Optimal. Leaf size=94 \[ -\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {3}{40} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {99}{400} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {1089 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]
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Rubi [A] time = 0.02, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac {1}{10} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {3}{40} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {99}{400} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {1089 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)}{\sqrt {3+5 x}} \, dx &=-\frac {1}{10} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {3}{4} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {3}{40} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {99}{80} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {99}{400} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {3}{40} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {1089}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {99}{400} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {3}{40} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {1089 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{400 \sqrt {5}}\\ &=\frac {99}{400} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {3}{40} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {1}{10} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {1089 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 69, normalized size = 0.73 \[ \frac {10 \sqrt {5 x+3} \left (320 x^3-360 x^2-78 x+89\right )+1089 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{4000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 67, normalized size = 0.71 \[ -\frac {1}{400} \, {\left (160 \, x^{2} - 100 \, x - 89\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1089}{8000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.08, size = 140, normalized size = 1.49 \[ -\frac {1}{20000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{2000} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 87, normalized size = 0.93 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-3200 \sqrt {-10 x^{2}-x +3}\, x^{2}+2000 \sqrt {-10 x^{2}-x +3}\, x +1089 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1780 \sqrt {-10 x^{2}-x +3}\right )}{8000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 58, normalized size = 0.62 \[ -\frac {2}{5} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {1}{4} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1089}{8000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {89}{400} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )}{\sqrt {5\,x+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 81.45, size = 223, normalized size = 2.37 \[ - \frac {7 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (\frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{968} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{2} + \frac {3 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} + \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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