Optimal. Leaf size=188 \[ \frac {575}{162} \sqrt {1-2 x} (5 x+3)^{5/2}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac {785}{36} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {34145 \sqrt {1-2 x} \sqrt {5 x+3}}{1944}+\frac {81733 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5832}+\frac {21935 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2916} \]
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Rubi [A] time = 0.08, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ \frac {575}{162} \sqrt {1-2 x} (5 x+3)^{5/2}+\frac {185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{36 (3 x+2)}-\frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{6 (3 x+2)^2}-\frac {785}{36} \sqrt {1-2 x} (5 x+3)^{3/2}+\frac {34145 \sqrt {1-2 x} \sqrt {5 x+3}}{1944}+\frac {81733 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5832}+\frac {21935 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{2916} \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 97
Rule 149
Rule 154
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {1}{6} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {1}{18} \int \frac {\left (-\frac {355}{4}-2875 x\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {1}{810} \int \frac {\left (\frac {202525}{4}-211950 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac {\int \frac {(84825-1024350 x) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)} \, dx}{9720}\\ &=\frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {\int \frac {-2551200-6129975 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{58320}\\ &=\frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {153545 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{5832}+\frac {408665 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{11664}\\ &=\frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}-\frac {153545 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{2916}+\frac {\left (81733 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5832}\\ &=\frac {34145 \sqrt {1-2 x} \sqrt {3+5 x}}{1944}-\frac {785}{36} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {575}{162} \sqrt {1-2 x} (3+5 x)^{5/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{6 (2+3 x)^2}+\frac {185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{36 (2+3 x)}+\frac {81733 \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5832}+\frac {21935 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{2916}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 144, normalized size = 0.77 \[ \frac {6 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (21600 x^4-28980 x^3+31731 x^2+120534 x+53204\right )+87740 \sqrt {14 x-7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-81733 \sqrt {10-20 x} (3 x+2)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{11664 \sqrt {2 x-1} (3 x+2)^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.14, size = 157, normalized size = 0.84 \[ -\frac {81733 \, \sqrt {5} \sqrt {2} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 87740 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 12 \, {\left (21600 \, x^{4} - 28980 \, x^{3} + 31731 \, x^{2} + 120534 \, x + 53204\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{23328 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.16, size = 364, normalized size = 1.94 \[ -\frac {4387}{11664} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{3240} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 155 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 5245 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {81733}{23328} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {77 \, \sqrt {10} {\left (263 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {92120 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {368480 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{486 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 242, normalized size = 1.29 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (259200 \sqrt {-10 x^{2}-x +3}\, x^{4}-347760 \sqrt {-10 x^{2}-x +3}\, x^{3}+735597 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-789660 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+380772 \sqrt {-10 x^{2}-x +3}\, x^{2}+980796 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1052880 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1446408 \sqrt {-10 x^{2}-x +3}\, x +326932 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-350960 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+638448 \sqrt {-10 x^{2}-x +3}\right )}{23328 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 159, normalized size = 0.85 \[ \frac {5}{21} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {925}{126} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {10135}{2268} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {37 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{28 \, {\left (3 \, x + 2\right )}} - \frac {925}{81} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {81733}{23328} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {21935}{5832} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {20825}{1944} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^3} \,d x \]
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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