Optimal. Leaf size=127 \[ -\frac{1}{5} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{23}{25} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{8}{125} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1597}{250} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
[Out]
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Rubi [A] time = 0.258665, antiderivative size = 127, 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 \[ -\frac{1}{5} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}-\frac{23}{25} \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}-\frac{8}{125} \sqrt{33} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{1597}{250} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[((2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x],x]
[Out]
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Rubi in Sympy [A] time = 25.3885, size = 114, normalized size = 0.9 \[ - \frac{\sqrt{- 2 x + 1} \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{5} - \frac{23 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{25} - \frac{1597 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{750} - \frac{8 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{125} \]
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)**(1/2),x)
[Out]
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Mathematica [A] time = 0.216752, size = 92, normalized size = 0.72 \[ \frac{-45 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} (5 x+11)-805 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+1597 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{375 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[((2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x],x]
[Out]
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Maple [C] time = 0.017, size = 169, normalized size = 1.3 \[{\frac{1}{22500\,{x}^{3}+17250\,{x}^{2}-5250\,x-4500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 805\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -1597\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -13500\,{x}^{4}-40050\,{x}^{3}-19620\,{x}^{2}+9630\,x+5940 \right ) } \]
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)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^(3/2)/sqrt(-2*x + 1),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{\sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^(3/2)/sqrt(-2*x + 1),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)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}}{\sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(5*x + 3)*(3*x + 2)^(3/2)/sqrt(-2*x + 1),x, algorithm="giac")
[Out]