3.1061 \(\int \frac{2-5 x}{x^{3/2} \sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=146 \[ \frac{2 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{3 x^2+5 x+2}}{\sqrt{x}}-\frac{5 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]

[Out]

(2*Sqrt[x]*(2 + 3*x))/Sqrt[2 + 5*x + 3*x^2] - (2*Sqrt[2 + 5*x + 3*x^2])/Sqrt[x]
- (2*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/S
qrt[2 + 5*x + 3*x^2] - (5*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcT
an[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi [A]  time = 0.224909, antiderivative size = 146, 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 \[ \frac{2 \sqrt{x} (3 x+2)}{\sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{3 x^2+5 x+2}}{\sqrt{x}}-\frac{5 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}}-\frac{2 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[x]*(2 + 3*x))/Sqrt[2 + 5*x + 3*x^2] - (2*Sqrt[2 + 5*x + 3*x^2])/Sqrt[x]
- (2*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/S
qrt[2 + 5*x + 3*x^2] - (5*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcT
an[Sqrt[x]], -1/2])/Sqrt[2 + 5*x + 3*x^2]

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Rubi in Sympy [A]  time = 25.6281, size = 133, normalized size = 0.91 \[ \frac{\sqrt{x} \left (6 x + 4\right )}{\sqrt{3 x^{2} + 5 x + 2}} - \frac{\sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{2 \sqrt{3 x^{2} + 5 x + 2}} - \frac{5 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2 \sqrt{3 x^{2} + 5 x + 2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)*(6*x + 4)/sqrt(3*x**2 + 5*x + 2) - sqrt((6*x + 4)/(x + 1))*(4*x + 4)*ell
iptic_e(atan(sqrt(x)), -1/2)/(2*sqrt(3*x**2 + 5*x + 2)) - 5*sqrt((6*x + 4)/(x +
1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(4*sqrt(3*x**2 + 5*x + 2)) - 2*sqr
t(3*x**2 + 5*x + 2)/sqrt(x)

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Mathematica [C]  time = 0.179691, size = 90, normalized size = 0.62 \[ \frac{i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} x \left (2 E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-7 F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )\right )}{\sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.026, size = 114, normalized size = 0.8 \[ -{\frac{1}{3} \left ( 8\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ({\frac{1}{2}\sqrt{6\,x+4}},i\sqrt{2} \right ) +18\,{x}^{2}+30\,x+12 \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(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))-(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Ell
ipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))+18*x^2+30*x+12)/x^(1/2)/(3*x^2+5*x+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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