Optimal. Leaf size=171 \[ -\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac {115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{108 (3 x+2)^2}+\frac {365 (5 x+3)^{3/2} \sqrt {1-2 x}}{216 (3 x+2)}-\frac {845}{648} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {362}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {215 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1944 \sqrt {7}} \]
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Rubi [A] time = 0.07, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {(5 x+3)^{3/2} (1-2 x)^{5/2}}{9 (3 x+2)^3}+\frac {115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{108 (3 x+2)^2}+\frac {365 (5 x+3)^{3/2} \sqrt {1-2 x}}{216 (3 x+2)}-\frac {845}{648} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {362}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {215 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1944 \sqrt {7}} \]
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)^{3/2}}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}-\frac {1}{54} \int \frac {\left (-\frac {2325}{4}-735 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}+\frac {1}{162} \int \frac {\sqrt {3+5 x} \left (\frac {6975}{8}+\frac {2535 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac {1}{972} \int \frac {-\frac {57705}{4}-21720 x}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac {215 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{3888}+\frac {1810}{243} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}-\frac {215 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1944}+\frac {1}{243} \left (724 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {845}{648} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac {115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{108 (2+3 x)^2}+\frac {365 \sqrt {1-2 x} (3+5 x)^{3/2}}{216 (2+3 x)}+\frac {362}{243} \sqrt {10} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {215 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1944 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 139, normalized size = 0.81 \[ \frac {21 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (4320 x^3+34341 x^2+36234 x+10304\right )+215 \sqrt {14 x-7} (3 x+2)^3 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-20272 \sqrt {10-20 x} (3 x+2)^3 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{13608 \sqrt {2 x-1} (3 x+2)^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.09, size = 161, normalized size = 0.94 \[ \frac {215 \, \sqrt {7} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20272 \, \sqrt {10} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \, {\left (4320 \, x^{3} + 34341 \, x^{2} + 36234 \, x + 10304\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{27216 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.26, size = 396, normalized size = 2.32 \[ -\frac {43}{54432} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {181}{243} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {4}{81} \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11 \, \sqrt {10} {\left (67 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 56000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {65464000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {261856000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{108 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 270, normalized size = 1.58 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-547344 \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+5805 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-181440 \sqrt {-10 x^{2}-x +3}\, x^{3}-1094688 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+11610 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1442322 \sqrt {-10 x^{2}-x +3}\, x^{2}-729792 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+7740 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1521828 \sqrt {-10 x^{2}-x +3}\, x -162176 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-432768 \sqrt {-10 x^{2}-x +3}\right )}{27216 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 161, normalized size = 0.94 \[ \frac {125}{378} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {25 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{84 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1825}{756} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {181}{243} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {215}{27216} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {655}{4536} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {65 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{504 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^4} \,d x \]
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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