Optimal. Leaf size=132 \[ \frac {(5 x+3)^{3/2} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {27}{16} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2+\frac {9 \sqrt {1-2 x} (5 x+3)^{3/2} (29320 x+62091)}{12800}+\frac {13246251 \sqrt {1-2 x} \sqrt {5 x+3}}{51200}-\frac {145708761 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 132, 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 = {97, 153, 147, 50, 54, 216} \[ \frac {(5 x+3)^{3/2} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {27}{16} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2+\frac {9 \sqrt {1-2 x} (5 x+3)^{3/2} (29320 x+62091)}{12800}+\frac {13246251 \sqrt {1-2 x} \sqrt {5 x+3}}{51200}-\frac {145708761 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200 \sqrt {10}} \]
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)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^2 \sqrt {3+5 x} \left (42+\frac {135 x}{2}\right )}{\sqrt {1-2 x}} \, dx\\ &=\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {1}{40} \int \frac {\left (-\frac {10365}{2}-\frac {32985 x}{4}\right ) (2+3 x) \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {13246251 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{25600}\\ &=\frac {13246251 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {145708761 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{102400}\\ &=\frac {13246251 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {145708761 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{51200 \sqrt {5}}\\ &=\frac {13246251 \sqrt {1-2 x} \sqrt {3+5 x}}{51200}+\frac {27}{16} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac {(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac {145708761 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 88, normalized size = 0.67 \[ \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (864000 x^4+3729600 x^3+8057880 x^2+15218818 x-22217679\right )-145708761 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{512000 \sqrt {-(1-2 x)^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.92, size = 91, normalized size = 0.69 \[ \frac {145708761 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (864000 \, x^{4} + 3729600 \, x^{3} + 8057880 \, x^{2} + 15218818 \, x - 22217679\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1024000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.00, size = 97, normalized size = 0.73 \[ -\frac {145708761}{512000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (2 \, {\left (36 \, {\left (8 \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} + 115 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 8919 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4415417 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 145708761 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1280000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 140, normalized size = 1.06 \[ -\frac {\left (-17280000 \sqrt {-10 x^{2}-x +3}\, x^{4}-74592000 \sqrt {-10 x^{2}-x +3}\, x^{3}-161157600 \sqrt {-10 x^{2}-x +3}\, x^{2}+291417522 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-304376360 \sqrt {-10 x^{2}-x +3}\, x -145708761 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+444353580 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{1024000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.20, size = 184, normalized size = 1.39 \[ -\frac {27}{32} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {155771121}{1024000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {251559}{25600} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x - \frac {21}{11}\right ) - \frac {2547}{640} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {2079}{64} \, \sqrt {10 \, x^{2} - 21 \, x + 8} x - \frac {9801}{2560} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {43659}{1280} \, \sqrt {10 \, x^{2} - 21 \, x + 8} + \frac {5811399}{51200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{16 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {441 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (2 \, x - 1\right )}} - \frac {11319 \, \sqrt {-10 \, x^{2} - x + 3}}{32 \, {\left (2 \, x - 1\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]
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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