3.1060 \(\int \frac{2-5 x}{\sqrt{x} \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{10 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{10 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-10*Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (10*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) + (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 = 0.185892, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ -\frac{10 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{10 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-10*Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (10*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) + (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 = 20.835, size = 116, normalized size = 0.9 \[ - \frac{5 \sqrt{x} \left (6 x + 4\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{5 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} + \frac{\sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{2 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.228005, size = 150, normalized size = 1.16 \[ -\frac{2 x^{3/2} \left (5 \left (\frac{2}{x^2}+\frac{5}{x}+3\right )-\frac{8 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{\sqrt{x}}+\frac{5 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )}{\sqrt{x}}\right )}{3 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*x^(3/2)*(5*(3 + 2/x^2 + 5/x) + ((5*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]
*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/Sqrt[x] - ((8*I)*Sqrt[2]*Sqrt[1 +
x^(-1)]*Sqrt[3 + 2/x]*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/Sqrt[x]))/(
3*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.023, size = 80, normalized size = 0.6 \[{\frac{\sqrt{3}\sqrt{2}}{9}\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{-x} \left ( 21\,{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -5\,{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)
^(1/2)*(21*EllipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))-5*EllipticE(1/2*(6*x+4)^(1/2),
I*2^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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