Optimal. Leaf size=191 \[ \frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{3/2}}-\frac{536 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{7/2}}-\frac{9124 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105}-\frac{184636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105} \]
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Rubi [A] time = 0.422659, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{184636 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 \sqrt{3 x+2}}+\frac{974 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{3/2}}-\frac{536 \sqrt{1-2 x} \sqrt{5 x+3}}{5145 (3 x+2)^{5/2}}+\frac{2 \sqrt{1-2 x} \sqrt{5 x+3}}{147 (3 x+2)^{7/2}}-\frac{9124 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105}-\frac{184636 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{252105} \]
Antiderivative was successfully verified.
[In] Int[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)),x]
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Rubi in Sympy [A] time = 40.4077, size = 172, normalized size = 0.9 \[ \frac{184636 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{252105 \sqrt{3 x + 2}} + \frac{974 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{36015 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{536 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{5145 \left (3 x + 2\right )^{\frac{5}{2}}} + \frac{2 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{147 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{184636 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{756315} - \frac{9124 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{756315} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((3+5*x)**(3/2)/(2+3*x)**(9/2)/(1-2*x)**(1/2),x)
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Mathematica [A] time = 0.264225, size = 104, normalized size = 0.54 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (2492586 x^3+5015853 x^2+3324960 x+727631\right )}{(3 x+2)^{7/2}}+\sqrt{2} \left (92318 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-17045 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{756315} \]
Antiderivative was successfully verified.
[In] Integrate[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)),x]
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Maple [C] time = 0.031, size = 505, normalized size = 2.6 \[{\frac{2}{7563150\,{x}^{2}+756315\,x-2268945} \left ( 460215\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}-2492586\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+920430\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-4985172\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+613620\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3323448\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+136360\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -738544\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +74777580\,{x}^{5}+157953348\,{x}^{4}+92363085\,{x}^{3}-13338867\,{x}^{2}-27741747\,x-6548679 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((3+5*x)^(3/2)/(2+3*x)^(9/2)/(1-2*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^(9/2)*sqrt(-2*x + 1)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^(9/2)*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((3+5*x)**(3/2)/(2+3*x)**(9/2)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(3/2)/((3*x + 2)^(9/2)*sqrt(-2*x + 1)),x, algorithm="giac")
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