3.1076 \(\int \frac{(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{3464 (3 x+2) \sqrt{x}}{27 \sqrt{3 x^2+5 x+2}}+\frac{4 (866 x+715) \sqrt{x}}{9 \sqrt{3 x^2+5 x+2}}-\frac{1430 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{3 x^2+5 x+2}}+\frac{3464 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{3/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}} \]

[Out]

(2*x^(3/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (3464*Sqrt[x]*(2 + 3*x))/(
27*Sqrt[2 + 5*x + 3*x^2]) + (4*Sqrt[x]*(715 + 866*x))/(9*Sqrt[2 + 5*x + 3*x^2])
+ (3464*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2]
)/(27*Sqrt[2 + 5*x + 3*x^2]) - (1430*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Ell
ipticF[ArcTan[Sqrt[x]], -1/2])/(9*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.302255, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{3464 (3 x+2) \sqrt{x}}{27 \sqrt{3 x^2+5 x+2}}+\frac{4 (866 x+715) \sqrt{x}}{9 \sqrt{3 x^2+5 x+2}}-\frac{1430 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{9 \sqrt{3 x^2+5 x+2}}+\frac{3464 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{3/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(74 + 95*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (3464*Sqrt[x]*(2 + 3*x))/(
27*Sqrt[2 + 5*x + 3*x^2]) + (4*Sqrt[x]*(715 + 866*x))/(9*Sqrt[2 + 5*x + 3*x^2])
+ (3464*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2]
)/(27*Sqrt[2 + 5*x + 3*x^2]) - (1430*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*Ell
ipticF[ArcTan[Sqrt[x]], -1/2])/(9*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 32.4452, size = 172, normalized size = 0.92 \[ \frac{2 x^{\frac{3}{2}} \left (95 x + 74\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} - \frac{1732 \sqrt{x} \left (6 x + 4\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} + \frac{4 \sqrt{x} \left (866 x + 715\right )}{9 \sqrt{3 x^{2} + 5 x + 2}} + \frac{866 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} - \frac{715 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{18 \sqrt{3 x^{2} + 5 x + 2}} \]

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]

2*x**(3/2)*(95*x + 74)/(9*(3*x**2 + 5*x + 2)**(3/2)) - 1732*sqrt(x)*(6*x + 4)/(2
7*sqrt(3*x**2 + 5*x + 2)) + 4*sqrt(x)*(866*x + 715)/(9*sqrt(3*x**2 + 5*x + 2)) +
866*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(27*sqrt(
3*x**2 + 5*x + 2)) - 715*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(
x)), -1/2)/(18*sqrt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.392333, size = 167, normalized size = 0.89 \[ \frac{-826 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-3464 i \sqrt{\frac{2}{x}+2} \sqrt{\frac{2}{x}+3} \left (3 x^2+5 x+2\right ) x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (12825 x^3+32020 x^2+26060 x+6928\right )}{27 \sqrt{x} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(6928 + 26060*x + 32020*x^2 + 12825*x^3) - (3464*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] - (82
6*I)*Sqrt[2 + 2/x]*Sqrt[3 + 2/x]*x^(3/2)*(2 + 5*x + 3*x^2)*EllipticF[I*ArcSinh[S
qrt[2/3]/Sqrt[x]], 3/2])/(27*Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))

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Maple [A]  time = 0.031, size = 315, normalized size = 1.7 \[{\frac{2}{81\, \left ( 2+3\,x \right ) ^{2} \left ( 1+x \right ) ^{2}} \left ( 1359\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}-2598\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ){x}^{2}+2265\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x-4330\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) x+906\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -1732\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +46764\,{x}^{4}+117405\,{x}^{3}+96192\,{x}^{2}+25740\,x \right ) \sqrt{3\,{x}^{2}+5\,x+2}{\frac{1}{\sqrt{x}}}} \]

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]

2/81*(1359*(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^2-2598*(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+2265*(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-4330*(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+906*(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))-1732*(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))+46764*x^4+117405*x^3+96192*x^
2+25740*x)*(3*x^2+5*x+2)^(1/2)/x^(1/2)/(2+3*x)^2/(1+x)^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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