Optimal. Leaf size=108 \[ \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {407 \sqrt {5 x+3}}{98 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{147 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {98, 150, 157, 54, 216, 93, 204} \[ \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {407 \sqrt {5 x+3}}{98 \sqrt {1-2 x}}+\frac {25}{6} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{147 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 98
Rule 150
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {1}{21} \int \frac {\sqrt {3+5 x} \left (174+\frac {525 x}{2}\right )}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=-\frac {407 \sqrt {3+5 x}}{98 \sqrt {1-2 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {1}{147} \int \frac {-\frac {6123}{2}-\frac {18375 x}{4}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {407 \sqrt {3+5 x}}{98 \sqrt {1-2 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {1}{147} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {125}{12} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {407 \sqrt {3+5 x}}{98 \sqrt {1-2 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}-\frac {2}{147} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {1}{6} \left (25 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {407 \sqrt {3+5 x}}{98 \sqrt {1-2 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2}}+\frac {25}{6} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{147 \sqrt {7}}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 97, normalized size = 0.90 \[ -\frac {-18865 \sqrt {22} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+56 \sqrt {5 x+3} (41 x+18)+24 \sqrt {7-14 x} (2 x-1) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{12348 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.38, size = 142, normalized size = 1.31 \[ -\frac {8575 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 8 \, \sqrt {7} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 308 \, {\left (292 \, x - 69\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{8232 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.52, size = 180, normalized size = 1.67 \[ -\frac {1}{10290} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {25}{24} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {11 \, {\left (292 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1221 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{7350 \, {\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 191, normalized size = 1.77 \[ \frac {\left (34300 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-32 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-34300 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+32 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+89936 \sqrt {-10 x^{2}-x +3}\, x +8575 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-8 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-21252 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{8232 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.31, size = 163, normalized size = 1.51 \[ -\frac {12233125 \, x^{2}}{3557763 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {625 \, x^{3}}{6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {25}{24} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {1}{1029} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {2446625}{7115526} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {12021894385 \, x}{697321548 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {16029625 \, x^{2}}{117612 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {6953014391}{697321548 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {12465295 \, x}{205821 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {2681981}{274428 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )} \,d x \]
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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