3.1080 \(\int \frac{2-5 x}{x^{3/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=208 \[ -\frac{838 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{838 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{2085 x+1717}{3 \sqrt{x} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}}-\frac{695 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{838 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]

[Out]

(2*(38 + 45*x))/(3*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2)) - (838*Sqrt[x]*(2 + 3*x))/(3
*Sqrt[2 + 5*x + 3*x^2]) - (1717 + 2085*x)/(3*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2]) + (8
38*Sqrt[2 + 5*x + 3*x^2])/(3*Sqrt[x]) + (838*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 +
x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) - (695*(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 = 0.338077, antiderivative size = 208, 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 \[ -\frac{838 \sqrt{x} (3 x+2)}{3 \sqrt{3 x^2+5 x+2}}+\frac{838 \sqrt{3 x^2+5 x+2}}{3 \sqrt{x}}-\frac{2085 x+1717}{3 \sqrt{x} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}}-\frac{695 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{838 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{3 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(38 + 45*x))/(3*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2)) - (838*Sqrt[x]*(2 + 3*x))/(3
*Sqrt[2 + 5*x + 3*x^2]) - (1717 + 2085*x)/(3*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2]) + (8
38*Sqrt[2 + 5*x + 3*x^2])/(3*Sqrt[x]) + (838*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 +
x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2]) - (695*(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 = 37.0602, size = 190, normalized size = 0.91 \[ - \frac{419 \sqrt{x} \left (6 x + 4\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{419 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} - \frac{695 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{2} + 5 x + 2}} + \frac{90 x + 76}{3 \sqrt{x} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{2085 x + 1717}{3 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}} + \frac{838 \sqrt{3 x^{2} + 5 x + 2}}{3 \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-419*sqrt(x)*(6*x + 4)/(3*sqrt(3*x**2 + 5*x + 2)) + 419*sqrt((6*x + 4)/(x + 1))*
(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(6*sqrt(3*x**2 + 5*x + 2)) - 695*sqrt(
(6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(8*sqrt(3*x**2 + 5*
x + 2)) + (90*x + 76)/(3*sqrt(x)*(3*x**2 + 5*x + 2)**(3/2)) - (2085*x + 1717)/(3
*sqrt(x)*sqrt(3*x**2 + 5*x + 2)) + 838*sqrt(3*x**2 + 5*x + 2)/(3*sqrt(x))

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Mathematica [C]  time = 0.393777, size = 167, normalized size = 0.8 \[ \frac{-409 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-1676 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (6255 x^3+15576 x^2+12665 x+3358\right )}{6 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3358 + 12665*x + 15576*x^2 + 6255*x^3) - (1676*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/
x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (409
*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticF[I*ArcSinh[Sq
rt[2/3]/Sqrt[x]], 3/2])/(6*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.036, size = 316, normalized size = 1.5 \[{\frac{1}{ \left ( 18+18\,x \right ) \left ( 2+3\,x \right ) } \left ( 1287\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-2514\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+2145\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-4190\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+858\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -1676\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +45252\,{x}^{4}+113310\,{x}^{3}+92580\,{x}^{2}+24570\,x-36 \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/18*(1287*(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^2-2514*(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^2+2145*(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-4190*(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+858*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Ellipti
cF(1/2*(6*x+4)^(1/2),I*2^(1/2))-1676*(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))+45252*x^4+113310*x^3+92580*x^
2+24570*x-36)/(1+x)/(2+3*x)/x^(1/2)/(3*x^2+5*x+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{5 \, x - 2}{{\left (9 \, x^{5} + 30 \, x^{4} + 37 \, x^{3} + 20 \, x^{2} + 4 \, x\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x - 2)/((9*x^5 + 30*x^4 + 37*x^3 + 20*x^2 + 4*x)*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**(3/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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