Optimal. Leaf size=209 \[ \frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}+\frac {17 (5 x+3)^{5/2} (1-2 x)^{5/2}}{28 (3 x+2)^5}+\frac {935 (5 x+3)^{5/2} (1-2 x)^{3/2}}{224 (3 x+2)^4}+\frac {10285 (5 x+3)^{5/2} \sqrt {1-2 x}}{448 (3 x+2)^3}-\frac {113135 (5 x+3)^{3/2} \sqrt {1-2 x}}{12544 (3 x+2)^2}-\frac {3733455 \sqrt {5 x+3} \sqrt {1-2 x}}{175616 (3 x+2)}-\frac {41068005 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \]
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Rubi [A] time = 0.07, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \begin {gather*} \frac {(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}+\frac {17 (5 x+3)^{5/2} (1-2 x)^{5/2}}{28 (3 x+2)^5}+\frac {935 (5 x+3)^{5/2} (1-2 x)^{3/2}}{224 (3 x+2)^4}+\frac {10285 (5 x+3)^{5/2} \sqrt {1-2 x}}{448 (3 x+2)^3}-\frac {113135 (5 x+3)^{3/2} \sqrt {1-2 x}}{12544 (3 x+2)^2}-\frac {3733455 \sqrt {5 x+3} \sqrt {1-2 x}}{175616 (3 x+2)}-\frac {41068005 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx &=\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {85}{28} \int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx\\ &=\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac {935}{56} \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac {935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac {30855}{448} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac {935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac {10285 \sqrt {1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}+\frac {113135}{896} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {113135 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac {935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac {10285 \sqrt {1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}+\frac {3733455 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{25088}\\ &=-\frac {3733455 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {113135 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac {935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac {10285 \sqrt {1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}+\frac {41068005 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232}\\ &=-\frac {3733455 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {113135 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac {935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac {10285 \sqrt {1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}+\frac {41068005 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616}\\ &=-\frac {3733455 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {113135 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac {(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac {17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac {935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac {10285 \sqrt {1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}-\frac {41068005 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 138, normalized size = 0.66 \begin {gather*} \frac {1}{28} \left (\frac {935 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{43904}+\frac {2 (5 x+3)^{5/2} (1-2 x)^{7/2}}{(3 x+2)^6}+\frac {17 (5 x+3)^{5/2} (1-2 x)^{5/2}}{(3 x+2)^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.49, size = 154, normalized size = 0.74 \begin {gather*} -\frac {161051 \sqrt {1-2 x} \left (\frac {255 (1-2 x)^5}{(5 x+3)^5}+\frac {10115 (1-2 x)^4}{(5 x+3)^4}-\frac {186298 (1-2 x)^3}{(5 x+3)^3}-\frac {1154538 (1-2 x)^2}{(5 x+3)^2}-\frac {3469445 (1-2 x)}{5 x+3}-4285785\right )}{175616 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^6}-\frac {41068005 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.73, size = 146, normalized size = 0.70 \begin {gather*} -\frac {41068005 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 (872316385 \, x^{5} + 2946673460 \, x^{4} + 3982356144 \, x^{3} + 2692519968 \, x^{2} + 910641904 \, x + 123208128\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2458624 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.28, size = 484, normalized size = 2.32 \begin {gather*} \frac {8213601}{4917248} \, \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 {805255 \, \sqrt {10} {\left (51 \, {\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 )}^{11} + 80920 \, {\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} - 59615360 \, {\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} - 14778086400 \, {\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} - 1776355840000 \, {\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 {87772876800000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {351091507200000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{87808 \, {\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 )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 346, normalized size = 1.66 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (29938575645 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+119754302580 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+12212429390 \sqrt {-10 x^{2}-x +3}\, x^{5}+199590504300 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+41253428440 \sqrt {-10 x^{2}-x +3}\, x^{4}+177413781600 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+55752986016 \sqrt {-10 x^{2}-x +3}\, x^{3}+88706890800 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+37695279552 \sqrt {-10 x^{2}-x +3}\, x^{2}+23655170880 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+12748986656 \sqrt {-10 x^{2}-x +3}\, x +2628352320 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1724913792 \sqrt {-10 x^{2}-x +3}\right )}{2458624 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 273, normalized size = 1.31 \begin {gather*} \frac {7709075}{921984} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{6 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {47 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{84 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {2805 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1568 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {103785 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {4625445 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {62789925}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {41068005}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {55323015}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {18300755 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3687936 \, {\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 = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^7} \,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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