Optimal. Leaf size=72 \[ -\frac {3}{20} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {41}{200} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \]
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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 = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \begin {gather*} -\frac {3}{20} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {41}{200} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 216
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)}{\sqrt {3+5 x}} \, dx &=-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {41}{40} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {41}{200} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {451}{400} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {41}{200} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {451 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{200 \sqrt {5}}\\ &=\frac {41}{200} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {451 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 64, normalized size = 0.89 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (-120 x^2+38 x+11\right )+451 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{2000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.19, size = 95, normalized size = 1.32 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (12 \sqrt {5} (5 x+3)^{3/2}-25 \sqrt {5} \sqrt {5 x+3}\right )}{1000}+\frac {451 i \log \left (\sqrt {11-2 (5 x+3)}-i \sqrt {2} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.62, size = 62, normalized size = 0.86 \begin {gather*} \frac {1}{200} \, {\left (60 \, x + 11\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {451}{4000} \, \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.02, size = 86, normalized size = 1.19 \begin {gather*} \frac {3}{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.97 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (1200 \sqrt {-10 x^{2}-x +3}\, x +451 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+220 \sqrt {-10 x^{2}-x +3}\right )}{4000 \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 44, normalized size = 0.61 \begin {gather*} \frac {451}{4000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {3}{10} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {11}{200} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.27, size = 550, normalized size = 7.64 \begin {gather*} \frac {\frac {4\,{\left (\sqrt {1-2\,x}-1\right )}^3}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {8\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {32\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}}{\frac {4\,{\left (\sqrt {1-2\,x}-1\right )}^2}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {4}{25}}-\frac {\frac {2427\,{\left (\sqrt {1-2\,x}-1\right )}^3}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {858\,\left (\sqrt {1-2\,x}-1\right )}{15625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {2427\,{\left (\sqrt {1-2\,x}-1\right )}^5}{1250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {429\,{\left (\sqrt {1-2\,x}-1\right )}^7}{500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {96\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {672\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {24\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}}{\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^2}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {24\,{\left (\sqrt {1-2\,x}-1\right )}^4}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {8\,{\left (\sqrt {1-2\,x}-1\right )}^6}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^8}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {16}{625}}-\frac {429\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{1000}+\frac {22\,\sqrt {2}\,\sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {5}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.43, size = 165, normalized size = 2.29 \begin {gather*} - \frac {7 \sqrt {2} \left (\begin {cases} \frac {11 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{2}\right )}{25} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{2} + \frac {3 \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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