3.1048 \(\int \frac{(2-5 x) \left (2+5 x+3 x^2\right )^{3/2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt{x}}+\frac{2}{105} \sqrt{x} (531 x+1045) \sqrt{3 x^2+5 x+2}+\frac{5848 \sqrt{x} (3 x+2)}{315 \sqrt{3 x^2+5 x+2}}+\frac{482 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{3 x^2+5 x+2}}-\frac{5848 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{3 x^2+5 x+2}} \]

[Out]

(5848*Sqrt[x]*(2 + 3*x))/(315*Sqrt[2 + 5*x + 3*x^2]) + (2*Sqrt[x]*(1045 + 531*x)
*Sqrt[2 + 5*x + 3*x^2])/105 - (2*(14 + 5*x)*(2 + 5*x + 3*x^2)^(3/2))/(7*Sqrt[x])
- (5848*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2
])/(315*Sqrt[2 + 5*x + 3*x^2]) + (482*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*El
lipticF[ArcTan[Sqrt[x]], -1/2])/(21*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.296857, 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 \[ -\frac{2 (5 x+14) \left (3 x^2+5 x+2\right )^{3/2}}{7 \sqrt{x}}+\frac{2}{105} \sqrt{x} (531 x+1045) \sqrt{3 x^2+5 x+2}+\frac{5848 \sqrt{x} (3 x+2)}{315 \sqrt{3 x^2+5 x+2}}+\frac{482 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} F\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{21 \sqrt{3 x^2+5 x+2}}-\frac{5848 \sqrt{2} (x+1) \sqrt{\frac{3 x+2}{x+1}} E\left (\tan ^{-1}\left (\sqrt{x}\right )|-\frac{1}{2}\right )}{315 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(5848*Sqrt[x]*(2 + 3*x))/(315*Sqrt[2 + 5*x + 3*x^2]) + (2*Sqrt[x]*(1045 + 531*x)
*Sqrt[2 + 5*x + 3*x^2])/105 - (2*(14 + 5*x)*(2 + 5*x + 3*x^2)^(3/2))/(7*Sqrt[x])
- (5848*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x]], -1/2
])/(315*Sqrt[2 + 5*x + 3*x^2]) + (482*Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*El
lipticF[ArcTan[Sqrt[x]], -1/2])/(21*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 32.1822, size = 177, normalized size = 0.95 \[ \frac{2924 \sqrt{x} \left (6 x + 4\right )}{315 \sqrt{3 x^{2} + 5 x + 2}} + \frac{8 \sqrt{x} \left (\frac{531 x}{4} + \frac{1045}{4}\right ) \sqrt{3 x^{2} + 5 x + 2}}{105} - \frac{1462 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) E\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{315 \sqrt{3 x^{2} + 5 x + 2}} + \frac{241 \sqrt{\frac{6 x + 4}{x + 1}} \left (4 x + 4\right ) F\left (\operatorname{atan}{\left (\sqrt{x} \right )}\middle | - \frac{1}{2}\right )}{42 \sqrt{3 x^{2} + 5 x + 2}} - \frac{4 \left (\frac{5 x}{2} + 7\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{7 \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2924*sqrt(x)*(6*x + 4)/(315*sqrt(3*x**2 + 5*x + 2)) + 8*sqrt(x)*(531*x/4 + 1045/
4)*sqrt(3*x**2 + 5*x + 2)/105 - 1462*sqrt((6*x + 4)/(x + 1))*(4*x + 4)*elliptic_
e(atan(sqrt(x)), -1/2)/(315*sqrt(3*x**2 + 5*x + 2)) + 241*sqrt((6*x + 4)/(x + 1)
)*(4*x + 4)*elliptic_f(atan(sqrt(x)), -1/2)/(42*sqrt(3*x**2 + 5*x + 2)) - 4*(5*x
/2 + 7)*(3*x**2 + 5*x + 2)**(3/2)/(7*sqrt(x))

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Mathematica [C]  time = 0.261036, size = 163, normalized size = 0.87 \[ \frac{1382 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )+5848 i \sqrt{2} \sqrt{\frac{1}{x}+1} \sqrt{\frac{2}{x}+3} x^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{2}{3}}}{\sqrt{x}}\right )|\frac{3}{2}\right )-2 \left (2025 x^5+7641 x^4+9855 x^3+177 x^2-7390 x-3328\right )}{315 \sqrt{x} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(-3328 - 7390*x + 177*x^2 + 9855*x^3 + 7641*x^4 + 2025*x^5) + (5848*I)*Sqrt[
2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]]
, 3/2] + (1382*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I*Arc
Sinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(315*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.023, size = 129, normalized size = 0.7 \[ -{\frac{2}{945} \left ( 6075\,{x}^{5}+771\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) -1462\,\sqrt{6\,x+4}\sqrt{3+3\,x}\sqrt{3}\sqrt{2}\sqrt{-x}{\it EllipticE} \left ( 1/2\,\sqrt{6\,x+4},i\sqrt{2} \right ) +22923\,{x}^{4}+29565\,{x}^{3}+26847\,{x}^{2}+21690\,x+7560 \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/945*(6075*x^5+771*(6*x+4)^(1/2)*(3+3*x)^(1/2)*3^(1/2)*2^(1/2)*(-x)^(1/2)*Elli
pticF(1/2*(6*x+4)^(1/2),I*2^(1/2))-1462*(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))+22923*x^4+29565*x^3+26847*
x^2+21690*x+7560)/(3*x^2+5*x+2)^(1/2)/x^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (15 \, x^{3} + 19 \, x^{2} - 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{x^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(15*x^3 + 19*x^2 - 4)*sqrt(3*x^2 + 5*x + 2)/x^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{4 \sqrt{3 x^{2} + 5 x + 2}}{x^{\frac{3}{2}}}\right )\, dx - \int 19 \sqrt{x} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 15 x^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-4*sqrt(3*x**2 + 5*x + 2)/x**(3/2), x) - Integral(19*sqrt(x)*sqrt(3*x*
*2 + 5*x + 2), x) - Integral(15*x**(3/2)*sqrt(3*x**2 + 5*x + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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