3.1058 \(\int \frac{(2-5 x) x^{3/2}}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=177 \[ \frac{52}{27} \sqrt{3 x^2+5 x+2} \sqrt{x}-\frac{412 (3 x+2) \sqrt{x}}{81 \sqrt{3 x^2+5 x+2}}-\frac{52 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}+\frac{412 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}-\frac{2}{3} \sqrt{3 x^2+5 x+2} x^{3/2} \]

[Out]

(-412*Sqrt[x]*(2 + 3*x))/(81*Sqrt[2 + 5*x + 3*x^2]) + (52*Sqrt[x]*Sqrt[2 + 5*x +
3*x^2])/27 - (2*x^(3/2)*Sqrt[2 + 5*x + 3*x^2])/3 + (412*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(81*Sqrt[2 + 5*x + 3*x^2]) -
(52*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(2
7*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.299806, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{52}{27} \sqrt{3 x^2+5 x+2} \sqrt{x}-\frac{412 (3 x+2) \sqrt{x}}{81 \sqrt{3 x^2+5 x+2}}-\frac{52 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{27 \sqrt{3 x^2+5 x+2}}+\frac{412 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{81 \sqrt{3 x^2+5 x+2}}-\frac{2}{3} \sqrt{3 x^2+5 x+2} x^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-412*Sqrt[x]*(2 + 3*x))/(81*Sqrt[2 + 5*x + 3*x^2]) + (52*Sqrt[x]*Sqrt[2 + 5*x +
3*x^2])/27 - (2*x^(3/2)*Sqrt[2 + 5*x + 3*x^2])/3 + (412*Sqrt[2]*(1 + x)*Sqrt[(2
+ 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2])/(81*Sqrt[2 + 5*x + 3*x^2]) -
(52*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(2
7*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 31.4133, size = 162, normalized size = 0.92 \[ - \frac{2 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}}{3} - \frac{206 \sqrt{x} \left (6 x + 4\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} + \frac{52 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}}{27} + \frac{103 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} - \frac{13 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2-5*x)*x**(3/2)/(3*x**2+5*x+2)**(1/2),x)

[Out]

-2*x**(3/2)*sqrt(3*x**2 + 5*x + 2)/3 - 206*sqrt(x)*(6*x + 4)/(81*sqrt(3*x**2 + 5
*x + 2)) + 52*sqrt(x)*sqrt(3*x**2 + 5*x + 2)/27 + 103*sqrt((6*x + 4)/(x + 1))*(4
*x + 4)*elliptic_e(atan(sqrt(x)), -1/2)/(81*sqrt(3*x**2 + 5*x + 2)) - 13*sqrt((6
*x + 4)/(x + 1))*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(27*sqrt(3*x**2 + 5*x
+ 2))

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Mathematica [C]  time = 0.248528, size = 158, normalized size = 0.89 \[ \frac{256 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-412 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (81 x^4-99 x^3+282 x^2+874 x+412\right )}{81 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(412 + 874*x + 282*x^2 - 99*x^3 + 81*x^4) - (412*I)*Sqrt[2]*Sqrt[1 + x^(-1)]
*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2] + (256*I)*Sq
rt[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 = 0.02, size = 123, normalized size = 0.7 \[{\frac{2}{243} \left ( 231\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -103\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -243\,{x}^{4}+297\,{x}^{3}+1008\,{x}^{2}+468\,x \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/243/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(231*(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))-103*(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))-243*x^4+2
97*x^3+1008*x^2+468*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2-5*x)*x**(3/2)/(3*x**2+5*x+2)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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