Optimal. Leaf size=122 \[ \frac{3895 \sqrt{1-2 x} \sqrt{5 x+3}}{8232 (3 x+2)}+\frac{25 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^2}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^3}-\frac{15235 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
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Rubi [A] time = 0.0389857, 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} \[ \frac{3895 \sqrt{1-2 x} \sqrt{5 x+3}}{8232 (3 x+2)}+\frac{25 \sqrt{1-2 x} \sqrt{5 x+3}}{588 (3 x+2)^2}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{21 (3 x+2)^3}-\frac{15235 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
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}}{\sqrt{1-2 x} (2+3 x)^4} \, dx &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^3}+\frac{1}{21} \int \frac{\frac{35}{2}+20 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^3}+\frac{25 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}+\frac{1}{294} \int \frac{\frac{965}{4}-125 x}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^3}+\frac{25 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}+\frac{3895 \sqrt{1-2 x} \sqrt{3+5 x}}{8232 (2+3 x)}+\frac{\int \frac{45705}{8 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2058}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^3}+\frac{25 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}+\frac{3895 \sqrt{1-2 x} \sqrt{3+5 x}}{8232 (2+3 x)}+\frac{15235 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5488}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^3}+\frac{25 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}+\frac{3895 \sqrt{1-2 x} \sqrt{3+5 x}}{8232 (2+3 x)}+\frac{15235 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2744}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{21 (2+3 x)^3}+\frac{25 \sqrt{1-2 x} \sqrt{3+5 x}}{588 (2+3 x)^2}+\frac{3895 \sqrt{1-2 x} \sqrt{3+5 x}}{8232 (2+3 x)}-\frac{15235 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2744 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0470148, size = 74, normalized size = 0.61 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (11685 x^2+15930 x+5296\right )}{(3 x+2)^3}-15235 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 202, normalized size = 1.7 \begin{align*}{\frac{1}{38416\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 411345\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+822690\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+548460\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+163590\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+121880\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +223020\,x\sqrt{-10\,{x}^{2}-x+3}+74144\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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 = 1.64651, size = 144, normalized size = 1.18 \begin{align*} \frac{15235}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{21 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{25 \, \sqrt{-10 \, x^{2} - x + 3}}{588 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{3895 \, \sqrt{-10 \, x^{2} - x + 3}}{8232 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78523, size = 305, normalized size = 2.5 \begin{align*} -\frac{15235 \, \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 (11685 \, x^{2} + 15930 \, x + 5296\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{38416 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 = 3.40047, size = 429, normalized size = 3.52 \begin{align*} \frac{3047}{76832} \, \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{55 \,{\left (277 \, \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 )}^{5} - 159040 \, \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} - 20713280 \, \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 )}}{1372 \,{\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 )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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