Optimal. Leaf size=72 \[ -\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {33}{16} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {363 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{16 \sqrt {10}} \]
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Rubi [A] time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ -\frac {1}{4} \sqrt {1-2 x} (5 x+3)^{3/2}-\frac {33}{16} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {363 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{16 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{4} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {33}{8} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {33}{16} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{4} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {363}{32} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {33}{16} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{4} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {363 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{16 \sqrt {5}}\\ &=-\frac {33}{16} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{4} \sqrt {1-2 x} (3+5 x)^{3/2}+\frac {363 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{16 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.01 \[ -\frac {\sqrt {1-2 x} \left (50 \sqrt {2 x-1} \sqrt {5 x+3} (4 x+9)+363 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{160 \sqrt {2 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 62, normalized size = 0.86 \[ -\frac {5}{16} \, \sqrt {5 \, x + 3} {\left (4 \, x + 9\right )} \sqrt {-2 \, x + 1} - \frac {363}{320} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.27, size = 45, normalized size = 0.62 \[ -\frac {1}{160} \, \sqrt {5} {\left (10 \, \sqrt {5 \, x + 3} {\left (4 \, x + 9\right )} \sqrt {-10 \, x + 5} - 363 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 72, normalized size = 1.00 \[ \frac {363 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{320 \sqrt {5 x +3}\, \sqrt {-2 x +1}}-\frac {\left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{4}-\frac {33 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 41, normalized size = 0.57 \[ -\frac {5}{4} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {363}{320} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {45}{16} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \frac {{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.30, size = 187, normalized size = 2.60 \[ \begin {cases} - \frac {25 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{2 \sqrt {10 x - 5}} - \frac {55 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{8 \sqrt {10 x - 5}} + \frac {363 i \sqrt {x + \frac {3}{5}}}{16 \sqrt {10 x - 5}} - \frac {363 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{160} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {363 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{160} + \frac {25 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{2 \sqrt {5 - 10 x}} + \frac {55 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{8 \sqrt {5 - 10 x}} - \frac {363 \sqrt {x + \frac {3}{5}}}{16 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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