Optimal. Leaf size=122 \[ \frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{\sqrt{1-2 x}}+2 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{23 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}+\frac{139}{10} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.24398, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{\sqrt{5 x+3} (3 x+2)^{3/2}}{\sqrt{1-2 x}}+2 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}+\frac{23 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5 \sqrt{33}}+\frac{139}{10} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]
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Rubi in Sympy [A] time = 24.2159, size = 109, normalized size = 0.89 \[ 2 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3} + \frac{139 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{30} + \frac{23 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{165} + \frac{\left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{\sqrt{- 2 x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(3/2)*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)
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Mathematica [A] time = 0.21864, size = 103, normalized size = 0.84 \[ \frac{-30 \sqrt{3 x+2} \sqrt{5 x+3} (x-4)+70 \sqrt{2-4 x} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-139 \sqrt{2-4 x} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{30 \sqrt{1-2 x}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/(1 - 2*x)^(3/2),x]
[Out]
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Maple [C] time = 0.023, size = 164, normalized size = 1.3 \[ -{\frac{1}{900\,{x}^{3}+690\,{x}^{2}-210\,x-180}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -139\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -450\,{x}^{3}+1230\,{x}^{2}+2100\,x+720 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(3/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^(3/2)/(-2*x + 1)^(3/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^(3/2)/(-2*x + 1)^(3/2),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((2+3*x)**(3/2)*(3+5*x)**(1/2)/(1-2*x)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^(3/2)/(-2*x + 1)^(3/2),x, algorithm="giac")
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