3.1068 \(\int \frac{2-5 x}{\sqrt{x} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=151 \[ -\frac{30 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (45 x+38)}{\sqrt{3 x^2+5 x+2}}-\frac{37 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{30 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]

[Out]

(-30*Sqrt[x]*(2 + 3*x))/Sqrt[2 + 5*x + 3*x^2] + (2*Sqrt[x]*(38 + 45*x))/Sqrt[2 +
5*x + 3*x^2] + (30*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqr
t[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2] - (37*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 = 0.228852, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{30 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}+\frac{2 \sqrt{x} (45 x+38)}{\sqrt{3 x^2+5 x+2}}-\frac{37 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}+\frac{30 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-30*Sqrt[x]*(2 + 3*x))/Sqrt[2 + 5*x + 3*x^2] + (2*Sqrt[x]*(38 + 45*x))/Sqrt[2 +
5*x + 3*x^2] + (30*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqr
t[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2] - (37*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 = 25.6955, size = 139, normalized size = 0.92 \[ - \frac{15 \sqrt{x} \left (6 x + 4\right )}{\sqrt{3 x^{2} + 5 x + 2}} + \frac{\sqrt{x} \left (90 x + 76\right )}{\sqrt{3 x^{2} + 5 x + 2}} + \frac{15 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{2 \sqrt{3 x^{2} + 5 x + 2}} - \frac{37 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-15*sqrt(x)*(6*x + 4)/sqrt(3*x**2 + 5*x + 2) + sqrt(x)*(90*x + 76)/sqrt(3*x**2 +
5*x + 2) + 15*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)
/(2*sqrt(3*x**2 + 5*x + 2)) - 37*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(at
an(sqrt(x)), -1/2)/(4*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.227815, size = 137, normalized size = 0.91 \[ \frac{-7 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-30 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-74 x-60}{\sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-60 - 74*x - (30*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*
ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (7*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*
x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(Sqrt[x]*Sqrt[2 + 5*x + 3*
x^2])

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Maple [A]  time = 0.028, size = 113, normalized size = 0.8 \[ -{\frac{1}{3} \left ( 15\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -8\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -270\,{x}^{2}-228\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(15*(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))-8*(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))-270*x^2-228*x)/(3*x^2+5*x+2)^(1/2)/x^(1/2
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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