3.1039 \(\int \frac{(2-5 x) \sqrt{2+5 x+3 x^2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=159 \[ \frac{22 \sqrt{x} (3 x+2)}{9 \sqrt{3 x^2+5 x+2}}-\frac{2 (5 x+6) \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}+\frac{10 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}}-\frac{22 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{3 x^2+5 x+2}} \]

[Out]

(22*Sqrt[x]*(2 + 3*x))/(9*Sqrt[2 + 5*x + 3*x^2]) - (2*(6 + 5*x)*Sqrt[2 + 5*x + 3
*x^2])/(3*Sqrt[x]) - (22*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTa
n[Sqrt[x]], -1/2])/(9*Sqrt[2 + 5*x + 3*x^2]) + (10*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x
)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.238947, antiderivative size = 159, 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 \[ \frac{22 \sqrt{x} (3 x+2)}{9 \sqrt{3 x^2+5 x+2}}-\frac{2 (5 x+6) \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}+\frac{10 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}}-\frac{22 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(22*Sqrt[x]*(2 + 3*x))/(9*Sqrt[2 + 5*x + 3*x^2]) - (2*(6 + 5*x)*Sqrt[2 + 5*x + 3
*x^2])/(3*Sqrt[x]) - (22*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTa
n[Sqrt[x]], -1/2])/(9*Sqrt[2 + 5*x + 3*x^2]) + (10*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x
)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 26.4099, size = 146, normalized size = 0.92 \[ \frac{11 \sqrt{x} \left (6 x + 4\right )}{9 \sqrt{3 x^{2} + 5 x + 2}} - \frac{11 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{18 \sqrt{3 x^{2} + 5 x + 2}} + \frac{5 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} - \frac{4 \left (\frac{5 x}{2} + 3\right ) \sqrt{3 x^{2} + 5 x + 2}}{3 \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11*sqrt(x)*(6*x + 4)/(9*sqrt(3*x**2 + 5*x + 2)) - 11*sqrt((6*x + 4)/(x + 1))*(4*
x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(18*sqrt(3*x**2 + 5*x + 2)) + 5*sqrt((6*x
+ 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(6*sqrt(3*x**2 + 5*x +
2)) - 4*(5*x/2 + 3)*sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x))

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Mathematica [C]  time = 0.256754, size = 153, normalized size = 0.96 \[ \frac{8 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+22 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (45 x^3+96 x^2+65 x+14\right )}{9 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(14 + 65*x + 96*x^2 + 45*x^3) + (22*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x
]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (8*I)*Sqrt[2]*Sqrt[1 +
x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(9*S
qrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.038, size = 119, normalized size = 0.8 \[ -{\frac{1}{27} \left ( 3\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -11\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +270\,{x}^{3}+774\,{x}^{2}+720\,x+216 \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/27*(3*(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))-11*(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))+270*x^3+774*x^2+720*x+216)/(3*x^2+5*x+2)
^(1/2)/x^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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