Optimal. Leaf size=122 \[ \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}+\frac {32 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)}-\frac {25}{9} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{441 \sqrt {7}} \]
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Rubi [A] time = 0.04, antiderivative size = 122, 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, 149, 157, 54, 216, 93, 204} \[ \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)}+\frac {32 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)}-\frac {25}{9} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{441 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 98
Rule 149
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^2} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {1}{7} \int \frac {\sqrt {3+5 x} \left (69+\frac {175 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {1}{147} \int \frac {\frac {4027}{2}+\frac {6125 x}{2}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {169}{882} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx-\frac {125}{18} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}+\frac {169}{441} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )-\frac {1}{9} \left (25 \sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {32 \sqrt {1-2 x} \sqrt {3+5 x}}{147 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)}-\frac {25}{9} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )-\frac {169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{441 \sqrt {7}}\\ \end {align*}
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Mathematica [C] time = 0.24, size = 173, normalized size = 1.42 \[ \frac {-468930 (3 x+2) \sqrt {44 x-22} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {5}{11} (2 x-1)\right )-42 \sqrt {2 x-1} \sqrt {5 x+3} \left (30450 x^2-70643 x-60610\right )-771995 \sqrt {10} \left (6 x^2+x-2\right ) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-2704 \sqrt {7} \sqrt {-(1-2 x)^2} (3 x+2) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49392 \sqrt {-(1-2 x)^2} (3 x+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 136, normalized size = 1.11 \[ \frac {8575 \, \sqrt {5} \sqrt {2} {\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 338 \, \sqrt {7} {\left (6 \, x^{2} + x - 2\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 84 \, {\left (1091 \, x + 725\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12348 \, {\left (6 \, x^{2} + x - 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.00, size = 286, normalized size = 2.34 \[ \frac {169}{61740} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {25}{36} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {121 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{245 \, {\left (2 \, x - 1\right )}} - \frac {22 \, \sqrt {10} {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{147 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 198, normalized size = 1.62 \[ -\frac {\left (51450 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2028 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+8575 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-338 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+91644 \sqrt {-10 x^{2}-x +3}\, x -17150 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+676 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+60900 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{12348 \left (3 x +2\right ) \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 103, normalized size = 0.84 \[ -\frac {25}{36} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {169}{6174} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {5455 \, x}{441 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {9784}{1323 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{189 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2} \,d x \]
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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