Optimal. Leaf size=185 \[ \frac{715 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}-\frac{5 \sqrt{x} (429 x+361)}{3 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{295 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{715 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]
[Out]
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Rubi [A] time = 0.287779, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{715 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}-\frac{5 \sqrt{x} (429 x+361)}{3 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (45 x+38)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{295 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{715 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
[In] Int[(2 - 5*x)/(Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 32.0141, size = 168, normalized size = 0.91 \[ \frac{715 \sqrt{x} \left (6 x + 4\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} + \frac{\sqrt{x} \left (90 x + 76\right )}{3 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{\sqrt{x} \left (2145 x + 1805\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} - \frac{715 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{12 \sqrt{3 x^{2} + 5 x + 2}} + \frac{295 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2-5*x)/(3*x**2+5*x+2)**(5/2)/x**(1/2),x)
[Out]
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Mathematica [C] time = 0.403299, size = 167, normalized size = 0.9 \[ \frac{170 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+715 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+2 \left (2655 x^3+6615 x^2+5383 x+1430\right )}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 - 5*x)/(Sqrt[x]*(2 + 5*x + 3*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.033, size = 315, normalized size = 1.7 \[ -{\frac{1}{18\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}} \left ( 1125\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-2145\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+1875\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-3575\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+750\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -1430\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +38610\,{x}^{4}+96840\,{x}^{3}+79350\,{x}^{2}+21204\,x \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2-5*x)/(3*x^2+5*x+2)^(5/2)/x^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} \sqrt{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*sqrt(x)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{5 \, x - 2}{{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*sqrt(x)),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((2-5*x)/(3*x**2+5*x+2)**(5/2)/x**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} \sqrt{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*sqrt(x)),x, algorithm="giac")
[Out]