Optimal. Leaf size=137 \[ -\frac{57595 \sqrt{5 x+3}}{249018 \sqrt{1-2 x}}+\frac{51 \sqrt{5 x+3}}{28 (1-2 x)^{3/2} (3 x+2)}-\frac{1735 \sqrt{5 x+3}}{3234 (1-2 x)^{3/2}}+\frac{3 \sqrt{5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}-\frac{5805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
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Rubi [A] time = 0.0447339, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ -\frac{57595 \sqrt{5 x+3}}{249018 \sqrt{1-2 x}}+\frac{51 \sqrt{5 x+3}}{28 (1-2 x)^{3/2} (3 x+2)}-\frac{1735 \sqrt{5 x+3}}{3234 (1-2 x)^{3/2}}+\frac{3 \sqrt{5 x+3}}{14 (1-2 x)^{3/2} (3 x+2)^2}-\frac{5805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 103
Rule 151
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx &=\frac{3 \sqrt{3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{1}{14} \int \frac{-\frac{1}{2}-90 x}{(1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=\frac{3 \sqrt{3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{51 \sqrt{3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}+\frac{1}{98} \int \frac{-\frac{5005}{4}-3570 x}{(1-2 x)^{5/2} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{1735 \sqrt{3+5 x}}{3234 (1-2 x)^{3/2}}+\frac{3 \sqrt{3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{51 \sqrt{3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}-\frac{\int \frac{\frac{38815}{8}+\frac{182175 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{11319}\\ &=-\frac{1735 \sqrt{3+5 x}}{3234 (1-2 x)^{3/2}}-\frac{57595 \sqrt{3+5 x}}{249018 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{51 \sqrt{3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}+\frac{2 \int \frac{14750505}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{871563}\\ &=-\frac{1735 \sqrt{3+5 x}}{3234 (1-2 x)^{3/2}}-\frac{57595 \sqrt{3+5 x}}{249018 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{51 \sqrt{3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}+\frac{5805 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=-\frac{1735 \sqrt{3+5 x}}{3234 (1-2 x)^{3/2}}-\frac{57595 \sqrt{3+5 x}}{249018 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{51 \sqrt{3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}+\frac{5805 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=-\frac{1735 \sqrt{3+5 x}}{3234 (1-2 x)^{3/2}}-\frac{57595 \sqrt{3+5 x}}{249018 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{14 (1-2 x)^{3/2} (2+3 x)^2}+\frac{51 \sqrt{3+5 x}}{28 (1-2 x)^{3/2} (2+3 x)}-\frac{5805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0624632, size = 95, normalized size = 0.69 \[ -\frac{-7 \sqrt{5 x+3} \left (2073420 x^3-676860 x^2-945629 x+391476\right )-2107215 \sqrt{7-14 x} (2 x-1) (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3486252 (1-2 x)^{3/2} (3 x+2)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.016, size = 257, normalized size = 1.9 \begin{align*}{\frac{1}{6972504\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) ^{2}} \left ( 75859740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+25286580\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-48465945\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+29027880\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-8428860\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-9476040\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+8428860\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -13238806\,x\sqrt{-10\,{x}^{2}-x+3}+5480664\,\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 )}^{3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61233, size = 356, normalized size = 2.6 \begin{align*} -\frac{2107215 \, \sqrt{7}{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, 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 (2073420 \, x^{3} - 676860 \, x^{2} - 945629 \, x + 391476\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{6972504 \,{\left (36 \, x^{4} + 12 \, x^{3} - 23 \, 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 = 3.47177, size = 400, normalized size = 2.92 \begin{align*} \frac{1161}{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{32 \,{\left (367 \, \sqrt{5}{\left (5 \, x + 3\right )} - 2211 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{21789075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{297 \,{\left (197 \, \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} + 36680 \, \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 )}}{4802 \,{\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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