Optimal. Leaf size=127 \[ \frac{\sqrt{3 x+2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{23 \sqrt{3 x+2} \sqrt{5 x+3}}{7 \sqrt{1-2 x}}-\frac{23 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{139}{14} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.254951, 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 \[ \frac{\sqrt{3 x+2} (5 x+3)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{23 \sqrt{3 x+2} \sqrt{5 x+3}}{7 \sqrt{1-2 x}}-\frac{23 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{7 \sqrt{33}}-\frac{139}{14} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/(1 - 2*x)^(5/2),x]
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Rubi in Sympy [A] time = 23.8978, size = 112, normalized size = 0.88 \[ - \frac{139 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{42} - \frac{23 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{231} - \frac{23 \sqrt{3 x + 2} \sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1}} + \frac{\sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{3 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(3/2)*(2+3*x)**(1/2)/(1-2*x)**(5/2),x)
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Mathematica [A] time = 0.189167, size = 115, normalized size = 0.91 \[ -\frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (48-173 x)-70 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+139 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{42 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/(1 - 2*x)^(5/2),x]
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Maple [C] time = 0.027, size = 276, normalized size = 2.2 \[{\frac{1}{42\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 140\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-278\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +139\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +5190\,{x}^{3}+5134\,{x}^{2}+252\,x-576 \right ) \sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(3/2)*(2+3*x)^(1/2)/(1-2*x)^(5/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*sqrt(3*x + 2)/(-2*x + 1)^(5/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*sqrt(3*x + 2)/(-2*x + 1)^(5/2),x, algorithm="fricas")
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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)**(3/2)*(2+3*x)**(1/2)/(1-2*x)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}}{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)*sqrt(3*x + 2)/(-2*x + 1)^(5/2),x, algorithm="giac")
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