3.1062 \(\int \frac{2-5 x}{x^{5/2} \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=175 \[ -\frac{25 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{25 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{\sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{25 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{3 x^2+5 x+2}}{3 x^{3/2}} \]

[Out]

(-25*Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (2*Sqrt[2 + 5*x + 3*x^2])/(3
*x^(3/2)) + (25*Sqrt[2 + 5*x + 3*x^2])/(3*Sqrt[x]) + (25*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) - (
Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2
+ 5*x + 3*x^2]

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Rubi [A]  time = 0.28525, antiderivative size = 175, 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 \[ -\frac{25 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{25 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{\sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{25 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{3 x^2+5 x+2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 - 5*x)/(x^(5/2)*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

(-25*Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (2*Sqrt[2 + 5*x + 3*x^2])/(3
*x^(3/2)) + (25*Sqrt[2 + 5*x + 3*x^2])/(3*Sqrt[x]) + (25*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) - (
Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/Sqrt[2
+ 5*x + 3*x^2]

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Rubi in Sympy [A]  time = 31.0074, size = 160, normalized size = 0.91 \[ - \frac{25 \sqrt{x} \left (6 x + 4\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} + \frac{25 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{12 \sqrt{3 x^{2} + 5 x + 2}} - \frac{\sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} + \frac{25 \sqrt{3 x^{2} + 5 x + 2}}{3 \sqrt{x}} - \frac{2 \sqrt{3 x^{2} + 5 x + 2}}{3 x^{\frac{3}{2}}} \]

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)**(1/2),x)

[Out]

-25*sqrt(x)*(6*x + 4)/(6*sqrt(3*x**2 + 5*x + 2)) + 25*sqrt((6*x + 4)/(x + 1))*(4
*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(12*sqrt(3*x**2 + 5*x + 2)) - sqrt((6*x
+ 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(4*sqrt(3*x**2 + 5*x + 2
)) + 25*sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x)) - 2*sqrt(3*x**2 + 5*x + 2)/(3*x**(3/2
))

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Mathematica [C]  time = 0.228059, size = 148, normalized size = 0.85 \[ \frac{22 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{5/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-25 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{5/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (3 x^2+5 x+2\right )}{3 x^{3/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - 5*x)/(x^(5/2)*Sqrt[2 + 5*x + 3*x^2]),x]

[Out]

(-2*(2 + 5*x + 3*x^2) - (25*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(5/2)*El
lipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (22*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqr
t[3 + 2/x]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(3*x^(3/2)*Sqrt
[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.027, size = 121, normalized size = 0.7 \[{\frac{1}{18} \left ( 69\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-25\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+450\,{x}^{3}+714\,{x}^{2}+240\,x-24 \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)/x^(5/2)/(3*x^2+5*x+2)^(1/2),x)

[Out]

1/18*(69*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*EllipticF(1/2*(6
*x+4)^(1/2),I*2^(1/2))*x-25*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/
2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))*x+450*x^3+714*x^2+240*x-24)/(3*x^2+5*x
+2)^(1/2)/x^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{5 \, x - 2}{\sqrt{3 \, x^{2} + 5 \, x + 2} x^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)/(sqrt(3*x^2 + 5*x + 2)*x^(5/2)),x, algorithm="maxima")

[Out]

-integrate((5*x - 2)/(sqrt(3*x^2 + 5*x + 2)*x^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{5 \, x - 2}{\sqrt{3 \, x^{2} + 5 \, x + 2} x^{\frac{5}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)/(sqrt(3*x^2 + 5*x + 2)*x^(5/2)),x, algorithm="fricas")

[Out]

integral(-(5*x - 2)/(sqrt(3*x^2 + 5*x + 2)*x^(5/2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{2}{x^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{5}{x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2-5*x)/x**(5/2)/(3*x**2+5*x+2)**(1/2),x)

[Out]

-Integral(-2/(x**(5/2)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(5/(x**(3/2)*sqrt(3
*x**2 + 5*x + 2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{5 \, x - 2}{\sqrt{3 \, x^{2} + 5 \, x + 2} x^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(5*x - 2)/(sqrt(3*x^2 + 5*x + 2)*x^(5/2)),x, algorithm="giac")

[Out]

integrate(-(5*x - 2)/(sqrt(3*x^2 + 5*x + 2)*x^(5/2)), x)