Optimal. Leaf size=122 \[ \frac{15 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}-\frac{15 \sqrt{1-2 x} \sqrt{5 x+3}}{98 (3 x+2)^2}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^2}-\frac{1585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
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Rubi [A] time = 0.0381462, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {99, 151, 12, 93, 204} \[ \frac{15 \sqrt{1-2 x} \sqrt{5 x+3}}{1372 (3 x+2)}-\frac{15 \sqrt{1-2 x} \sqrt{5 x+3}}{98 (3 x+2)^2}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^2}-\frac{1585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 99
Rule 151
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^3} \, dx &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{2}{7} \int \frac{-\frac{35}{2}-30 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{98 (2+3 x)^2}-\frac{1}{49} \int \frac{-\frac{205}{4}-75 x}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{98 (2+3 x)^2}+\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}-\frac{1}{343} \int -\frac{1585}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{98 (2+3 x)^2}+\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}+\frac{1585 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{98 (2+3 x)^2}+\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}+\frac{1585 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^2}-\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{98 (2+3 x)^2}+\frac{15 \sqrt{1-2 x} \sqrt{3+5 x}}{1372 (2+3 x)}-\frac{1585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0455029, size = 85, normalized size = 0.7 \[ \frac{7 \sqrt{5 x+3} \left (-90 x^2+405 x+212\right )-1585 \sqrt{7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9604 \sqrt{1-2 x} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 209, normalized size = 1.7 \begin{align*}{\frac{1}{19208\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) } \left ( 28530\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+23775\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-6340\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1260\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-6340\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -5670\,x\sqrt{-10\,{x}^{2}-x+3}-2968\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,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 [A] time = 3.59048, size = 193, normalized size = 1.58 \begin{align*} \frac{1585}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{25 \, x}{686 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{785}{4116 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{42 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} - \frac{95}{588 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76944, size = 293, normalized size = 2.4 \begin{align*} -\frac{1585 \, \sqrt{7}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 (90 \, x^{2} - 405 \, x - 212\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{19208 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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 = 2.74585, size = 382, normalized size = 3.13 \begin{align*} \frac{317}{38416} \, \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}}{1715 \,{\left (2 \, x - 1\right )}} - \frac{33 \,{\left (7 \, \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 )}^{3} - 680 \, \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 )}\right )}}{98 \,{\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 )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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