Optimal. Leaf size=171 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac{331 \sqrt{1-2 x} (5 x+3)^{3/2}}{168 (3 x+2)}-\frac{39745 \sqrt{1-2 x} \sqrt{5 x+3}}{4536}-\frac{575}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{326717 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{13608 \sqrt{7}} \]
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Rubi [A] time = 0.0671226, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac{331 \sqrt{1-2 x} (5 x+3)^{3/2}}{168 (3 x+2)}-\frac{39745 \sqrt{1-2 x} \sqrt{5 x+3}}{4536}-\frac{575}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{326717 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{13608 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 149
Rule 154
Rule 157
Rule 54
Rule 216
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{1}{9} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{1}{54} \int \frac{\left (\frac{139}{4}-1065 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=\frac{331 \sqrt{1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{\int \frac{\left (\frac{8757}{8}-\frac{119235 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx}{1134}\\ &=-\frac{39745 \sqrt{1-2 x} \sqrt{3+5 x}}{4536}+\frac{331 \sqrt{1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}+\frac{\int \frac{-\frac{317283}{4}-241500 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{6804}\\ &=-\frac{39745 \sqrt{1-2 x} \sqrt{3+5 x}}{4536}+\frac{331 \sqrt{1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{2875}{243} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{326717 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{27216}\\ &=-\frac{39745 \sqrt{1-2 x} \sqrt{3+5 x}}{4536}+\frac{331 \sqrt{1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}+\frac{326717 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{13608}-\frac{1}{243} \left (1150 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{39745 \sqrt{1-2 x} \sqrt{3+5 x}}{4536}+\frac{331 \sqrt{1-2 x} (3+5 x)^{3/2}}{168 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{575}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{326717 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{13608 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.168675, size = 131, normalized size = 0.77 \[ \frac{21 \sqrt{5 x+3} \left (151200 x^4+497982 x^3+263253 x^2-118190 x-78416\right )+225400 \sqrt{10-20 x} (3 x+2)^3 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-326717 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{95256 \sqrt{1-2 x} (3 x+2)^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 270, normalized size = 1.6 \begin{align*}{\frac{1}{190512\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 8821359\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-6085800\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+17642718\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-12171600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-3175200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+11761812\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-8114400\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-12045222\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2613736\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -1803200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -11550924\,x\sqrt{-10\,{x}^{2}-x+3}-3293472\,\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 = 3.54774, size = 217, normalized size = 1.27 \begin{align*} \frac{865}{2646} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{173 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{588 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{34805}{5292} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{575}{486} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{326717}{190512} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{152917}{31752} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{2507 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3528 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59996, size = 498, normalized size = 2.91 \begin{align*} -\frac{326717 \, \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 ) - 225400 \, \sqrt{10}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 42 \,{\left (75600 \, x^{3} + 286791 \, x^{2} + 275022 \, x + 78416\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{190512 \,{\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.07223, size = 545, normalized size = 3.19 \begin{align*} \frac{326717}{1905120} \, \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{575}{486} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{10}{81} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{11 \,{\left (2463 \, \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} + 1767360 \, \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} + 377652800 \, \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 )}}{756 \,{\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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