Optimal. Leaf size=180 \[ \frac {87374783 \sqrt {1-2 x} \sqrt {5 x+3}}{131712 (3 x+2)}+\frac {835409 \sqrt {1-2 x} \sqrt {5 x+3}}{9408 (3 x+2)^2}+\frac {23909 \sqrt {1-2 x} \sqrt {5 x+3}}{1680 (3 x+2)^3}+\frac {293 \sqrt {1-2 x} \sqrt {5 x+3}}{120 (3 x+2)^4}+\frac {7 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}-\frac {333216939 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]
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Rubi [A] time = 0.07, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 151, 12, 93, 204} \begin {gather*} \frac {87374783 \sqrt {1-2 x} \sqrt {5 x+3}}{131712 (3 x+2)}+\frac {835409 \sqrt {1-2 x} \sqrt {5 x+3}}{9408 (3 x+2)^2}+\frac {23909 \sqrt {1-2 x} \sqrt {5 x+3}}{1680 (3 x+2)^3}+\frac {293 \sqrt {1-2 x} \sqrt {5 x+3}}{120 (3 x+2)^4}+\frac {7 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^5}-\frac {333216939 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^6 \sqrt {3+5 x}} \, dx &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {\frac {337}{2}-260 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {1}{420} \int \frac {\frac {85323}{4}-30765 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {\int \frac {\frac {15850275}{8}-2510445 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{8820}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {\int \frac {\frac {1888544805}{16}-\frac {438589725 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{123480}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {\int \frac {104963335785}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{864360}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {333216939 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}+\frac {333216939 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=\frac {7 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^5}+\frac {293 \sqrt {1-2 x} \sqrt {3+5 x}}{120 (2+3 x)^4}+\frac {23909 \sqrt {1-2 x} \sqrt {3+5 x}}{1680 (2+3 x)^3}+\frac {835409 \sqrt {1-2 x} \sqrt {3+5 x}}{9408 (2+3 x)^2}+\frac {87374783 \sqrt {1-2 x} \sqrt {3+5 x}}{131712 (2+3 x)}-\frac {333216939 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{43904 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 135, normalized size = 0.75 \begin {gather*} \frac {1}{35} \left (\frac {5 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (262124349 x^2+361165738 x+124968544\right )}{(3 x+2)^3}-333216939 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{43904}+\frac {975 \sqrt {5 x+3} (1-2 x)^{5/2}}{56 (3 x+2)^4}+\frac {3 \sqrt {5 x+3} (1-2 x)^{5/2}}{(3 x+2)^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 138, normalized size = 0.77 \begin {gather*} \frac {121 \sqrt {1-2 x} \left (\frac {41110705 (1-2 x)^4}{(5 x+3)^4}+\frac {724635030 (1-2 x)^3}{(5 x+3)^3}+\frac {5757300864 (1-2 x)^2}{(5 x+3)^2}+\frac {22040051530 (1-2 x)}{5 x+3}+33060077295\right )}{219520 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^5}-\frac {333216939 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.21, size = 131, normalized size = 0.73 \begin {gather*} -\frac {1666084695 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 ) - 14 \, {\left (11795595705 \, x^{4} + 31981229550 \, x^{3} + 32535654204 \, x^{2} + 14720806136 \, x + 2499608096\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{3073280 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.96, size = 426, normalized size = 2.37 \begin {gather*} \frac {333216939}{6146560} \, \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 {121 \, \sqrt {10} {\left (8222141 \, {\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 )}^{9} + 5797080240 \, {\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 )}^{7} + 1842336276480 \, {\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} + 282112659584000 \, {\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 {16926759575040000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {67707038300160000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{21952 \, {\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 )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 298, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (404858580885 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1349528602950 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+165138339870 \sqrt {-10 x^{2}-x +3}\, x^{4}+1799371470600 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+447737213700 \sqrt {-10 x^{2}-x +3}\, x^{3}+1199580980400 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+455499158856 \sqrt {-10 x^{2}-x +3}\, x^{2}+399860326800 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+206091285904 \sqrt {-10 x^{2}-x +3}\, x +53314710240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+34994513344 \sqrt {-10 x^{2}-x +3}\right )}{3073280 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.49, size = 184, normalized size = 1.02 \begin {gather*} \frac {333216939}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{15 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {293 \, \sqrt {-10 \, x^{2} - x + 3}}{120 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {23909 \, \sqrt {-10 \, x^{2} - x + 3}}{1680 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {835409 \, \sqrt {-10 \, x^{2} - x + 3}}{9408 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {87374783 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \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 (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^6\,\sqrt {5\,x+3}} \,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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