Optimal. Leaf size=113 \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^3}{15 (5 x+3)^{3/2}}-\frac {392 \sqrt {1-2 x} (3 x+2)^2}{825 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (1740 x+1243)}{11000}+\frac {1071 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1000 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {97, 150, 147, 54, 216} \begin {gather*} -\frac {2 \sqrt {1-2 x} (3 x+2)^3}{15 (5 x+3)^{3/2}}-\frac {392 \sqrt {1-2 x} (3 x+2)^2}{825 \sqrt {5 x+3}}+\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (1740 x+1243)}{11000}+\frac {1071 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 97
Rule 147
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {(7-21 x) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {392 \sqrt {1-2 x} (2+3 x)^2}{825 \sqrt {3+5 x}}+\frac {4}{825} \int \frac {\left (357-\frac {3045 x}{2}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {392 \sqrt {1-2 x} (2+3 x)^2}{825 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (1243+1740 x)}{11000}+\frac {1071 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2000}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {392 \sqrt {1-2 x} (2+3 x)^2}{825 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (1243+1740 x)}{11000}+\frac {1071 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1000 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {392 \sqrt {1-2 x} (2+3 x)^2}{825 \sqrt {3+5 x}}+\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (1243+1740 x)}{11000}+\frac {1071 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 83, normalized size = 0.73 \begin {gather*} \frac {35343 (5 x+3)^{3/2} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \left (178200 x^4+204930 x^3+3925 x^2-52336 x-11567\right )}{330000 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 125, normalized size = 1.11 \begin {gather*} -\frac {\sqrt {1-2 x} \left (\frac {400 (1-2 x)^3}{(5 x+3)^3}+\frac {24080 (1-2 x)^2}{(5 x+3)^2}+\frac {48475 (1-2 x)}{5 x+3}-70686\right )}{33000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {1071 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{1000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.57, size = 96, normalized size = 0.85 \begin {gather*} -\frac {35343 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \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 ) - 20 \, {\left (89100 \, x^{3} + 147015 \, x^{2} + 75470 \, x + 11567\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{660000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.91, size = 171, normalized size = 1.51 \begin {gather*} \frac {27}{25000} \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {1}{1650000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2364 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {1071}{10000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {591 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{103125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 130, normalized size = 1.15 \begin {gather*} \frac {\left (1782000 \sqrt {-10 x^{2}-x +3}\, x^{3}+883575 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2940300 \sqrt {-10 x^{2}-x +3}\, x^{2}+1060290 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1509400 \sqrt {-10 x^{2}-x +3}\, x +318087 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+231340 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{660000 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^3}{{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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