Optimal. Leaf size=187 \[ \frac{598660 \sqrt{1-2 x} \sqrt{3 x+2}}{2152227 \sqrt{5 x+3}}-\frac{18470 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 (5 x+3)^{3/2}}+\frac{368 \sqrt{3 x+2}}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{7388 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}}-\frac{119732 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}} \]
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Rubi [A] time = 0.431198, antiderivative size = 187, 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 \[ \frac{598660 \sqrt{1-2 x} \sqrt{3 x+2}}{2152227 \sqrt{5 x+3}}-\frac{18470 \sqrt{1-2 x} \sqrt{3 x+2}}{195657 (5 x+3)^{3/2}}+\frac{368 \sqrt{3 x+2}}{5929 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{4 \sqrt{3 x+2}}{231 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{7388 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}}-\frac{119732 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{65219 \sqrt{33}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2)),x]
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Rubi in Sympy [A] time = 41.6047, size = 172, normalized size = 0.92 \[ - \frac{119732 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2152227} - \frac{7388 \sqrt{33} F\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{2152227} - \frac{239464 \sqrt{3 x + 2} \sqrt{5 x + 3}}{2152227 \sqrt{- 2 x + 1}} + \frac{18160 \sqrt{3 x + 2}}{27951 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} - \frac{370 \sqrt{3 x + 2}}{2541 \sqrt{- 2 x + 1} \left (5 x + 3\right )^{\frac{3}{2}}} + \frac{4 \sqrt{3 x + 2}}{231 \left (- 2 x + 1\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(5/2)/(3+5*x)**(5/2)/(2+3*x)**(1/2),x)
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Mathematica [A] time = 0.257881, size = 103, normalized size = 0.55 \[ \frac{2 \left (\frac{\sqrt{3 x+2} \left (5986600 x^3-2800980 x^2-1822554 x+881831\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}+\sqrt{2} \left (1085 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )+59866 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right )}{2152227} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2)),x]
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Maple [C] time = 0.038, size = 383, normalized size = 2.1 \[ -{\frac{2}{2152227\, \left ( -1+2\,x \right ) ^{2}}\sqrt{1-2\,x} \left ( 10850\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+598660\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1085\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+59866\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-3255\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -179598\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -17959800\,{x}^{4}-3570260\,{x}^{3}+11069622\,{x}^{2}+999615\,x-1763662 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(5/2)/(3+5*x)^(5/2)/(2+3*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(5/2)*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. \[{\rm integral}\left (\frac{1}{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(5/2)*sqrt(3*x + 2)*(-2*x + 1)^(5/2)),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(1/(1-2*x)**(5/2)/(3+5*x)**(5/2)/(2+3*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(5/2)*sqrt(3*x + 2)*(-2*x + 1)^(5/2)),x, algorithm="giac")
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