Optimal. Leaf size=127 \[ -\frac{74 \sqrt{3 x+2} \sqrt{5 x+3}}{147 \sqrt{1-2 x}}+\frac{11 \sqrt{3 x+2} \sqrt{5 x+3}}{21 (1-2 x)^{3/2}}-\frac{13 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}-\frac{37}{49} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.256445, 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 \[ -\frac{74 \sqrt{3 x+2} \sqrt{5 x+3}}{147 \sqrt{1-2 x}}+\frac{11 \sqrt{3 x+2} \sqrt{5 x+3}}{21 (1-2 x)^{3/2}}-\frac{13 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{49 \sqrt{33}}-\frac{37}{49} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(3/2)/((1 - 2*x)^(5/2)*Sqrt[2 + 3*x]),x]
[Out]
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Rubi in Sympy [A] time = 23.791, size = 114, normalized size = 0.9 \[ - \frac{37 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{147} - \frac{13 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{1617} - \frac{74 \sqrt{3 x + 2} \sqrt{5 x + 3}}{147 \sqrt{- 2 x + 1}} + \frac{11 \sqrt{3 x + 2} \sqrt{5 x + 3}}{21 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(3/2)/(1-2*x)**(5/2)/(2+3*x)**(1/2),x)
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Mathematica [A] time = 0.25014, size = 115, normalized size = 0.91 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (148 x+3)+35 \sqrt{2-4 x} (2 x-1) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-74 \sqrt{2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{294 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(3/2)/((1 - 2*x)^(5/2)*Sqrt[2 + 3*x]),x]
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Maple [C] time = 0.03, size = 276, normalized size = 2.2 \[{\frac{1}{294\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 70\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-148\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-35\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +74\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4440\,{x}^{3}+5714\,{x}^{2}+1890\,x+36 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^(1/2),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. \[{\rm integral}\left (\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]
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)/(1-2*x)**(5/2)/(2+3*x)**(1/2),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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