Optimal. Leaf size=86 \[ -\frac{58 \sqrt{5 x+3}}{539 \sqrt{1-2 x}}+\frac{3 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)}-\frac{123 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
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Rubi [A] time = 0.024538, antiderivative size = 86, normalized size of antiderivative = 1., 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 = {103, 152, 12, 93, 204} \[ -\frac{58 \sqrt{5 x+3}}{539 \sqrt{1-2 x}}+\frac{3 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)}-\frac{123 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{49 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 103
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx &=\frac{3 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)}+\frac{1}{7} \int \frac{\frac{1}{2}-30 x}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{58 \sqrt{3+5 x}}{539 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)}-\frac{2}{539} \int -\frac{1353}{4 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{58 \sqrt{3+5 x}}{539 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)}+\frac{123}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{58 \sqrt{3+5 x}}{539 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)}+\frac{123}{49} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{58 \sqrt{3+5 x}}{539 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)}-\frac{123 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{49 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.035837, size = 78, normalized size = 0.91 \[ \frac{7 (115-174 x) \sqrt{5 x+3}-1353 \sqrt{7-14 x} (3 x+2) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3773 \sqrt{1-2 x} (3 x+2)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 161, normalized size = 1.9 \begin{align*}{\frac{1}{ \left ( 15092+22638\,x \right ) \left ( 2\,x-1 \right ) } \left ( 8118\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1353\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-2706\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +2436\,x\sqrt{-10\,{x}^{2}-x+3}-1610\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \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}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76459, size = 247, normalized size = 2.87 \begin{align*} -\frac{1353 \, \sqrt{7}{\left (6 \, x^{2} + x - 2\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 ) - 14 \,{\left (174 \, x - 115\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{7546 \,{\left (6 \, x^{2} + x - 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.09304, size = 296, normalized size = 3.44 \begin{align*} \frac{123}{6860} \, \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{8 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{2695 \,{\left (2 \, x - 1\right )}} + \frac{198 \, \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 )}}{49 \,{\left ({\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 )}^{2} + 280\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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