Optimal. Leaf size=142 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{11 \sqrt{1-2 x}}+\frac{939}{880} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3+\frac{76587 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{17600}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (7645620 x+18424549)}{2816000}-\frac{291096141 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]
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Rubi [A] time = 0.25934, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{11 \sqrt{1-2 x}}+\frac{939}{880} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^3+\frac{76587 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{17600}+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (7645620 x+18424549)}{2816000}-\frac{291096141 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{256000 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^5/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 26.3267, size = 133, normalized size = 0.94 \[ \frac{939 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{880} + \frac{76587 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{17600} + \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{602092575 x}{4} + \frac{5803732935}{16}\right )}{2640000} - \frac{291096141 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{2560000} + \frac{7 \left (3 x + 2\right )^{4} \sqrt{5 x + 3}}{11 \sqrt{- 2 x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**5/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)
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Mathematica [A] time = 0.122438, size = 74, normalized size = 0.52 \[ \frac{3202057551 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (17107200 x^4+76887360 x^3+171939240 x^2+332129358 x-488641609\right )}{28160000 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^5/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x]),x]
[Out]
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Maple [A] time = 0.021, size = 140, normalized size = 1. \[ -{\frac{1}{-56320000+112640000\,x} \left ( -342144000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1537747200\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+6404115102\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3438784800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-3202057551\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -6642587160\,x\sqrt{-10\,{x}^{2}-x+3}+9772832180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^5/(1-2*x)^(3/2)/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.50312, size = 134, normalized size = 0.94 \[ \frac{243}{80} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + \frac{24273}{1600} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{291096141}{5120000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{487863}{12800} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{19975419}{256000} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{16807 \, \sqrt{-10 \, x^{2} - x + 3}}{176 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^5/(sqrt(5*x + 3)*(-2*x + 1)^(3/2)),x, algorithm="maxima")
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Fricas [A] time = 0.2358, size = 113, normalized size = 0.8 \[ \frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (17107200 \, x^{4} + 76887360 \, x^{3} + 171939240 \, x^{2} + 332129358 \, x - 488641609\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 3202057551 \,{\left (2 \, x - 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{56320000 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^5/(sqrt(5*x + 3)*(-2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{5}}{\left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**5/(1-2*x)**(3/2)/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.23662, size = 131, normalized size = 0.92 \[ -\frac{291096141}{2560000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (198 \,{\left (12 \,{\left (8 \,{\left (36 \, \sqrt{5}{\left (5 \, x + 3\right )} + 377 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 29669 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4900505 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 16010291851 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{352000000 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^5/(sqrt(5*x + 3)*(-2*x + 1)^(3/2)),x, algorithm="giac")
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