Optimal. Leaf size=195 \[ \frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {5 x+3}}+\frac {2992825 \sqrt {1-2 x}}{1344 (3 x+2) (5 x+3)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {25024175 \sqrt {1-2 x}}{1344 (5 x+3)^{3/2}}-\frac {519421265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}} \]
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Rubi [A] time = 0.07, antiderivative size = 195, 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} \[ \frac {7 (1-2 x)^{3/2}}{12 (3 x+2)^4 (5 x+3)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {5 x+3}}+\frac {2992825 \sqrt {1-2 x}}{1344 (3 x+2) (5 x+3)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (3 x+2)^2 (5 x+3)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {25024175 \sqrt {1-2 x}}{1344 (5 x+3)^{3/2}}-\frac {519421265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{448 \sqrt {7}} \]
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)^5 (3+5 x)^{5/2}} \, dx &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {1}{12} \int \frac {\left (\frac {495}{2}-264 x\right ) \sqrt {1-2 x}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {1}{108} \int \frac {-\frac {126423}{4}+49236 x}{\sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {31923045}{8}+\frac {11597355 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx}{1512}\\ &=\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}-\frac {\int \frac {-\frac {5879900565}{16}+\frac {942739875 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{10584}\\ &=-\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {\int \frac {-\frac {663674731335}{32}+\frac {156075779475 x}{8}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{174636}\\ &=-\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}-\frac {\int -\frac {35635934727855}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{960498}\\ &=-\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}+\frac {519421265}{896} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}+\frac {519421265}{448} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {25024175 \sqrt {1-2 x}}{1344 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{12 (2+3 x)^4 (3+5 x)^{3/2}}+\frac {847 \sqrt {1-2 x}}{72 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {36817 \sqrt {1-2 x}}{288 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {2992825 \sqrt {1-2 x}}{1344 (2+3 x) (3+5 x)^{3/2}}+\frac {227000875 \sqrt {1-2 x}}{1344 \sqrt {3+5 x}}-\frac {519421265 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{448 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 152, normalized size = 0.78 \[ \frac {65016 (3 x+2) (1-2 x)^{7/2}+7056 (1-2 x)^{7/2}+(3 x+2)^2 \left (716706 (1-2 x)^{7/2}+9444023 (3 x+2) \left (3 (1-2 x)^{5/2}-55 (3 x+2) \left (21 \sqrt {7} (5 x+3)^{3/2} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-\sqrt {1-2 x} (107 x+62)\right )\right )\right )}{65856 (3 x+2)^4 (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 146, normalized size = 0.75 \[ -\frac {1558263795 \, \sqrt {7} {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (91935354375 \, x^{5} + 298295199450 \, x^{4} + 386933096475 \, x^{3} + 250814924064 \, x^{2} + 81243850516 \, x + 10520317456\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{18816 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.63, size = 495, normalized size = 2.54 \[ \frac {103884253}{12544} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {55}{48} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {4056 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {16224 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {55 \, {\left (6089929 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 4375094808 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 1081495934400 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 90973105216000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{224 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 346, normalized size = 1.77 \[ \frac {\left (3155484184875 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+12201205514850 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1287094961250 \sqrt {-10 x^{2}-x +3}\, x^{5}+19648148191155 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4176132792300 \sqrt {-10 x^{2}-x +3}\, x^{4}+16866647317080 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+5417063350650 \sqrt {-10 x^{2}-x +3}\, x^{3}+8140370065080 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+3511408936896 \sqrt {-10 x^{2}-x +3}\, x^{2}+2094306540480 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1137413907224 \sqrt {-10 x^{2}-x +3}\, x +224389986480 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+147284444384 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{18816 \left (3 x +2\right )^{4} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.61, size = 325, normalized size = 1.67 \[ \frac {519421265}{6272} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {227000875 \, x}{672 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {79003515}{448 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {24449315 \, x}{288 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{324 \, {\left (81 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 216 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 96 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 16 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {37387}{648 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {571291}{864 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {60813781}{5184 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {237706249}{5184 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^5\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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