3.2614 \(\int \frac{(5-x) (3+2 x)^{3/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=143 \[ -\frac{2 \sqrt{2 x+3} (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}-\frac{350 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{274 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*Sqrt[3 + 2*x]*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (274*Sqrt[-2 - 5*x
- 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(3*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2]) - (350*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2
/3])/(3*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.260709, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{2 \sqrt{2 x+3} (139 x+121)}{3 \sqrt{3 x^2+5 x+2}}-\frac{350 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{274 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 2*x]*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (274*Sqrt[-2 - 5*x
- 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(3*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2]) - (350*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2
/3])/(3*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 39.9086, size = 138, normalized size = 0.97 \[ - \frac{2 \sqrt{2 x + 3} \left (139 x + 121\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{274 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{9 \sqrt{3 x^{2} + 5 x + 2}} - \frac{350 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{9 \sqrt{3 x^{2} + 5 x + 2}} \]

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]

-2*sqrt(2*x + 3)*(139*x + 121)/(3*sqrt(3*x**2 + 5*x + 2)) + 274*sqrt(-9*x**2 - 1
5*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(9*sqrt(3*x**2 + 5*x +
2)) - 350*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/
3)/(9*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.504022, size = 183, normalized size = 1.28 \[ -\frac{2 \left (12 x^2+607 x+541\right ) \sqrt{2 x+3}+64 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-274 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{9 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(541 + 607*x + 12*x^2) - 274*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(
3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]],
3/5] + 64*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*
EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(9*(3 + 2*x)*Sqrt[2 + 5*x + 3*x
^2])

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Maple [A]  time = 0.026, size = 137, normalized size = 1. \[ -{\frac{1}{270\,{x}^{3}+855\,{x}^{2}+855\,x+270}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 38\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +137\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +8340\,{x}^{2}+19770\,x+10890 \right ) } \]

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]

-1/45*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(38*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2
)*(-30*x-20)^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+137*(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))+8340*x^2+19770*x+10890)/(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (2 \, x^{2} - 7 \, x - 15\right )} \sqrt{2 \, x + 3}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(2*x^2 - 7*x - 15)*sqrt(2*x + 3)/(3*x^2 + 5*x + 2)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{15 \sqrt{2 x + 3}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{7 x \sqrt{2 x + 3}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{2 x^{2} \sqrt{2 x + 3}}{3 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 5 x \sqrt{3 x^{2} + 5 x + 2} + 2 \sqrt{3 x^{2} + 5 x + 2}}\, dx \]

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(-15*sqrt(2*x + 3)/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5
*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-7*x*sqrt(2*x + 3)/(3*x**2*sq
rt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x
) - Integral(2*x**2*sqrt(2*x + 3)/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x*
*2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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