Optimal. Leaf size=116 \[ -\frac{2 (1-2 x)^{7/2}}{55 \sqrt{5 x+3}}+\frac{7}{275} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{7}{100} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{231 \sqrt{5 x+3} \sqrt{1-2 x}}{1000}+\frac{2541 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1000 \sqrt{10}} \]
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Rubi [A] time = 0.0293435, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {78, 50, 54, 216} \[ -\frac{2 (1-2 x)^{7/2}}{55 \sqrt{5 x+3}}+\frac{7}{275} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{7}{100} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{231 \sqrt{5 x+3} \sqrt{1-2 x}}{1000}+\frac{2541 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1000 \sqrt{10}} \]
Antiderivative was successfully verified.
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Rule 78
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{7/2}}{55 \sqrt{3+5 x}}+\frac{21}{55} \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{7/2}}{55 \sqrt{3+5 x}}+\frac{7}{275} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{7}{10} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{7/2}}{55 \sqrt{3+5 x}}+\frac{7}{100} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{7}{275} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{231}{200} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{7/2}}{55 \sqrt{3+5 x}}+\frac{231 \sqrt{1-2 x} \sqrt{3+5 x}}{1000}+\frac{7}{100} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{7}{275} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{2541 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2000}\\ &=-\frac{2 (1-2 x)^{7/2}}{55 \sqrt{3+5 x}}+\frac{231 \sqrt{1-2 x} \sqrt{3+5 x}}{1000}+\frac{7}{100} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{7}{275} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{2541 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{7/2}}{55 \sqrt{3+5 x}}+\frac{231 \sqrt{1-2 x} \sqrt{3+5 x}}{1000}+\frac{7}{100} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{7}{275} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{2541 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1000 \sqrt{10}}\\ \end{align*}
Mathematica [A] time = 0.032681, size = 83, normalized size = 0.72 \[ \frac{-10 \left (1600 x^4-3480 x^3+3590 x^2+761 x-943\right )-2541 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{10000 \sqrt{1-2 x} \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 116, normalized size = 1. \begin{align*}{\frac{1}{20000} \left ( 16000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+12705\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-26800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+7623\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +22500\,x\sqrt{-10\,{x}^{2}-x+3}+18860\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.35017, size = 124, normalized size = 1.07 \begin{align*} -\frac{8 \, x^{4}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{87 \, x^{3}}{25 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{359 \, x^{2}}{100 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{2541}{20000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{761 \, x}{1000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{943}{1000 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7666, size = 263, normalized size = 2.27 \begin{align*} -\frac{2541 \, \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 (800 \, x^{3} - 1340 \, x^{2} + 1125 \, x + 943\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20000 \,{\left (5 \, x + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.35898, size = 167, normalized size = 1.44 \begin{align*} \frac{1}{25000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 139 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 3597 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{2541}{10000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{6250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{3125 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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