Optimal. Leaf size=225 \[ \frac{625 \sqrt{x} (3 x+2)}{2 \sqrt{3 x^2+5 x+2}}-\frac{625 \sqrt{3 x^2+5 x+2}}{2 \sqrt{x}}+\frac{795 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{625 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{265 \sqrt{3 x^2+5 x+2}}{x^{3/2}}-\frac{3 (225 x+181)}{x^{3/2} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}} \]
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Rubi [A] time = 0.397149, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{625 \sqrt{x} (3 x+2)}{2 \sqrt{3 x^2+5 x+2}}-\frac{625 \sqrt{3 x^2+5 x+2}}{2 \sqrt{x}}+\frac{795 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{625 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{265 \sqrt{3 x^2+5 x+2}}{x^{3/2}}-\frac{3 (225 x+181)}{x^{3/2} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(2 - 5*x)/(x^(5/2)*(2 + 5*x + 3*x^2)^(5/2)),x]
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Rubi in Sympy [A] time = 43.2237, size = 211, normalized size = 0.94 \[ \frac{625 \sqrt{x} \left (6 x + 4\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} - \frac{625 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{2} + 5 x + 2}} + \frac{795 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{2} + 5 x + 2}} - \frac{625 \sqrt{3 x^{2} + 5 x + 2}}{2 \sqrt{x}} + \frac{90 x + 76}{3 x^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{2025 x + 1629}{3 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}} + \frac{265 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2-5*x)/x**(5/2)/(3*x**2+5*x+2)**(5/2),x)
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Mathematica [C] time = 0.399231, size = 169, normalized size = 0.75 \[ \frac{14310 x^4+35550 x^3+28806 x^2+510 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{5/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+1875 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{5/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+7590 x-4}{6 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 - 5*x)/(x^(5/2)*(2 + 5*x + 3*x^2)^(5/2)),x]
[Out]
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Maple [A] time = 0.037, size = 327, normalized size = 1.5 \[ -{\frac{1}{ \left ( 12+12\,x \right ) \left ( 2+3\,x \right ) } \left ( 855\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}-1875\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}+1425\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-3125\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+570\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-1250\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+33750\,{x}^{5}+83880\,{x}^{4}+67650\,{x}^{3}+17388\,{x}^{2}-180\,x+8 \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2-5*x)/x^(5/2)/(3*x^2+5*x+2)^(5/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(5/2)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{5 \, x - 2}{{\left (9 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} + 20 \, x^{3} + 4 \, x^{2}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(5/2)),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((2-5*x)/x**(5/2)/(3*x**2+5*x+2)**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{5 \, x - 2}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(5*x - 2)/((3*x^2 + 5*x + 2)^(5/2)*x^(5/2)),x, algorithm="giac")
[Out]