Optimal. Leaf size=142 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{33 (1-2 x)^{3/2}}-\frac{2051 \sqrt{5 x+3} (3 x+2)^3}{726 \sqrt{1-2 x}}-\frac{23909 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{4840}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (50124540 x+120791143)}{774400}+\frac{8261577 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]
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Rubi [A] time = 0.263685, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^4}{33 (1-2 x)^{3/2}}-\frac{2051 \sqrt{5 x+3} (3 x+2)^3}{726 \sqrt{1-2 x}}-\frac{23909 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{4840}-\frac{\sqrt{1-2 x} \sqrt{5 x+3} (50124540 x+120791143)}{774400}+\frac{8261577 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^5/((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 25.8082, size = 133, normalized size = 0.94 \[ - \frac{23909 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2} \sqrt{5 x + 3}}{4840} - \frac{\sqrt{- 2 x + 1} \sqrt{5 x + 3} \left (\frac{563901075 x}{4} + \frac{5435601435}{16}\right )}{2178000} + \frac{8261577 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{64000} - \frac{2051 \left (3 x + 2\right )^{3} \sqrt{5 x + 3}}{726 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{4} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**5/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)
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Mathematica [A] time = 0.177408, size = 79, normalized size = 0.56 \[ \frac{2998952451 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (18817920 x^4+101146320 x^3+359461476 x^2-1261070176 x+452899509\right )}{23232000 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^5/((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]),x]
[Out]
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Maple [A] time = 0.023, size = 154, normalized size = 1.1 \[{\frac{1}{46464000\, \left ( -1+2\,x \right ) ^{2}} \left ( -376358400\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+11995809804\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-2022926400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-11995809804\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-7189229520\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2998952451\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +25221403520\,x\sqrt{-10\,{x}^{2}-x+3}-9057990180\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^5/(1-2*x)^(5/2)/(3+5*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.49746, size = 146, normalized size = 1.03 \[ -\frac{81}{40} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{8261577}{128000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{4131}{320} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{326943}{6400} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{16807 \, \sqrt{-10 \, x^{2} - x + 3}}{528 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1020425 \, \sqrt{-10 \, x^{2} - x + 3}}{5808 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^5/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225309, size = 127, normalized size = 0.89 \[ -\frac{\sqrt{10}{\left (2 \, \sqrt{10}{\left (18817920 \, x^{4} + 101146320 \, x^{3} + 359461476 \, x^{2} - 1261070176 \, x + 452899509\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - 2998952451 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )}}{20 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{46464000 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^5/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (3 x + 2\right )^{5}}{\left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+3*x)**5/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.244393, size = 131, normalized size = 0.92 \[ \frac{8261577}{64000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (9801 \,{\left (12 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 119 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 27809 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 9996528778 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 164942367909 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1452000000 \,{\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^5/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="giac")
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