318 x-16267424049\right )-107121749031 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{16384000 \sqrt {-(1-2 x)^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.33, size = 173, normalized size = 0.95 \begin {gather*} \frac {121 \sqrt {5 x+3} \left (\frac {2766574096875 (1-2 x)^6}{(5 x+3)^6}+\frac {6270901286250 (1-2 x)^5}{(5 x+3)^5}+\frac {5843004492600 (1-2 x)^4}{(5 x+3)^4}+\frac {2843089725520 (1-2 x)^3}{(5 x+3)^3}+\frac {749840250480 (1-2 x)^2}{(5 x+3)^2}+\frac {97270844448 (1-2 x)}{5 x+3}+3933798400\right )}{1638400 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}+\frac {107121749031 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{1638400 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 101, normalized size = 0.55 \begin {gather*} \frac {107121749031 \, \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 (276480000 \, x^{6} + 1479168000 \, x^{5} + 3687379200 \, x^{4} + 5945485120 \, x^{3} + 7755469800 \, x^{2} + 11734056318 \, x - 16267424049\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{32768000 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 123, normalized size = 0.67 \begin {gather*} -\frac {107121749031}{16384000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 35 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4299 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 3832457 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 295101237 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 16230568035 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 535608745155 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{204800000 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 174, normalized size = 0.95 \begin {gather*} -\frac {\left (-5529600000 \sqrt {-10 x^{2}-x +3}\, x^{6}-29583360000 \sqrt {-10 x^{2}-x +3}\, x^{5}-73747584000 \sqrt {-10 x^{2}-x +3}\, x^{4}-118909702400 \sqrt {-10 x^{2}-x +3}\, x^{3}-155109396000 \sqrt {-10 x^{2}-x +3}\, x^{2}+214243498062 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-234681126360 \sqrt {-10 x^{2}-x +3}\, x -107121749031 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+325348480980 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{32768000 \left (2 x -1\right ) \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 = 1.48, size = 143, normalized size = 0.78 \begin {gather*} -\frac {3375 \, x^{7}}{4 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {80325 \, x^{6}}{16 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {3574125 \, x^{5}}{256 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {25493477 \, x^{4}}{1024 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1415345109 \, x^{3}}{40960 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {8193669099 \, x^{2}}{163840 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {107121749031}{32768000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {46134951291 \, x}{1638400 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {48802272147}{1638400 \, \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 )}^4\,{\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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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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