3.1042 \(\int \frac{(2-5 x) \sqrt{2+5 x+3 x^2}}{x^{9/2}} \, dx\)

Optimal. Leaf size=205 \[ -\frac{62 \sqrt{3 x^2+5 x+2}}{21 \sqrt{x}}+\frac{62 \sqrt{x} (3 x+2)}{21 \sqrt{3 x^2+5 x+2}}+\frac{43 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{62 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{3 x^2+5 x+2}}-\frac{4 \sqrt{3 x^2+5 x+2} (1-3 x)}{7 x^{7/2}}+\frac{43 \sqrt{3 x^2+5 x+2}}{21 x^{3/2}} \]

[Out]

(62*Sqrt[x]*(2 + 3*x))/(21*Sqrt[2 + 5*x + 3*x^2]) - (4*(1 - 3*x)*Sqrt[2 + 5*x +
3*x^2])/(7*x^(7/2)) + (43*Sqrt[2 + 5*x + 3*x^2])/(21*x^(3/2)) - (62*Sqrt[2 + 5*x
+ 3*x^2])/(21*Sqrt[x]) - (62*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[
ArcTan[Sqrt[x]], -1/2])/(21*Sqrt[2 + 5*x + 3*x^2]) + (43*(1 + x)*Sqrt[(2 + 3*x)/
(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(7*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.341394, antiderivative size = 205, 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 \[ -\frac{62 \sqrt{3 x^2+5 x+2}}{21 \sqrt{x}}+\frac{62 \sqrt{x} (3 x+2)}{21 \sqrt{3 x^2+5 x+2}}+\frac{43 (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{7 \sqrt{2} \sqrt{3 x^2+5 x+2}}-\frac{62 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{3 x^2+5 x+2}}-\frac{4 \sqrt{3 x^2+5 x+2} (1-3 x)}{7 x^{7/2}}+\frac{43 \sqrt{3 x^2+5 x+2}}{21 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((2 - 5*x)*Sqrt[2 + 5*x + 3*x^2])/x^(9/2),x]

[Out]

(62*Sqrt[x]*(2 + 3*x))/(21*Sqrt[2 + 5*x + 3*x^2]) - (4*(1 - 3*x)*Sqrt[2 + 5*x +
3*x^2])/(7*x^(7/2)) + (43*Sqrt[2 + 5*x + 3*x^2])/(21*x^(3/2)) - (62*Sqrt[2 + 5*x
+ 3*x^2])/(21*Sqrt[x]) - (62*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[
ArcTan[Sqrt[x]], -1/2])/(21*Sqrt[2 + 5*x + 3*x^2]) + (43*(1 + x)*Sqrt[(2 + 3*x)/
(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(7*Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 36.8083, size = 189, normalized size = 0.92 \[ \frac{31 \sqrt{x} \left (6 x + 4\right )}{21 \sqrt{3 x^{2} + 5 x + 2}} - \frac{31 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{42 \sqrt{3 x^{2} + 5 x + 2}} + \frac{43 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{56 \sqrt{3 x^{2} + 5 x + 2}} - \frac{62 \sqrt{3 x^{2} + 5 x + 2}}{21 \sqrt{x}} + \frac{43 \sqrt{3 x^{2} + 5 x + 2}}{21 x^{\frac{3}{2}}} - \frac{2 \left (- 30 x + 10\right ) \sqrt{3 x^{2} + 5 x + 2}}{35 x^{\frac{7}{2}}} \]

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**(9/2),x)

[Out]

31*sqrt(x)*(6*x + 4)/(21*sqrt(3*x**2 + 5*x + 2)) - 31*sqrt((6*x + 4)/(x + 1))*(4
*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(42*sqrt(3*x**2 + 5*x + 2)) + 43*sqrt((6
*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(56*sqrt(3*x**2 + 5*x
+ 2)) - 62*sqrt(3*x**2 + 5*x + 2)/(21*sqrt(x)) + 43*sqrt(3*x**2 + 5*x + 2)/(21*
x**(3/2)) - 2*(-30*x + 10)*sqrt(3*x**2 + 5*x + 2)/(35*x**(7/2))

_______________________________________________________________________________________

Mathematica [C]  time = 0.237872, size = 155, normalized size = 0.76 \[ \frac{5 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{9/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+124 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{9/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+258 x^4+646 x^3+460 x^2+24 x-48}{42 x^{7/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((2 - 5*x)*Sqrt[2 + 5*x + 3*x^2])/x^(9/2),x]

[Out]

(-48 + 24*x + 460*x^2 + 646*x^3 + 258*x^4 + (124*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqr
t[3 + 2/x]*x^(9/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (5*I)*Sqrt[2]*
Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(9/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3
/2])/(42*x^(7/2)*Sqrt[2 + 5*x + 3*x^2])

_______________________________________________________________________________________

Maple [A]  time = 0.043, size = 135, normalized size = 0.7 \[ -{\frac{1}{126} \left ( 57\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}-62\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{3}+1116\,{x}^{5}+1086\,{x}^{4}-1194\,{x}^{3}-1380\,{x}^{2}-72\,x+144 \right ){x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2-5*x)*(3*x^2+5*x+2)^(1/2)/x^(9/2),x)

[Out]

-1/126*(57*(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^3-62*(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+1116*x^5+1086*x^4-1194*x^3-138
0*x^2-72*x+144)/(3*x^2+5*x+2)^(1/2)/x^(7/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(9/2),x, algorithm="maxima")

[Out]

-integrate(sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(9/2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(9/2),x, algorithm="fricas")

[Out]

integral(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(9/2), 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)*(3*x**2+5*x+2)**(1/2)/x**(9/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (5 \, x - 2\right )}}{x^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(9/2),x, algorithm="giac")

[Out]

integrate(-sqrt(3*x^2 + 5*x + 2)*(5*x - 2)/x^(9/2), x)