3.1081 \(\int \frac{2-5 x}{x^{5/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=225 \[ \frac{625 \sqrt{x} (3 x+2)}{2 \sqrt{3 x^2+5 x+2}}-\frac{625 \sqrt{3 x^2+5 x+2}}{2 \sqrt{x}}+\frac{795 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{625 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{265 \sqrt{3 x^2+5 x+2}}{x^{3/2}}-\frac{3 (225 x+181)}{x^{3/2} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}} \]

[Out]

(2*(38 + 45*x))/(3*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2)) + (625*Sqrt[x]*(2 + 3*x))/(2
*Sqrt[2 + 5*x + 3*x^2]) - (3*(181 + 225*x))/(x^(3/2)*Sqrt[2 + 5*x + 3*x^2]) + (2
65*Sqrt[2 + 5*x + 3*x^2])/x^(3/2) - (625*Sqrt[2 + 5*x + 3*x^2])/(2*Sqrt[x]) - (6
25*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sq
rt[2 + 5*x + 3*x^2]) + (795*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqr
t[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.397149, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{625 \sqrt{x} (3 x+2)}{2 \sqrt{3 x^2+5 x+2}}-\frac{625 \sqrt{3 x^2+5 x+2}}{2 \sqrt{x}}+\frac{795 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{625 (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{\sqrt{2} \sqrt{3 x^2+5 x+2}}+\frac{265 \sqrt{3 x^2+5 x+2}}{x^{3/2}}-\frac{3 (225 x+181)}{x^{3/2} \sqrt{3 x^2+5 x+2}}+\frac{2 (45 x+38)}{3 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(38 + 45*x))/(3*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2)) + (625*Sqrt[x]*(2 + 3*x))/(2
*Sqrt[2 + 5*x + 3*x^2]) - (3*(181 + 225*x))/(x^(3/2)*Sqrt[2 + 5*x + 3*x^2]) + (2
65*Sqrt[2 + 5*x + 3*x^2])/x^(3/2) - (625*Sqrt[2 + 5*x + 3*x^2])/(2*Sqrt[x]) - (6
25*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sq
rt[2 + 5*x + 3*x^2]) + (795*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqr
t[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 43.2237, size = 211, normalized size = 0.94 \[ \frac{625 \sqrt{x} \left (6 x + 4\right )}{4 \sqrt{3 x^{2} + 5 x + 2}} - \frac{625 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{2} + 5 x + 2}} + \frac{795 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{8 \sqrt{3 x^{2} + 5 x + 2}} - \frac{625 \sqrt{3 x^{2} + 5 x + 2}}{2 \sqrt{x}} + \frac{90 x + 76}{3 x^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{2025 x + 1629}{3 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}} + \frac{265 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

625*sqrt(x)*(6*x + 4)/(4*sqrt(3*x**2 + 5*x + 2)) - 625*sqrt((6*x + 4)/(x + 1))*(
4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(8*sqrt(3*x**2 + 5*x + 2)) + 795*sqrt((
6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(8*sqrt(3*x**2 + 5*x
+ 2)) - 625*sqrt(3*x**2 + 5*x + 2)/(2*sqrt(x)) + (90*x + 76)/(3*x**(3/2)*(3*x**
2 + 5*x + 2)**(3/2)) - (2025*x + 1629)/(3*x**(3/2)*sqrt(3*x**2 + 5*x + 2)) + 265
*sqrt(3*x**2 + 5*x + 2)/x**(3/2)

_______________________________________________________________________________________

Mathematica [C]  time = 0.399231, size = 169, normalized size = 0.75 \[ \frac{14310 x^4+35550 x^3+28806 x^2+510 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{5/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+1875 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{5/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+7590 x-4}{6 x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-4 + 7590*x + 28806*x^2 + 35550*x^3 + 14310*x^4 + (1875*I)*Sqrt[2 + 2/x]*Sqrt[3
+ 2/x]*x^(5/2)*(2 + 5*x + 3*x^2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] +
(510*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(5/2)*(2 + 5*x + 3*x^2)*EllipticF[I*ArcSi
nh[Sqrt[2/3]/Sqrt[x]], 3/2])/(6*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.037, size = 327, normalized size = 1.5 \[ -{\frac{1}{ \left ( 12+12\,x \right ) \left ( 2+3\,x \right ) } \left ( 855\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}-1875\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}+1425\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-3125\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+570\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-1250\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+33750\,{x}^{5}+83880\,{x}^{4}+67650\,{x}^{3}+17388\,{x}^{2}-180\,x+8 \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12/x^(3/2)*(855*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Ellipt
icF(1/2*(6*x+4)^(1/2),I*2^(1/2))*x^3-1875*(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^3+1425*(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
^2-3125*(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+570*(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-1250*(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+33750*x^5+838
80*x^4+67650*x^3+17388*x^2-180*x+8)/(1+x)/(2+3*x)/(3*x^2+5*x+2)^(1/2)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{5 \, x - 2}{{\left (9 \, x^{6} + 30 \, x^{5} + 37 \, x^{4} + 20 \, x^{3} + 4 \, x^{2}\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x - 2)/((9*x^6 + 30*x^5 + 37*x^4 + 20*x^3 + 4*x^2)*sqrt(3*x^2 + 5*x
+ 2)*sqrt(x)), x)

_______________________________________________________________________________________

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)**(5/2),x)

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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