Optimal. Leaf size=74 \[ -\frac{404 \sqrt{1-2 x}}{9075 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{825 (5 x+3)^{3/2}}+\frac{9}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
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Rubi [A] time = 0.017441, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {89, 78, 54, 216} \[ -\frac{404 \sqrt{1-2 x}}{9075 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{825 (5 x+3)^{3/2}}+\frac{9}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ) \]
Antiderivative was successfully verified.
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Rule 89
Rule 78
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{825 (3+5 x)^{3/2}}+\frac{2}{825} \int \frac{\frac{1093}{2}+\frac{1485 x}{2}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{825 (3+5 x)^{3/2}}-\frac{404 \sqrt{1-2 x}}{9075 \sqrt{3+5 x}}+\frac{9}{25} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{825 (3+5 x)^{3/2}}-\frac{404 \sqrt{1-2 x}}{9075 \sqrt{3+5 x}}+\frac{18 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{25 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x}}{825 (3+5 x)^{3/2}}-\frac{404 \sqrt{1-2 x}}{9075 \sqrt{3+5 x}}+\frac{9}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )\\ \end{align*}
Mathematica [A] time = 0.0584654, size = 57, normalized size = 0.77 \[ -\frac{2 \sqrt{1-2 x} (1010 x+617)}{9075 (5 x+3)^{3/2}}-\frac{9}{25} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 96, normalized size = 1.3 \begin{align*}{\frac{1}{90750} \left ( 81675\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+98010\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+29403\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -20200\,x\sqrt{-10\,{x}^{2}-x+3}-12340\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.65448, size = 84, normalized size = 1.14 \begin{align*} \frac{9}{250} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{825 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{404 \, \sqrt{-10 \, x^{2} - x + 3}}{9075 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75211, size = 281, normalized size = 3.8 \begin{align*} -\frac{3267 \, \sqrt{5} \sqrt{2}{\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \,{\left (1010 \, x + 617\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{90750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right )^{2}}{\sqrt{1 - 2 x} \left (5 x + 3\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.90458, size = 194, normalized size = 2.62 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{726000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{9}{125} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{27 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{12100 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{405 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{45375 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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