Optimal. Leaf size=180 \[ -\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}+\frac {1852307 \sqrt {1-2 x} \sqrt {5 x+3}}{1185408 (3 x+2)}+\frac {17981 \sqrt {1-2 x} \sqrt {5 x+3}}{84672 (3 x+2)^2}+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{15120 (3 x+2)^3}-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{2520 (3 x+2)^4}-\frac {783959 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \[ -\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}+\frac {1852307 \sqrt {1-2 x} \sqrt {5 x+3}}{1185408 (3 x+2)}+\frac {17981 \sqrt {1-2 x} \sqrt {5 x+3}}{84672 (3 x+2)^2}+\frac {641 \sqrt {1-2 x} \sqrt {5 x+3}}{15120 (3 x+2)^3}-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{2520 (3 x+2)^4}-\frac {783959 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{43904 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 97
Rule 149
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac {1}{15} \int \frac {\left (\frac {9}{2}-20 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{2520 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac {\int \frac {-\frac {1691}{4}-1195 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{1260}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{2520 (2+3 x)^4}+\frac {641 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac {\int \frac {\frac {90125}{8}-22435 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{26460}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{2520 (2+3 x)^4}+\frac {641 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {17981 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac {\int \frac {\frac {13219115}{16}-\frac {3146675 x}{4}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{370440}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{2520 (2+3 x)^4}+\frac {641 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {17981 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {1852307 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac {\int \frac {740841255}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2593080}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{2520 (2+3 x)^4}+\frac {641 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {17981 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {1852307 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac {783959 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{87808}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{2520 (2+3 x)^4}+\frac {641 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {17981 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {1852307 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac {783959 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{43904}\\ &=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{2520 (2+3 x)^4}+\frac {641 \sqrt {1-2 x} \sqrt {3+5 x}}{15120 (2+3 x)^3}+\frac {17981 \sqrt {1-2 x} \sqrt {3+5 x}}{84672 (2+3 x)^2}+\frac {1852307 \sqrt {1-2 x} \sqrt {3+5 x}}{1185408 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}-\frac {783959 \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 = 0.12, size = 133, normalized size = 0.74 \[ \frac {589 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (4223 x^2+4478 x+1152\right )}{(3 x+2)^3}-3993 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{921984}+\frac {81 (1-2 x)^{3/2} (5 x+3)^{5/2}}{280 (3 x+2)^4}+\frac {3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{35 (3 x+2)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 131, normalized size = 0.73 \[ -\frac {11759385 \, \sqrt {7} {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (83353815 \, x^{4} + 226052850 \, x^{3} + 230080132 \, x^{2} + 103856008 \, x + 17507808\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{9219840 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.15, size = 426, normalized size = 2.37 \[ \frac {783959}{6146560} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (1767 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 2308880 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 925245440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 177804928000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {10860971520000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {43443886080000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{65856 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 298, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (2857530555 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+9525101850 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1166953410 \sqrt {-10 x^{2}-x +3}\, x^{4}+12700135800 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3164739900 \sqrt {-10 x^{2}-x +3}\, x^{3}+8466757200 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3221121848 \sqrt {-10 x^{2}-x +3}\, x^{2}+2822252400 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1453984112 \sqrt {-10 x^{2}-x +3}\, x +376300320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+245109312 \sqrt {-10 x^{2}-x +3}\right )}{9219840 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 198, normalized size = 1.10 \[ \frac {783959}{614656} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {32395}{32928} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{35 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {13 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{280 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {545 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2352 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {19437 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{21952 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {239723 \, \sqrt {-10 \, x^{2} - x + 3}}{131712 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^6} \,d x \]
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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