Optimal. Leaf size=160 \[ -\frac{31940 \sqrt{1-2 x} \sqrt{3 x+2}}{539 \sqrt{5 x+3}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{192}{49} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{6388}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
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Rubi [A] time = 0.35179, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{31940 \sqrt{1-2 x} \sqrt{3 x+2}}{539 \sqrt{5 x+3}}+\frac{288 \sqrt{1-2 x}}{49 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2 \sqrt{1-2 x}}{7 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{192}{49} \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{6388}{49} \sqrt{\frac{3}{11}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 32.7397, size = 143, normalized size = 0.89 \[ - \frac{31940 \sqrt{- 2 x + 1} \sqrt{3 x + 2}}{539 \sqrt{5 x + 3}} + \frac{288 \sqrt{- 2 x + 1}}{49 \sqrt{3 x + 2} \sqrt{5 x + 3}} + \frac{2 \sqrt{- 2 x + 1}}{7 \left (3 x + 2\right )^{\frac{3}{2}} \sqrt{5 x + 3}} + \frac{6388 \sqrt{33} E\left (\operatorname{asin}{\left (\frac{\sqrt{21} \sqrt{- 2 x + 1}}{7} \right )}\middle | \frac{35}{33}\right )}{539} + \frac{576 \sqrt{35} F\left (\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{- 2 x + 1}}{11} \right )}\middle | \frac{33}{35}\right )}{1715} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2+3*x)**(5/2)/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.196633, size = 100, normalized size = 0.62 \[ \frac{2}{539} \left (-\frac{\sqrt{1-2 x} \left (143730 x^2+186888 x+60635\right )}{(3 x+2)^{3/2} \sqrt{5 x+3}}-2 \sqrt{2} \left (1597 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-805 F\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)),x]
[Out]
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Maple [C] time = 0.034, size = 267, normalized size = 1.7 \[ -{\frac{2}{5390\,{x}^{2}+539\,x-1617}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 4830\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-9582\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+3220\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) -6388\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{11}\sqrt{2}\sqrt{3+5\,x},i/2\sqrt{11}\sqrt{3}\sqrt{2} \right ) +287460\,{x}^{3}+230046\,{x}^{2}-65618\,x-60635 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)*sqrt(-2*x + 1)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)*sqrt(-2*x + 1)),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/(2+3*x)**(5/2)/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(5/2)*sqrt(-2*x + 1)),x, algorithm="giac")
[Out]