Optimal. Leaf size=127 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac{\sqrt{1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac{11237 \sqrt{1-2 x}}{111132 (3 x+2)}+\frac{11237 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{55566 \sqrt{21}} \]
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Rubi [A] time = 0.179008, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{1-2 x} (5 x+3)^3}{15 (3 x+2)^5}-\frac{53 \sqrt{1-2 x} (5 x+3)^2}{630 (3 x+2)^4}-\frac{\sqrt{1-2 x} (59665 x+37224)}{79380 (3 x+2)^3}+\frac{11237 \sqrt{1-2 x}}{111132 (3 x+2)}+\frac{11237 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{55566 \sqrt{21}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 20.8472, size = 110, normalized size = 0.87 \[ \frac{11237 \sqrt{- 2 x + 1}}{111132 \left (3 x + 2\right )} - \frac{\sqrt{- 2 x + 1} \left (2505930 x + 1563408\right )}{3333960 \left (3 x + 2\right )^{3}} - \frac{53 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{2}}{630 \left (3 x + 2\right )^{4}} - \frac{\sqrt{- 2 x + 1} \left (5 x + 3\right )^{3}}{15 \left (3 x + 2\right )^{5}} + \frac{11237 \sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}}{1166886} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**3*(1-2*x)**(1/2)/(2+3*x)**6,x)
[Out]
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Mathematica [A] time = 0.116667, size = 68, normalized size = 0.54 \[ \frac{\frac{21 \sqrt{1-2 x} \left (4550985 x^4+240615 x^3-10100352 x^2-8471518 x-1984928\right )}{(3 x+2)^5}+112370 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{11668860} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^6,x]
[Out]
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Maple [A] time = 0.017, size = 75, normalized size = 0.6 \[ 1944\,{\frac{1}{ \left ( -4-6\,x \right ) ^{5}} \left ( -{\frac{11237\, \left ( 1-2\,x \right ) ^{9/2}}{1333584}}+{\frac{4237\, \left ( 1-2\,x \right ) ^{7/2}}{122472}}+{\frac{4954\, \left ( 1-2\,x \right ) ^{5/2}}{229635}}-{\frac{263117\, \left ( 1-2\,x \right ) ^{3/2}}{1102248}}+{\frac{78659\,\sqrt{1-2\,x}}{314928}} \right ) }+{\frac{11237\,\sqrt{21}}{1166886}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^6,x)
[Out]
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Maxima [A] time = 1.52382, size = 173, normalized size = 1.36 \[ -\frac{11237}{2333772} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4550985 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 18685170 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 11651808 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 128927330 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 134900185 \, \sqrt{-2 \, x + 1}}{277830 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.210595, size = 161, normalized size = 1.27 \[ \frac{\sqrt{21}{\left (\sqrt{21}{\left (4550985 \, x^{4} + 240615 \, x^{3} - 10100352 \, x^{2} - 8471518 \, x - 1984928\right )} \sqrt{-2 \, x + 1} + 56185 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{\sqrt{21}{\left (3 \, x - 5\right )} - 21 \, \sqrt{-2 \, x + 1}}{3 \, x + 2}\right )\right )}}{11668860 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^6,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3+5*x)**3*(1-2*x)**(1/2)/(2+3*x)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.236427, size = 157, normalized size = 1.24 \[ -\frac{11237}{2333772} \, \sqrt{21}{\rm ln}\left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{4550985 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 18685170 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - 11651808 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + 128927330 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 134900185 \, \sqrt{-2 \, x + 1}}{8890560 \,{\left (3 \, x + 2\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^3*sqrt(-2*x + 1)/(3*x + 2)^6,x, algorithm="giac")
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