Optimal. Leaf size=94 \[ -\frac {2 (1-2 x)^{5/2}}{55 \sqrt {5 x+3}}+\frac {1}{22} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {3}{20} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {33 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{20 \sqrt {10}} \]
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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 = {78, 50, 54, 216} \begin {gather*} -\frac {2 (1-2 x)^{5/2}}{55 \sqrt {5 x+3}}+\frac {1}{22} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {3}{20} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {33 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{20 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 78
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {5}{11} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {3}{4} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33}{40} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{20 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2}}{55 \sqrt {3+5 x}}+\frac {3}{20} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {1}{22} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {33 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 78, normalized size = 0.83 \begin {gather*} \frac {10 \left (24 x^3-46 x^2-5 x+11\right )+33 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{200 \sqrt {1-2 x} \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.39, size = 109, normalized size = 1.16 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (-12 \sqrt {5} (5 x+3)^2+157 \sqrt {5} (5 x+3)-88 \sqrt {5}\right )}{2500 \sqrt {5 x+3}}-\frac {33 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{10 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.45, size = 81, normalized size = 0.86 \begin {gather*} -\frac {33 \, \sqrt {10} {\left (5 \, x + 3\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 (12 \, x^{2} - 17 \, x - 11\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{400 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.39, size = 111, normalized size = 1.18 \begin {gather*} -\frac {1}{2500} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 157 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {33}{200} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{1250 \, \sqrt {5 \, x + 3}} + \frac {22 \, \sqrt {10} \sqrt {5 \, x + 3}}{625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 99, normalized size = 1.05 \begin {gather*} \frac {\left (-240 \sqrt {-10 x^{2}-x +3}\, x^{2}+165 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+340 \sqrt {-10 x^{2}-x +3}\, x +99 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+220 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{400 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 97, normalized size = 1.03 \begin {gather*} \frac {33}{400} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {99}{500} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{25 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{50 \, {\left (5 \, x + 3\right )}} - \frac {33 \, \sqrt {-10 \, x^{2} - x + 3}}{125 \, {\left (5 \, x + 3\right )}} \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 {{\left (1-2\,x\right )}^{3/2}\,\left (3\,x+2\right )}{{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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