3.1070 \(\int \frac{2-5 x}{x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=201 \[ -\frac{170 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{170 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{115 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{170 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}}-\frac{115 \sqrt{3 x^2+5 x+2}}{3 x^{3/2}}+\frac{2 (45 x+38)}{x^{3/2} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-170*Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (2*(38 + 45*x))/(x^(3/2)*Sq
rt[2 + 5*x + 3*x^2]) - (115*Sqrt[2 + 5*x + 3*x^2])/(3*x^(3/2)) + (170*Sqrt[2 + 5
*x + 3*x^2])/(3*Sqrt[x]) + (170*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Elliptic
E[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) - (115*(1 + x)*Sqrt[(2 + 3*x
)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.331722, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{170 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{170 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{115 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{170 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}}-\frac{115 \sqrt{3 x^2+5 x+2}}{3 x^{3/2}}+\frac{2 (45 x+38)}{x^{3/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-170*Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (2*(38 + 45*x))/(x^(3/2)*Sq
rt[2 + 5*x + 3*x^2]) - (115*Sqrt[2 + 5*x + 3*x^2])/(3*x^(3/2)) + (170*Sqrt[2 + 5
*x + 3*x^2])/(3*Sqrt[x]) + (170*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Elliptic
E[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) - (115*(1 + x)*Sqrt[(2 + 3*x
)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 36.4581, size = 185, normalized size = 0.92 \[ - \frac{85 \sqrt{x} \left (6 x + 4\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{85 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} - \frac{115 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{2} + 5 x + 2}} + \frac{170 \sqrt{3 x^{2} + 5 x + 2}}{3 \sqrt{x}} + \frac{90 x + 76}{x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}} - \frac{115 \sqrt{3 x^{2} + 5 x + 2}}{3 x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-85*sqrt(x)*(6*x + 4)/(3*sqrt(3*x**2 + 5*x + 2)) + 85*sqrt((6*x + 4)/(x + 1))*(4
*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(6*sqrt(3*x**2 + 5*x + 2)) - 115*sqrt((6
*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(8*sqrt(3*x**2 + 5*x
+ 2)) + 170*sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x)) + (90*x + 76)/(x**(3/2)*sqrt(3*x*
*2 + 5*x + 2)) - 115*sqrt(3*x**2 + 5*x + 2)/(3*x**(3/2))

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Mathematica [C]  time = 0.246343, size = 145, normalized size = 0.72 \[ \frac{-5 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{5/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-340 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{5/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-690 x^2-610 x-4}{6 x^{3/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4 - 610*x - 690*x^2 - (340*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(5/2)*E
llipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (5*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqr
t[3 + 2/x]*x^(5/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(6*x^(3/2)*Sqrt
[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.033, size = 121, normalized size = 0.6 \[{\frac{1}{18} \left ( 165\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-170\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+3060\,{x}^{3}+3030\,{x}^{2}+210\,x-12 \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(165*(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))*x-170*(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))*x+3060*x^3+3030*x^2+210*x-12)/x^(3/2
)/(3*x^2+5*x+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{5 \, x - 2}{{\left (3 \, x^{4} + 5 \, x^{3} + 2 \, x^{2}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x - 2)/((3*x^4 + 5*x^3 + 2*x^2)*sqrt(3*x^2 + 5*x + 2)*sqrt(x)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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