3.1065 \(\int \frac{(2-5 x) x^{5/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=182 \[ -\frac{580}{27} \sqrt{3 x^2+5 x+2} \sqrt{x}+\frac{1804 (3 x+2) \sqrt{x}}{81 \sqrt{3 x^2+5 x+2}}+\frac{580 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}-\frac{1804 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{3/2}}{3 \sqrt{3 x^2+5 x+2}} \]

[Out]

(1804*Sqrt[x]*(2 + 3*x))/(81*Sqrt[2 + 5*x + 3*x^2]) + (2*x^(3/2)*(74 + 95*x))/(3
*Sqrt[2 + 5*x + 3*x^2]) - (580*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/27 - (1804*Sqrt[2]
*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(81*Sqrt[2 +
5*x + 3*x^2]) + (580*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sq
rt[x]], -1/2])/(27*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.302392, antiderivative size = 182, 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 \[ -\frac{580}{27} \sqrt{3 x^2+5 x+2} \sqrt{x}+\frac{1804 (3 x+2) \sqrt{x}}{81 \sqrt{3 x^2+5 x+2}}+\frac{580 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}-\frac{1804 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}+\frac{2 (95 x+74) x^{3/2}}{3 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(1804*Sqrt[x]*(2 + 3*x))/(81*Sqrt[2 + 5*x + 3*x^2]) + (2*x^(3/2)*(74 + 95*x))/(3
*Sqrt[2 + 5*x + 3*x^2]) - (580*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])/27 - (1804*Sqrt[2]
*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(81*Sqrt[2 +
5*x + 3*x^2]) + (580*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sq
rt[x]], -1/2])/(27*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 31.7926, size = 167, normalized size = 0.92 \[ \frac{2 x^{\frac{3}{2}} \left (95 x + 74\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{902 \sqrt{x} \left (6 x + 4\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} - \frac{580 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}}{27} - \frac{451 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} + \frac{145 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(3/2),x)

[Out]

2*x**(3/2)*(95*x + 74)/(3*sqrt(3*x**2 + 5*x + 2)) + 902*sqrt(x)*(6*x + 4)/(81*sq
rt(3*x**2 + 5*x + 2)) - 580*sqrt(x)*sqrt(3*x**2 + 5*x + 2)/27 - 451*sqrt((6*x +
4)/(x + 1))*(4*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(81*sqrt(3*x**2 + 5*x + 2)
) + 145*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(27*sq
rt(3*x**2 + 5*x + 2))

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Mathematica [C]  time = 0.244685, size = 150, normalized size = 0.82 \[ \frac{-64 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+1804 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-90 x^3+708 x^2+5540 x+3608}{81 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(3608 + 5540*x + 708*x^2 - 90*x^3 + (1804*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2
/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] - (64*I)*Sqrt[2]*Sqrt[1
+ x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(
81*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.028, size = 118, normalized size = 0.7 \[ -{\frac{2}{243} \left ( 483\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -451\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +135\,{x}^{3}+7056\,{x}^{2}+5220\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/243*(483*(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))-451*(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))+135*x^3+7056*x^2+5220*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. \[ -\int \frac{{\left (5 \, x - 2\right )} x^{\frac{5}{2}}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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