Optimal. Leaf size=144 \[ -\frac{32735 \sqrt{5 x+3}}{15092 \sqrt{1-2 x}}+\frac{2865 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)}+\frac{27 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{102345 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
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Rubi [A] time = 0.326419, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{32735 \sqrt{5 x+3}}{15092 \sqrt{1-2 x}}+\frac{2865 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)}+\frac{27 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^2}+\frac{\sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^3}-\frac{102345 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]
Antiderivative was successfully verified.
[In] Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^4*Sqrt[3 + 5*x]),x]
[Out]
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Rubi in Sympy [A] time = 29.7378, size = 131, normalized size = 0.91 \[ - \frac{102345 \sqrt{7} \operatorname{atan}{\left (\frac{\sqrt{7} \sqrt{- 2 x + 1}}{7 \sqrt{5 x + 3}} \right )}}{19208} - \frac{32735 \sqrt{5 x + 3}}{15092 \sqrt{- 2 x + 1}} + \frac{2865 \sqrt{5 x + 3}}{392 \sqrt{- 2 x + 1} \left (3 x + 2\right )} + \frac{27 \sqrt{5 x + 3}}{28 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{2}} + \frac{\sqrt{5 x + 3}}{7 \sqrt{- 2 x + 1} \left (3 x + 2\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**(1/2),x)
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Mathematica [A] time = 0.118313, size = 82, normalized size = 0.57 \[ \frac{\frac{14 \sqrt{5 x+3} \left (-1767690 x^3-1549935 x^2+377658 x+421184\right )}{\sqrt{1-2 x} (3 x+2)^3}-1125795 \sqrt{7} \tan ^{-1}\left (\frac{-37 x-20}{2 \sqrt{7-14 x} \sqrt{5 x+3}}\right )}{422576} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^4*Sqrt[3 + 5*x]),x]
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Maple [B] time = 0.024, size = 257, normalized size = 1.8 \[{\frac{1}{422576\, \left ( 2+3\,x \right ) ^{3} \left ( -1+2\,x \right ) } \left ( 60792930\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}+91189395\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+20264310\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+24747660\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-22515900\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+21699090\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-9006360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -5287212\,x\sqrt{-10\,{x}^{2}-x+3}-5896576\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{4}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="maxima")
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Fricas [A] time = 0.229879, size = 147, normalized size = 1.02 \[ \frac{\sqrt{7}{\left (2 \, \sqrt{7}{\left (1767690 \, x^{3} + 1549935 \, x^{2} - 377658 \, x - 421184\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} + 1125795 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )}}{14 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}\right )\right )}}{422576 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="fricas")
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-2*x)**(3/2)/(2+3*x)**4/(3+5*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.430915, size = 464, normalized size = 3.22 \[ \frac{20469}{76832} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{32 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{132055 \,{\left (2 \, x - 1\right )}} + \frac{297 \,{\left (4937 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 1785280 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 188708800 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{9604 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(5*x + 3)*(3*x + 2)^4*(-2*x + 1)^(3/2)),x, algorithm="giac")
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