Optimal. Leaf size=222 \[ \frac{2776 \sqrt{1-2 x} (5 x+3)^{5/2}}{1701 (3 x+2)^{5/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{13316 \sqrt{1-2 x} (5 x+3)^{3/2}}{35721 (3 x+2)^{3/2}}-\frac{1241596 \sqrt{1-2 x} \sqrt{5 x+3}}{750141 \sqrt{3 x+2}}-\frac{1241596 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141}-\frac{100444 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141} \]
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Rubi [A] time = 0.494021, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{2776 \sqrt{1-2 x} (5 x+3)^{5/2}}{1701 (3 x+2)^{5/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{567 (3 x+2)^{7/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac{13316 \sqrt{1-2 x} (5 x+3)^{3/2}}{35721 (3 x+2)^{3/2}}-\frac{1241596 \sqrt{1-2 x} \sqrt{5 x+3}}{750141 \sqrt{3 x+2}}-\frac{1241596 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141}-\frac{100444 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{750141} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(11/2),x]
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Rubi in Sympy [A] time = 48.9722, size = 201, normalized size = 0.91 \[ - \frac{9434 \left (- 2 x + 1\right )^{\frac{5}{2}} \sqrt{5 x + 3}}{83349 \left (3 x + 2\right )^{\frac{5}{2}}} - \frac{370 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3969 \left (3 x + 2\right )^{\frac{7}{2}}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{27 \left (3 x + 2\right )^{\frac{9}{2}}} + \frac{33290 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{35721 \left (3 x + 2\right )^{\frac{3}{2}}} + \frac{191720 \sqrt{- 2 x + 1} \sqrt{5 x + 3}}{107163 \sqrt{3 x + 2}} - \frac{100444 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2250423} - \frac{1241596 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2250423} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(11/2),x)
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Mathematica [A] time = 0.31405, size = 109, normalized size = 0.49 \[ \frac{2 \left (\frac{3 \sqrt{1-2 x} \sqrt{5 x+3} \left (29072682 x^4+115002639 x^3+142557831 x^2+71920155 x+12903031\right )}{(3 x+2)^{9/2}}+\sqrt{2} \left (10192945 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+50222 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{2250423} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(11/2),x]
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Maple [C] time = 0.03, size = 624, normalized size = 2.8 \[ -{\frac{2}{22504230\,{x}^{2}+2250423\,x-6751269} \left ( 825628545\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4067982\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{4}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2201676120\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+10847952\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{3}\sqrt{1-2\,x}\sqrt{3+5\,x}\sqrt{2+3\,x}+2201676120\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+10847952\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+978522720\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+4821312\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-872180460\,{x}^{6}+163087120\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +803552\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -3537297216\,{x}^{5}-4360088709\,{x}^{4}-1550254392\,{x}^{3}+680169084\,{x}^{2}+608572302\,x+116127279 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(11/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \sqrt{3 \, x + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2),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((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(11/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(11/2),x, algorithm="giac")
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