Optimal. Leaf size=55 \[ \frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}}-\frac{10 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]
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Rubi [A] time = 0.0131668, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {96, 93, 204} \[ \frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}}-\frac{10 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Rule 96
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx &=-\frac{10 \sqrt{1-2 x}}{11 \sqrt{3+5 x}}-3 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{10 \sqrt{1-2 x}}{11 \sqrt{3+5 x}}-6 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{10 \sqrt{1-2 x}}{11 \sqrt{3+5 x}}+\frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{\sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0183409, size = 55, normalized size = 1. \[ \frac{6 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{\sqrt{7}}-\frac{10 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 101, normalized size = 1.8 \begin{align*} -{\frac{1}{77} \left ( 165\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+99\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +70\,\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )} \sqrt{-2 \, x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53002, size = 205, normalized size = 3.73 \begin{align*} \frac{33 \, \sqrt{7}{\left (5 \, x + 3\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 70 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{77 \,{\left (5 \, x + 3\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{1}{\sqrt{1 - 2 x} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.90377, size = 180, normalized size = 3.27 \begin{align*} -\frac{3}{70} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{1}{22} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\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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