Optimal. Leaf size=79 \[ -\frac{370 \sqrt{1-2 x}}{847 \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{18 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
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Rubi [A] time = 0.165133, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ -\frac{370 \sqrt{1-2 x}}{847 \sqrt{5 x+3}}+\frac{4}{77 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{18 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 15.1956, size = 73, normalized size = 0.92 \[ - \frac{370 \sqrt{- 2 x + 1}}{847 \sqrt{5 x + 3}} + \frac{18 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{49} + \frac{4}{77 \sqrt{- 2 x + 1} \sqrt{5 x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)/(3+5*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.113767, size = 65, normalized size = 0.82 \[ \frac{2 (370 x-163)}{847 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{9 \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{7 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2)),x]
[Out]
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Maple [B] time = 0.023, size = 154, normalized size = 2. \[ -{\frac{1}{-5929+11858\,x}\sqrt{1-2\,x} \left ( 10890\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+1089\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-3267\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +5180\,x\sqrt{-10\,{x}^{2}-x+3}-2282\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(3/2)/(2+3*x)/(3+5*x)^(3/2),x)
[Out]
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Maxima [A] time = 1.49764, size = 78, normalized size = 0.99 \[ -\frac{9}{49} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{740 \, x}{847 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{326}{847 \, \sqrt{-10 \, x^{2} - x + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)*(-2*x + 1)^(3/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228698, size = 101, normalized size = 1.28 \[ -\frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (370 \, x - 163\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1089 \,{\left (10 \, x^{2} + x - 3\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{5929 \,{\left (10 \, x^{2} + x - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)*(-2*x + 1)^(3/2)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- 2 x + 1\right )^{\frac{3}{2}} \left (3 x + 2\right ) \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-2*x)**(3/2)/(2+3*x)/(3+5*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.250956, size = 215, normalized size = 2.72 \[ -\frac{9}{490} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{5}{242} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{8 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{4235 \,{\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((5*x + 3)^(3/2)*(3*x + 2)*(-2*x + 1)^(3/2)),x, algorithm="giac")
[Out]