Optimal. Leaf size=79 \[ \frac {4 (5 x+3)^{3/2}}{231 (1-2 x)^{3/2}}+\frac {6 \sqrt {5 x+3}}{49 \sqrt {1-2 x}}+\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
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Rubi [A] time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac {4 (5 x+3)^{3/2}}{231 (1-2 x)^{3/2}}+\frac {6 \sqrt {5 x+3}}{49 \sqrt {1-2 x}}+\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{49 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 204
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\frac {4 (3+5 x)^{3/2}}{231 (1-2 x)^{3/2}}+\frac {3}{7} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2} (2+3 x)} \, dx\\ &=\frac {6 \sqrt {3+5 x}}{49 \sqrt {1-2 x}}+\frac {4 (3+5 x)^{3/2}}{231 (1-2 x)^{3/2}}-\frac {3}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {6 \sqrt {3+5 x}}{49 \sqrt {1-2 x}}+\frac {4 (3+5 x)^{3/2}}{231 (1-2 x)^{3/2}}-\frac {6}{49} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=\frac {6 \sqrt {3+5 x}}{49 \sqrt {1-2 x}}+\frac {4 (3+5 x)^{3/2}}{231 (1-2 x)^{3/2}}+\frac {6 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{49 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 0.90 \[ -\frac {2 \left (7 \sqrt {5 x+3} (128 x-141)+99 \sqrt {7-14 x} (2 x-1) \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{11319 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 86, normalized size = 1.09 \[ \frac {99 \, \sqrt {7} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (128 \, x - 141\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{11319 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.38, size = 113, normalized size = 1.43 \[ -\frac {3}{3430} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {2 \, {\left (128 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1089 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{40425 \, {\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 154, normalized size = 1.95 \[ -\frac {\left (396 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-396 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1792 \sqrt {-10 x^{2}-x +3}\, x +99 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-1974 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{11319 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 87, normalized size = 1.10 \[ -\frac {3}{343} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {640 \, x}{1617 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1}{1617 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {55 \, x}{21 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {11}{7 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \]
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 \[ \int \frac {\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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