Optimal. Leaf size=222 \[ \frac{5260}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{135334 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{5103}+\frac{135334 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515}-\frac{452399 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515} \]
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Rubi [A] time = 0.484889, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{5260}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{27 \sqrt{3 x+2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}-\frac{31298}{567} \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}+\frac{135334 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{5103}+\frac{135334 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515}-\frac{452399 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{25515} \]
Antiderivative was successfully verified.
[In] Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(5/2),x]
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Rubi in Sympy [A] time = 49.7736, size = 201, normalized size = 0.91 \[ - \frac{370 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{189 \sqrt{3 x + 2}} - \frac{2 \left (- 2 x + 1\right )^{\frac{5}{2}} \left (5 x + 3\right )^{\frac{5}{2}}}{9 \left (3 x + 2\right )^{\frac{3}{2}}} - \frac{940 \left (- 2 x + 1\right )^{\frac{3}{2}} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{567} - \frac{2368 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \left (5 x + 3\right )^{\frac{3}{2}}}{567} + \frac{135334 \sqrt{- 2 x + 1} \sqrt{3 x + 2} \sqrt{5 x + 3}}{5103} - \frac{452399 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{76545} + \frac{135334 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{76545} \]
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)**(5/2),x)
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Mathematica [A] time = 0.418589, size = 112, normalized size = 0.5 \[ \frac{\frac{30 \sqrt{1-2 x} \sqrt{5 x+3} \left (24300 x^4-25110 x^3+5949 x^2+108285 x+56963\right )}{(3 x+2)^{3/2}}-2685410 \sqrt{2} F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+452399 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{76545} \]
Antiderivative was successfully verified.
[In] Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(5/2),x]
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Maple [C] time = 0.029, size = 282, normalized size = 1.3 \[{\frac{1}{765450\,{x}^{2}+76545\,x-229635} \left ( 8056230\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-1357197\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+7290000\,{x}^{6}+5370820\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -904798\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -6804000\,{x}^{5}-1155600\,{x}^{4}+34923870\,{x}^{3}+19802040\,{x}^{2}-8036760\,x-5126670 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]
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)^(5/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]
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)^(5/2),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}, x\right ) \]
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)^(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((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(5/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}\,{d x} \]
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)^(5/2),x, algorithm="giac")
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