Optimal. Leaf size=202 \[ \frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}+\frac {102293609 \sqrt {1-2 x}}{18816 (3 x+2) \sqrt {5 x+3}}+\frac {587477 \sqrt {1-2 x}}{1344 (3 x+2)^2 \sqrt {5 x+3}}+\frac {12023 \sqrt {1-2 x}}{240 (3 x+2)^3 \sqrt {5 x+3}}+\frac {2513 \sqrt {1-2 x}}{360 (3 x+2)^4 \sqrt {5 x+3}}-\frac {4639661185 \sqrt {1-2 x}}{56448 \sqrt {5 x+3}}+\frac {3538809681 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \]
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Rubi [A] time = 0.08, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {98, 149, 151, 152, 12, 93, 204} \[ \frac {7 (1-2 x)^{3/2}}{15 (3 x+2)^5 \sqrt {5 x+3}}+\frac {102293609 \sqrt {1-2 x}}{18816 (3 x+2) \sqrt {5 x+3}}+\frac {587477 \sqrt {1-2 x}}{1344 (3 x+2)^2 \sqrt {5 x+3}}+\frac {12023 \sqrt {1-2 x}}{240 (3 x+2)^3 \sqrt {5 x+3}}+\frac {2513 \sqrt {1-2 x}}{360 (3 x+2)^4 \sqrt {5 x+3}}-\frac {4639661185 \sqrt {1-2 x}}{56448 \sqrt {5 x+3}}+\frac {3538809681 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{6272 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 149
Rule 151
Rule 152
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^6 (3+5 x)^{3/2}} \, dx &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {1}{15} \int \frac {\left (\frac {491}{2}-260 x\right ) \sqrt {1-2 x}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}-\frac {1}{180} \int \frac {-\frac {124003}{4}+48180 x}{\sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}+\frac {12023 \sqrt {1-2 x}}{240 (2+3 x)^3 \sqrt {3+5 x}}-\frac {\int \frac {-\frac {31387125}{8}+\frac {11361735 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx}{3780}\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}+\frac {12023 \sqrt {1-2 x}}{240 (2+3 x)^3 \sqrt {3+5 x}}+\frac {587477 \sqrt {1-2 x}}{1344 (2+3 x)^2 \sqrt {3+5 x}}-\frac {\int \frac {-\frac {5806022145}{16}+\frac {925276275 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx}{52920}\\ &=\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}+\frac {12023 \sqrt {1-2 x}}{240 (2+3 x)^3 \sqrt {3+5 x}}+\frac {587477 \sqrt {1-2 x}}{1344 (2+3 x)^2 \sqrt {3+5 x}}+\frac {102293609 \sqrt {1-2 x}}{18816 (2+3 x) \sqrt {3+5 x}}-\frac {\int \frac {-\frac {685091891715}{32}+\frac {161112434175 x}{8}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{370440}\\ &=-\frac {4639661185 \sqrt {1-2 x}}{56448 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}+\frac {12023 \sqrt {1-2 x}}{240 (2+3 x)^3 \sqrt {3+5 x}}+\frac {587477 \sqrt {1-2 x}}{1344 (2+3 x)^2 \sqrt {3+5 x}}+\frac {102293609 \sqrt {1-2 x}}{18816 (2+3 x) \sqrt {3+5 x}}+\frac {\int -\frac {36785926633995}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2037420}\\ &=-\frac {4639661185 \sqrt {1-2 x}}{56448 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}+\frac {12023 \sqrt {1-2 x}}{240 (2+3 x)^3 \sqrt {3+5 x}}+\frac {587477 \sqrt {1-2 x}}{1344 (2+3 x)^2 \sqrt {3+5 x}}+\frac {102293609 \sqrt {1-2 x}}{18816 (2+3 x) \sqrt {3+5 x}}-\frac {3538809681 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{12544}\\ &=-\frac {4639661185 \sqrt {1-2 x}}{56448 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}+\frac {12023 \sqrt {1-2 x}}{240 (2+3 x)^3 \sqrt {3+5 x}}+\frac {587477 \sqrt {1-2 x}}{1344 (2+3 x)^2 \sqrt {3+5 x}}+\frac {102293609 \sqrt {1-2 x}}{18816 (2+3 x) \sqrt {3+5 x}}-\frac {3538809681 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{6272}\\ &=-\frac {4639661185 \sqrt {1-2 x}}{56448 \sqrt {3+5 x}}+\frac {7 (1-2 x)^{3/2}}{15 (2+3 x)^5 \sqrt {3+5 x}}+\frac {2513 \sqrt {1-2 x}}{360 (2+3 x)^4 \sqrt {3+5 x}}+\frac {12023 \sqrt {1-2 x}}{240 (2+3 x)^3 \sqrt {3+5 x}}+\frac {587477 \sqrt {1-2 x}}{1344 (2+3 x)^2 \sqrt {3+5 x}}+\frac {102293609 \sqrt {1-2 x}}{18816 (2+3 x) \sqrt {3+5 x}}+\frac {3538809681 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6272 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 157, normalized size = 0.78 \[ \frac {131376 (3 x+2) (1-2 x)^{7/2}+18816 (1-2 x)^{7/2}+(3 x+2)^2 \left (973656 (1-2 x)^{7/2}+9748787 (3 x+2) \left (2 (1-2 x)^{5/2}+55 (3 x+2) \left (33 \sqrt {7} (3 x+2) \sqrt {5 x+3} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\sqrt {1-2 x} (101 x+65)\right )\right )\right )}{219520 (3 x+2)^5 \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 146, normalized size = 0.72 \[ \frac {17694048405 \, \sqrt {7} {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (626354259975 \, x^{5} + 2074037896035 \, x^{4} + 2746600901250 \, x^{3} + 1818284414692 \, x^{2} + 601741553688 \, x + 79638637088\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{439040 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.02, size = 499, normalized size = 2.47 \[ -\frac {3538809681}{878080} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {3025}{2} \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} - \frac {121 \, {\left (34728039 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 30879615760 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 10961021460480 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1791349451136000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 112299870108160000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{3136 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 346, normalized size = 1.71 \[ -\frac {\left (21498268812075 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+84559857327495 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8768959639650 \sqrt {-10 x^{2}-x +3}\, x^{5}+138544399011150 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+29036530544490 \sqrt {-10 x^{2}-x +3}\, x^{4}+121027291090200 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+38452412617500 \sqrt {-10 x^{2}-x +3}\, x^{3}+59452002640800 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+25455981805688 \sqrt {-10 x^{2}-x +3}\, x^{2}+15570762596400 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8424381751632 \sqrt {-10 x^{2}-x +3}\, x +1698628646880 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1114940919232 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{439040 \left (3 x +2\right )^{5} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.37, size = 398, normalized size = 1.97 \[ -\frac {3538809681}{87808} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {4639661185 \, x}{28224 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {4844248403}{56448 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {343}{135 \, {\left (243 \, \sqrt {-10 \, x^{2} - x + 3} x^{5} + 810 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 1080 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 720 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 240 \, \sqrt {-10 \, x^{2} - x + 3} x + 32 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {5341}{360 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {242879}{2160 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {315689}{320 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {33314567}{2688 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^6\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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