Optimal. Leaf size=209 \[ \frac {297 \sqrt {1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac {9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac {1089 \sqrt {1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac {11979 \sqrt {1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac {395307 \sqrt {1-2 x} \sqrt {5 x+3}}{175616 (3 x+2)}-\frac {4348377 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \]
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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} \[ \frac {297 \sqrt {1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac {9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac {1089 \sqrt {1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac {11979 \sqrt {1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac {395307 \sqrt {1-2 x} \sqrt {5 x+3}}{175616 (3 x+2)}-\frac {4348377 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}} \]
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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9}{4} \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297}{40} \int \frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {3267}{320} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^4} \, dx\\ &=-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {11979}{896} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {395307 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{25088}\\ &=-\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {4348377 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{351232}\\ &=-\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac {4348377 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{175616}\\ &=-\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}-\frac {4348377 \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 \[ \frac {99 \left (\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} \left (814395 x^3+1285720 x^2+654436 x+105552\right )}{(3 x+2)^4}-219615 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{6146560}+\frac {9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 146, normalized size = 0.70 \[ -\frac {21741885 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (460633945 \, x^{5} + 1555340180 \, x^{4} + 2108117296 \, x^{3} + 1428134688 \, x^{2} + 482263920 \, x + 64829376\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12293120 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.08, size = 484, normalized size = 2.32 \[ \frac {4348377}{24586240} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {161051 \, \sqrt {10} {\left (27 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 42840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 27941760 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 6539187200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 940423680000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {46467993600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {185871974400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{87808 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]
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 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (15849834165 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+63399336660 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6448875230 \sqrt {-10 x^{2}-x +3}\, x^{5}+105665561100 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+21774762520 \sqrt {-10 x^{2}-x +3}\, x^{4}+93924943200 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+29513642144 \sqrt {-10 x^{2}-x +3}\, x^{3}+46962471600 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+19993885632 \sqrt {-10 x^{2}-x +3}\, x^{2}+12523325760 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+6751694880 \sqrt {-10 x^{2}-x +3}\, x +1391480640 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+907611264 \sqrt {-10 x^{2}-x +3}\right )}{12293120 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.57, size = 273, normalized size = 1.31 \[ \frac {272085}{307328} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{42 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {23 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{420 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {297 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1568 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {10989 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {489753 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {6648345}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {4348377}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {5857731}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {645909 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1229312 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^7} \,d x \]
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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