Optimal. Leaf size=186 \[ \frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac {439 (5 x+3)^{5/2} (3 x+2)^3}{66 \sqrt {1-2 x}}-\frac {4819}{440} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^2-\frac {4270537963 \sqrt {1-2 x} (5 x+3)^{3/2}}{3379200}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)}{140800}-\frac {4270537963 \sqrt {1-2 x} \sqrt {5 x+3}}{409600}+\frac {46975917593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{409600 \sqrt {10}} \]
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Rubi [A] time = 0.06, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 150, 153, 147, 50, 54, 216} \[ \frac {(5 x+3)^{5/2} (3 x+2)^4}{3 (1-2 x)^{3/2}}-\frac {439 (5 x+3)^{5/2} (3 x+2)^3}{66 \sqrt {1-2 x}}-\frac {4819}{440} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^2-\frac {4270537963 \sqrt {1-2 x} (5 x+3)^{3/2}}{3379200}-\frac {\sqrt {1-2 x} (5 x+3)^{5/2} (18161940 x+36714139)}{140800}-\frac {4270537963 \sqrt {1-2 x} \sqrt {5 x+3}}{409600}+\frac {46975917593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{409600 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 97
Rule 147
Rule 150
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^3 (3+5 x)^{3/2} \left (61+\frac {195 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-11389-\frac {72285 x}{4}\right ) (2+3 x)^2 (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {4819}{440} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {(2+3 x) (3+5 x)^{3/2} \left (\frac {7230145}{4}+\frac {22702425 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{1650}\\ &=-\frac {4819}{440} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac {4270537963 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{844800}\\ &=-\frac {4270537963 \sqrt {1-2 x} (3+5 x)^{3/2}}{3379200}-\frac {4819}{440} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac {4270537963 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{204800}\\ &=-\frac {4270537963 \sqrt {1-2 x} \sqrt {3+5 x}}{409600}-\frac {4270537963 \sqrt {1-2 x} (3+5 x)^{3/2}}{3379200}-\frac {4819}{440} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac {46975917593 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{819200}\\ &=-\frac {4270537963 \sqrt {1-2 x} \sqrt {3+5 x}}{409600}-\frac {4270537963 \sqrt {1-2 x} (3+5 x)^{3/2}}{3379200}-\frac {4819}{440} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac {46975917593 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{409600 \sqrt {5}}\\ &=-\frac {4270537963 \sqrt {1-2 x} \sqrt {3+5 x}}{409600}-\frac {4270537963 \sqrt {1-2 x} (3+5 x)^{3/2}}{3379200}-\frac {4819}{440} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}-\frac {439 (2+3 x)^3 (3+5 x)^{5/2}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^4 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2} (36714139+18161940 x)}{140800}+\frac {46975917593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{409600 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 105, normalized size = 0.56 \[ \frac {10 \sqrt {2 x-1} \sqrt {5 x+3} \left (248832000 x^6+1423526400 x^5+4002203520 x^4+8217694800 x^3+18987469764 x^2-58600061024 x+21368105901\right )+140927752779 \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{12288000 \sqrt {1-2 x} (2 x-1)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 111, normalized size = 0.60 \[ -\frac {140927752779 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (248832000 \, x^{6} + 1423526400 \, x^{5} + 4002203520 \, x^{4} + 8217694800 \, x^{3} + 18987469764 \, x^{2} - 58600061024 \, x + 21368105901\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24576000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 123, normalized size = 0.66 \[ \frac {46975917593}{4096000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (3 \, {\left (12 \, {\left (72 \, {\left (4 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} + 509 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 20743 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 18487133 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4270537963 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 469759175930 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 7751026402845 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{768000000 \, {\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 188, normalized size = 1.01 \[ \frac {\left (-4976640000 \sqrt {-10 x^{2}-x +3}\, x^{6}-28470528000 \sqrt {-10 x^{2}-x +3}\, x^{5}-80044070400 \sqrt {-10 x^{2}-x +3}\, x^{4}-164353896000 \sqrt {-10 x^{2}-x +3}\, x^{3}+563711011116 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-379749395280 \sqrt {-10 x^{2}-x +3}\, x^{2}-563711011116 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1172001220480 \sqrt {-10 x^{2}-x +3}\, x +140927752779 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-427362118020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{24576000 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.44, size = 354, normalized size = 1.90 \[ -\frac {81}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {891}{256} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {11872553}{2048} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {514294407}{8192000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) + \frac {139491}{5120} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {2401 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32 \, {\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac {1029 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {189 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{32 \, {\left (2 \, x - 1\right )}} - \frac {4250367}{20480} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x + \frac {89257707}{409600} \, \sqrt {10 \, x^{2} - 21 \, x + 8} - \frac {800415}{512} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {132055 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{384 \, {\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac {56595 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{64 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {24255 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{128 \, {\left (2 \, x - 1\right )}} + \frac {1452605 \, \sqrt {-10 \, x^{2} - x + 3}}{768 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {15827735 \, \sqrt {-10 \, x^{2} - x + 3}}{768 \, {\left (2 \, x - 1\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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