Optimal. Leaf size=144 \[ -\frac{608185 \sqrt{1-2 x}}{504 \sqrt{5 x+3}}+\frac{13409 \sqrt{1-2 x}}{168 (3 x+2) \sqrt{5 x+3}}+\frac{77 \sqrt{1-2 x}}{12 (3 x+2)^2 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 \sqrt{5 x+3}}+\frac{463881 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]
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Rubi [A] time = 0.0475551, antiderivative size = 144, 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 = {98, 151, 152, 12, 93, 204} \[ -\frac{608185 \sqrt{1-2 x}}{504 \sqrt{5 x+3}}+\frac{13409 \sqrt{1-2 x}}{168 (3 x+2) \sqrt{5 x+3}}+\frac{77 \sqrt{1-2 x}}{12 (3 x+2)^2 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 \sqrt{5 x+3}}+\frac{463881 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 98
Rule 151
Rule 152
Rule 12
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{3/2}} \, dx &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 \sqrt{3+5 x}}+\frac{1}{9} \int \frac{\frac{275}{2}-198 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 \sqrt{3+5 x}}+\frac{77 \sqrt{1-2 x}}{12 (2+3 x)^2 \sqrt{3+5 x}}+\frac{1}{126} \int \frac{\frac{50743}{4}-16170 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 \sqrt{3+5 x}}+\frac{77 \sqrt{1-2 x}}{12 (2+3 x)^2 \sqrt{3+5 x}}+\frac{13409 \sqrt{1-2 x}}{168 (2+3 x) \sqrt{3+5 x}}+\frac{1}{882} \int \frac{\frac{5986981}{8}-\frac{1407945 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{608185 \sqrt{1-2 x}}{504 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 \sqrt{3+5 x}}+\frac{77 \sqrt{1-2 x}}{12 (2+3 x)^2 \sqrt{3+5 x}}+\frac{13409 \sqrt{1-2 x}}{168 (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{321469533}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{4851}\\ &=-\frac{608185 \sqrt{1-2 x}}{504 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 \sqrt{3+5 x}}+\frac{77 \sqrt{1-2 x}}{12 (2+3 x)^2 \sqrt{3+5 x}}+\frac{13409 \sqrt{1-2 x}}{168 (2+3 x) \sqrt{3+5 x}}-\frac{463881}{112} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{608185 \sqrt{1-2 x}}{504 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 \sqrt{3+5 x}}+\frac{77 \sqrt{1-2 x}}{12 (2+3 x)^2 \sqrt{3+5 x}}+\frac{13409 \sqrt{1-2 x}}{168 (2+3 x) \sqrt{3+5 x}}-\frac{463881}{56} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{608185 \sqrt{1-2 x}}{504 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 \sqrt{3+5 x}}+\frac{77 \sqrt{1-2 x}}{12 (2+3 x)^2 \sqrt{3+5 x}}+\frac{13409 \sqrt{1-2 x}}{168 (2+3 x) \sqrt{3+5 x}}+\frac{463881 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{56 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.0607951, size = 79, normalized size = 0.55 \[ \frac{1}{392} \left (463881 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\frac{7 \sqrt{1-2 x} \left (1824555 x^3+3608883 x^2+2378026 x+521968\right )}{(3 x+2)^3 \sqrt{5 x+3}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 250, normalized size = 1.7 \begin{align*} -{\frac{1}{784\, \left ( 2+3\,x \right ) ^{3}} \left ( 62623935\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+162822231\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+158647302\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+25543770\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+68654388\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+50524362\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+11133144\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +33292364\,x\sqrt{-10\,{x}^{2}-x+3}+7307552\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 3.15281, size = 285, normalized size = 1.98 \begin{align*} -\frac{463881}{784} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{608185 \, x}{252 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{635003}{504 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{49}{27 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{1561}{108 \,{\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{4367}{24 \,{\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 = 1.53336, size = 367, normalized size = 2.55 \begin{align*} \frac{463881 \, \sqrt{7}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 (1824555 \, x^{3} + 3608883 \, x^{2} + 2378026 \, x + 521968\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{784 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\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 = 2.53812, size = 509, normalized size = 3.53 \begin{align*} -\frac{463881}{7840} \, \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}{2} \, \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 )} - \frac{11 \,{\left (33989 \, \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} + 15023680 \, \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} + 1769566400 \, \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 )}}{28 \,{\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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