Optimal. Leaf size=166 \[ \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}+\frac {1784635 \sqrt {1-2 x}}{72 \sqrt {5 x+3}}+\frac {7843 \sqrt {1-2 x}}{24 (3 x+2) (5 x+3)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {196735 \sqrt {1-2 x}}{72 (5 x+3)^{3/2}}-\frac {1361195 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{8 \sqrt {7}} \]
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Rubi [A] time = 0.06, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}+\frac {1784635 \sqrt {1-2 x}}{72 \sqrt {5 x+3}}+\frac {7843 \sqrt {1-2 x}}{24 (3 x+2) (5 x+3)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (3 x+2)^2 (5 x+3)^{3/2}}-\frac {196735 \sqrt {1-2 x}}{72 (5 x+3)^{3/2}}-\frac {1361195 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{8 \sqrt {7}} \]
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)^4 (3+5 x)^{5/2}} \, dx &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {1}{9} \int \frac {\left (\frac {429}{2}-198 x\right ) \sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}-\frac {1}{54} \int \frac {-\frac {83655}{4}+30393 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {7843 \sqrt {1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}-\frac {1}{378} \int \frac {-\frac {15408855}{8}+2470545 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac {196735 \sqrt {1-2 x}}{72 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {7843 \sqrt {1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac {\int \frac {-\frac {1739225565}{16}+\frac {409012065 x}{4}}{\sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{6237}\\ &=-\frac {196735 \sqrt {1-2 x}}{72 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {7843 \sqrt {1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac {1784635 \sqrt {1-2 x}}{72 \sqrt {3+5 x}}-\frac {2 \int -\frac {93387505365}{32 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{68607}\\ &=-\frac {196735 \sqrt {1-2 x}}{72 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {7843 \sqrt {1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac {1784635 \sqrt {1-2 x}}{72 \sqrt {3+5 x}}+\frac {1361195}{16} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {196735 \sqrt {1-2 x}}{72 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {7843 \sqrt {1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac {1784635 \sqrt {1-2 x}}{72 \sqrt {3+5 x}}+\frac {1361195}{8} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {196735 \sqrt {1-2 x}}{72 (3+5 x)^{3/2}}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac {77 \sqrt {1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac {7843 \sqrt {1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac {1784635 \sqrt {1-2 x}}{72 \sqrt {3+5 x}}-\frac {1361195 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{8 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 84, normalized size = 0.51 \[ \frac {\sqrt {1-2 x} \left (80308575 x^4+207031680 x^3+199977747 x^2+85776638 x+13784768\right )}{24 (3 x+2)^3 (5 x+3)^{3/2}}-\frac {1361195 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{8 \sqrt {7}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 131, normalized size = 0.79 \[ -\frac {4083585 \, \sqrt {7} {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (80308575 \, x^{4} + 207031680 \, x^{3} + 199977747 \, x^{2} + 85776638 \, x + 13784768\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{336 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.11, size = 434, normalized size = 2.61 \[ \frac {272239}{224} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11}{48} \, \sqrt {10} {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {3264 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {13056 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} + \frac {11 \, {\left (63359 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 30251200 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + 3730664000 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}\right )}}{4 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 298, normalized size = 1.80 \[ \frac {\left (2756419875 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8820543600 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1124320050 \sqrt {-10 x^{2}-x +3}\, x^{4}+11282945355 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2898443520 \sqrt {-10 x^{2}-x +3}\, x^{3}+7211611110 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2799688458 \sqrt {-10 x^{2}-x +3}\, x^{2}+2303141940 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1200872932 \sqrt {-10 x^{2}-x +3}\, x +294018120 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+192986752 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{336 \left (3 x +2\right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 240, normalized size = 1.45 \[ \frac {1361195}{112} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {1784635 \, x}{36 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1863329}{72 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {149501 \, x}{12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2401}{243 \, {\left (27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + 54 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 36 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 8 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {31213}{324 \, {\left (9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 4 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} + \frac {1115681}{648 \, {\left (3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + 2 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}\right )}} - \frac {13081615}{1944 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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