Optimal. Leaf size=180 \[ \frac{121 \sqrt{1-2 x} (5 x+3)^{5/2}}{16 (3 x+2)^3}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{5/2}}{8 (3 x+2)^4}+\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{3/2}}{448 (3 x+2)^2}-\frac{43923 \sqrt{1-2 x} \sqrt{5 x+3}}{6272 (3 x+2)}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]
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Rubi [A] time = 0.0545405, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{121 \sqrt{1-2 x} (5 x+3)^{5/2}}{16 (3 x+2)^3}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{5/2}}{8 (3 x+2)^4}+\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{3/2}}{448 (3 x+2)^2}-\frac{43923 \sqrt{1-2 x} \sqrt{5 x+3}}{6272 (3 x+2)}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{6272 \sqrt{7}} \]
Antiderivative was successfully verified.
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Rule 94
Rule 93
Rule 204
Rubi steps
\begin{align*} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx &=\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac{11}{2} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac{363}{16} \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}+\frac{1331}{32} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{1331 \sqrt{1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}+\frac{43923}{896} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{6272 (2+3 x)}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}+\frac{483153 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{12544}\\ &=-\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{6272 (2+3 x)}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}+\frac{483153 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{6272}\\ &=-\frac{43923 \sqrt{1-2 x} \sqrt{3+5 x}}{6272 (2+3 x)}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{3/2}}{448 (2+3 x)^2}+\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{5 (2+3 x)^5}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{5/2}}{8 (2+3 x)^4}+\frac{121 \sqrt{1-2 x} (3+5 x)^{5/2}}{16 (2+3 x)^3}-\frac{483153 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{6272 \sqrt{7}}\\ \end{align*}
Mathematica [A] time = 0.100654, size = 109, normalized size = 0.61 \[ \frac{11 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{43904}+\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{5 (3 x+2)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.013, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{439040\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 587030895\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+1956769650\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+2609026200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+240148090\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+1739350800\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+648585420\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+579783600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+660330328\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+77304480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +299064752\,x\sqrt{-10\,{x}^{2}-x+3}+50686272\,\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 = 4.67218, size = 306, normalized size = 1.7 \begin{align*} \frac{90695}{32928} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{5 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{56 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1221 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{784 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{54417 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{738705}{21952} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{483153}{87808} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{650859}{43904} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{215303 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{131712 \,{\left (3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58723, size = 429, normalized size = 2.38 \begin{align*} -\frac{2415765 \, \sqrt{7}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 (17153435 \, x^{4} + 46327530 \, x^{3} + 47166452 \, x^{2} + 21361768 \, x + 3620448\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{439040 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 = 3.11787, size = 594, normalized size = 3.3 \begin{align*} \frac{483153}{878080} \, \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{161051 \,{\left (3 \, \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 )}^{9} + 3920 \, \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 )}^{7} - 2007040 \, \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} - 307328000 \, \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} - 18439680000 \, \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 )}}{3136 \,{\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 )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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