3.2617 \(\int \frac{5-x}{(3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=170 \[ -\frac{6 (47 x+37)}{5 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}-\frac{908 \sqrt{3 x^2+5 x+2}}{25 \sqrt{2 x+3}}-\frac{94 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}}+\frac{454 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-6*(37 + 47*x))/(5*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2]) - (908*Sqrt[2 + 5*x + 3
*x^2])/(25*Sqrt[3 + 2*x]) + (454*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[2 + 5*x + 3*x^2]) - (94*Sqrt[3]*Sqrt[-2 -
5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*Sqrt[2 + 5*x + 3*
x^2])

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Rubi [A]  time = 0.342398, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{6 (47 x+37)}{5 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}-\frac{908 \sqrt{3 x^2+5 x+2}}{25 \sqrt{2 x+3}}-\frac{94 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3 x^2+5 x+2}}+\frac{454 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(37 + 47*x))/(5*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2]) - (908*Sqrt[2 + 5*x + 3
*x^2])/(25*Sqrt[3 + 2*x]) + (454*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[2 + 5*x + 3*x^2]) - (94*Sqrt[3]*Sqrt[-2 -
5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*Sqrt[2 + 5*x + 3*
x^2])

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Rubi in Sympy [A]  time = 46.1929, size = 162, normalized size = 0.95 \[ \frac{454 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{25 \sqrt{3 x^{2} + 5 x + 2}} - \frac{94 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{5 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2 \left (141 x + 111\right )}{5 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}} - \frac{908 \sqrt{3 x^{2} + 5 x + 2}}{25 \sqrt{2 x + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

454*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(25
*sqrt(3*x**2 + 5*x + 2)) - 94*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*s
qrt(6*x + 6)/2), -2/3)/(5*sqrt(3*x**2 + 5*x + 2)) - 2*(141*x + 111)/(5*sqrt(2*x
+ 3)*sqrt(3*x**2 + 5*x + 2)) - 908*sqrt(3*x**2 + 5*x + 2)/(25*sqrt(2*x + 3))

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Mathematica [A]  time = 0.507658, size = 172, normalized size = 1.01 \[ -\frac{2 \left (705 x+86 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-227 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+555\right )}{25 \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(555 + 705*x - 227*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 +
3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 86*Sqrt[5]*Sq
rt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin
[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/(25*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.032, size = 137, normalized size = 0.8 \[ -{\frac{1}{750\,{x}^{3}+2375\,{x}^{2}+2375\,x+750} \left ( 8\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +227\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +13620\,{x}^{2}+29750\,x+14630 \right ) \sqrt{3\,{x}^{2}+5\,x+2}\sqrt{3+2\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/125*(8*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*EllipticF(1/5*1
5^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+227*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-
30*x-20)^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+13620*x^2+2975
0*x+14630)*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x - 5}{{\left (6 \, x^{3} + 19 \, x^{2} + 19 \, x + 6\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(x - 5)/((6*x^3 + 19*x^2 + 19*x + 6)*sqrt(3*x^2 + 5*x + 2)*sqrt(2*x +
3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{6 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 19 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 19 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 6 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{6 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 19 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 19 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 6 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(6*x**3*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 19*x**2*sqrt(2*x + 3)
*sqrt(3*x**2 + 5*x + 2) + 19*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 6*sqrt(2*x
+ 3)*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-5/(6*x**3*sqrt(2*x + 3)*sqrt(3*x**
2 + 5*x + 2) + 19*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 19*x*sqrt(2*x + 3)
*sqrt(3*x**2 + 5*x + 2) + 6*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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