Optimal. Leaf size=128 \[ \frac {1}{9} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {59}{180} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {6401 \sqrt {5 x+3} \sqrt {1-2 x}}{5400}+\frac {250433 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{16200 \sqrt {10}}+\frac {98}{81} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 128, 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 = {101, 154, 157, 54, 216, 93, 204} \[ \frac {1}{9} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {59}{180} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {6401 \sqrt {5 x+3} \sqrt {1-2 x}}{5400}+\frac {250433 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{16200 \sqrt {10}}+\frac {98}{81} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Rule 54
Rule 93
Rule 101
Rule 154
Rule 157
Rule 204
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{2+3 x} \, dx &=\frac {1}{9} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {1}{9} \int \frac {\left (-52-\frac {177 x}{2}\right ) (1-2 x)^{3/2}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {59}{180} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {1}{9} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {1}{270} \int \frac {\left (-\frac {5421}{2}-\frac {19203 x}{4}\right ) \sqrt {1-2 x}}{(2+3 x) \sqrt {3+5 x}} \, dx\\ &=\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{5400}+\frac {59}{180} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {1}{9} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {\int \frac {-\frac {181833}{4}-\frac {751299 x}{8}}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{4050}\\ &=\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{5400}+\frac {59}{180} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {1}{9} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {343}{81} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx+\frac {250433 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{32400}\\ &=\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{5400}+\frac {59}{180} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {1}{9} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {686}{81} \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )+\frac {250433 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{16200 \sqrt {5}}\\ &=\frac {6401 \sqrt {1-2 x} \sqrt {3+5 x}}{5400}+\frac {59}{180} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {1}{9} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {250433 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{16200 \sqrt {10}}+\frac {98}{81} \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.15, size = 113, normalized size = 0.88 \[ \frac {30 \sqrt {-(1-2 x)^2} \sqrt {5 x+3} \left (2400 x^2-5940 x+8771\right )+196000 \sqrt {14 x-7} \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )-250433 \sqrt {10-20 x} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{162000 \sqrt {2 x-1}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.95, size = 107, normalized size = 0.84 \[ \frac {1}{5400} \, {\left (2400 \, x^{2} - 5940 \, x + 8771\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} + \frac {49}{81} \, \sqrt {7} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {250433}{324000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.44, size = 186, normalized size = 1.45 \[ -\frac {49}{810} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} + \frac {1}{27000} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 147 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 13199 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {250433}{324000} \, \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 115, normalized size = 0.90 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-144000 \sqrt {-10 x^{2}-x +3}\, x^{2}+356400 \sqrt {-10 x^{2}-x +3}\, x -250433 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+196000 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-526260 \sqrt {-10 x^{2}-x +3}\right )}{324000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.40, size = 83, normalized size = 0.65 \[ -\frac {2}{45} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {103}{90} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {250433}{324000} \, \sqrt {10} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {49}{81} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {9491}{5400} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{3\,x+2} \,d x \]
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 \[ \int \frac {\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}{3 x + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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