3.1038 \(\int \frac{(2-5 x) \sqrt{2+5 x+3 x^2}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=159 \[ \frac{2}{9} \sqrt{x} \sqrt{3 x^2+5 x+2} (1-9 x)+\frac{88 \sqrt{x} (3 x+2)}{27 \sqrt{3 x^2+5 x+2}}+\frac{34 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{3 x^2+5 x+2}}-\frac{88 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}} \]

[Out]

(88*Sqrt[x]*(2 + 3*x))/(27*Sqrt[2 + 5*x + 3*x^2]) + (2*(1 - 9*x)*Sqrt[x]*Sqrt[2
+ 5*x + 3*x^2])/9 - (88*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan
[Sqrt[x]], -1/2])/(27*Sqrt[2 + 5*x + 3*x^2]) + (34*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x
)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(9*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.239249, 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 \[ \frac{2}{9} \sqrt{x} \sqrt{3 x^2+5 x+2} (1-9 x)+\frac{88 \sqrt{x} (3 x+2)}{27 \sqrt{3 x^2+5 x+2}}+\frac{34 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{3 x^2+5 x+2}}-\frac{88 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(88*Sqrt[x]*(2 + 3*x))/(27*Sqrt[2 + 5*x + 3*x^2]) + (2*(1 - 9*x)*Sqrt[x]*Sqrt[2
+ 5*x + 3*x^2])/9 - (88*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan
[Sqrt[x]], -1/2])/(27*Sqrt[2 + 5*x + 3*x^2]) + (34*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x
)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(9*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 26.4099, size = 148, normalized size = 0.93 \[ \frac{4 \sqrt{x} \left (- \frac{45 x}{2} + \frac{5}{2}\right ) \sqrt{3 x^{2} + 5 x + 2}}{45} + \frac{44 \sqrt{x} \left (6 x + 4\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} - \frac{22 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} + \frac{17 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{18 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(x)*(-45*x/2 + 5/2)*sqrt(3*x**2 + 5*x + 2)/45 + 44*sqrt(x)*(6*x + 4)/(27*s
qrt(3*x**2 + 5*x + 2)) - 22*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sq
rt(x)), -1/2)/(27*sqrt(3*x**2 + 5*x + 2)) + 17*sqrt((6*x + 4)/(x + 1))*(4*x + 4)
*elliptic_f(atan(sqrt(x)), -1/2)/(18*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.232928, size = 158, normalized size = 0.99 \[ \frac{14 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+88 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+2 \left (-81 x^4-126 x^3+93 x^2+226 x+88\right )}{27 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(88 + 226*x + 93*x^2 - 126*x^3 - 81*x^4) + (88*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sq
rt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (14*I)*Sqrt[2
]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]],
3/2])/(27*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.027, size = 123, normalized size = 0.8 \[ -{\frac{2}{81} \left ( 15\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -22\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +243\,{x}^{4}+378\,{x}^{3}+117\,{x}^{2}-18\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/81/(3*x^2+5*x+2)^(1/2)/x^(1/2)*(15*(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))-22*(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))+243*x^4+378
*x^3+117*x^2-18*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-2*sqrt(3*x**2 + 5*x + 2)/sqrt(x), 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. \[ \int -\frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{\sqrt{x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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