Optimal. Leaf size=125 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{448 \sqrt{5 x+3} \sqrt{3 x+2}}{363 \sqrt{1-2 x}}-\frac{67 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55 \sqrt{33}}-\frac{4451 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{110 \sqrt{33}} \]
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Rubi [A] time = 0.260752, antiderivative size = 125, 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 \[ \frac{7 \sqrt{5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}-\frac{448 \sqrt{5 x+3} \sqrt{3 x+2}}{363 \sqrt{1-2 x}}-\frac{67 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55 \sqrt{33}}-\frac{4451 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{110 \sqrt{33}} \]
Antiderivative was successfully verified.
[In] Int[(2 + 3*x)^(5/2)/((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 23.8243, size = 114, normalized size = 0.91 \[ - \frac{4451 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{3630} - \frac{67 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1925} - \frac{448 \sqrt{3 x + 2} \sqrt{5 x + 3}}{363 \sqrt{- 2 x + 1}} + \frac{7 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}}{33 \left (- 2 x + 1\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+3*x)**(5/2)/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)
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Mathematica [A] time = 0.358694, size = 120, normalized size = 0.96 \[ -\frac{1}{2} \sqrt{11-5 (1-2 x)} \sqrt{7-3 (1-2 x)} \left (\frac{1127}{726 \sqrt{1-2 x}}-\frac{49}{66 (1-2 x)^{3/2}}\right )-\frac{2240 F\left (\sin ^{-1}\left (\frac{\sqrt{11-5 (1-2 x)}}{\sqrt{11}}\right )|-\frac{33}{2}\right )-4451 E\left (\sin ^{-1}\left (\frac{\sqrt{11-5 (1-2 x)}}{\sqrt{11}}\right )|-\frac{33}{2}\right )}{1815 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + 3*x)^(5/2)/((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]),x]
[Out]
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Maple [C] time = 0.03, size = 276, normalized size = 2.2 \[{\frac{1}{3630\, \left ( -1+2\,x \right ) ^{2} \left ( 15\,{x}^{2}+19\,x+6 \right ) } \left ( 4480\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-8902\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-2240\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +4451\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +169050\,{x}^{3}+170030\,{x}^{2}+11760\,x-17640 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}\sqrt{2+3\,x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+3*x)^(5/2)/(1-2*x)^(5/2)/(3+5*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(5/2)/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (9 \, x^{2} + 12 \, x + 4\right )} \sqrt{3 \, x + 2}}{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(5/2)/(sqrt(5*x + 3)*(-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((2+3*x)**(5/2)/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (3 \, x + 2\right )}^{\frac{5}{2}}}{\sqrt{5 \, x + 3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((3*x + 2)^(5/2)/(sqrt(5*x + 3)*(-2*x + 1)^(5/2)),x, algorithm="giac")
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