3.2580 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{5/2}} \, dx\)

Optimal. Leaf size=143 \[ \frac{\sqrt{3 x^2+5 x+2} (119 x+146)}{15 (2 x+3)^{3/2}}+\frac{17 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{67 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{10 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

((146 + 119*x)*Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^(3/2)) - (67*Sqrt[-2 - 5*x -
3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(10*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2]) + (17*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/
3])/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.258628, 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 \[ \frac{\sqrt{3 x^2+5 x+2} (119 x+146)}{15 (2 x+3)^{3/2}}+\frac{17 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{67 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{10 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

((146 + 119*x)*Sqrt[2 + 5*x + 3*x^2])/(15*(3 + 2*x)^(3/2)) - (67*Sqrt[-2 - 5*x -
3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(10*Sqrt[3]*Sqrt[2 + 5*x +
3*x^2]) + (17*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/
3])/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 39.9638, size = 136, normalized size = 0.95 \[ - \frac{67 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{30 \sqrt{3 x^{2} + 5 x + 2}} + \frac{17 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{6 \sqrt{3 x^{2} + 5 x + 2}} + \frac{\left (119 x + 146\right ) \sqrt{3 x^{2} + 5 x + 2}}{15 \left (2 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-67*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(30
*sqrt(3*x**2 + 5*x + 2)) + 17*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*s
qrt(6*x + 6)/2), -2/3)/(6*sqrt(3*x**2 + 5*x + 2)) + (119*x + 146)*sqrt(3*x**2 +
5*x + 2)/(15*(2*x + 3)**(3/2))

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Mathematica [A]  time = 0.462186, size = 182, normalized size = 1.27 \[ -\frac{90 x^3+480 x^2+610 x-16 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+67 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+220}{30 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(220 + 610*x + 480*x^2 + 90*x^3 + 67*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^
(5/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]
- 16*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(5/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*E
llipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(30*(3 + 2*x)^(3/2)*Sqrt[2 + 5*x
+ 3*x^2])

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Maple [A]  time = 0.045, size = 215, normalized size = 1.5 \[{\frac{1}{300} \left ( 36\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+134\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+54\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +201\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +7140\,{x}^{3}+20660\,{x}^{2}+19360\,x+5840 \right ) \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/300*(36*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x*(-2-2*x)
^(1/2)*(-30*x-20)^(1/2)*(3+2*x)^(1/2)+134*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x
)^(1/2),1/3*15^(1/2))*x*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*(3+2*x)^(1/2)+54*(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))+201*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*El
lipticE(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+7140*x^3+20660*x^2+19360*x+5840
)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{{\left (4 \, x^{2} + 12 \, x + 9\right )} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/((4*x^2 + 12*x + 9)*sqrt(2*x + 3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{5 \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \frac{x \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-5*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) +
9*sqrt(2*x + 3)), x) - Integral(x*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3)
+ 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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