Optimal. Leaf size=129 \[ -\frac{370 \sqrt{1-2 x} \sqrt{3 x+2}}{847 \sqrt{5 x+3}}+\frac{4 \sqrt{3 x+2}}{77 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{4}{77} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{74}{77} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.260963, antiderivative size = 129, 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 \[ -\frac{370 \sqrt{1-2 x} \sqrt{3 x+2}}{847 \sqrt{5 x+3}}+\frac{4 \sqrt{3 x+2}}{77 \sqrt{1-2 x} \sqrt{5 x+3}}-\frac{4}{77} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{74}{77} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)),x]
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Rubi in Sympy [A] time = 24.7778, size = 114, normalized size = 0.88 \[ - \frac{370 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{847 \sqrt{5 x + 3}} + \frac{74 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{847} - \frac{12 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{2695} + \frac{4 \sqrt{3 x + 2}}{77 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(3/2)/(3+5*x)**(3/2)/(2+3*x)**(1/2),x)
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Mathematica [A] time = 0.201761, size = 122, normalized size = 0.95 \[ \frac{2 \sqrt{3 x+2} \sqrt{5 x+3} (370 x-163)+140 \sqrt{2-4 x} (5 x+3) F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-74 \sqrt{2-4 x} (5 x+3) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{847 \sqrt{1-2 x} (5 x+3)} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)),x]
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Maple [C] time = 0.032, size = 159, normalized size = 1.2 \[ -{\frac{2}{25410\,{x}^{3}+19481\,{x}^{2}-5929\,x-5082}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 70\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -37\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +1110\,{x}^{2}+251\,x-326 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(3/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*sqrt(3*x + 2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{1}{{\left (10 \, x^{2} + x - 3\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*sqrt(3*x + 2)*(-2*x + 1)^(3/2)),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)**(3/2)/(3+5*x)**(3/2)/(2+3*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}} \sqrt{3 \, x + 2}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*sqrt(3*x + 2)*(-2*x + 1)^(3/2)),x, algorithm="giac")
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