3.2605 \(\int \frac{(5-x) (3+2 x)^{5/2}}{\sqrt{2+5 x+3 x^2}} \, dx\)

Optimal. Leaf size=192 \[ -\frac{2}{21} \sqrt{3 x^2+5 x+2} (2 x+3)^{5/2}+\frac{10}{7} \sqrt{3 x^2+5 x+2} (2 x+3)^{3/2}+\frac{1010}{189} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}-\frac{2525 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{189 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{865 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(1010*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/189 + (10*(3 + 2*x)^(3/2)*Sqrt[2 + 5*
x + 3*x^2])/7 - (2*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2])/21 + (865*Sqrt[-2 - 5*
x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(27*Sqrt[3]*Sqrt[2 + 5*
x + 3*x^2]) - (2525*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]]
, -2/3])/(189*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.39999, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{2}{21} \sqrt{3 x^2+5 x+2} (2 x+3)^{5/2}+\frac{10}{7} \sqrt{3 x^2+5 x+2} (2 x+3)^{3/2}+\frac{1010}{189} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}-\frac{2525 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{189 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{865 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(1010*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/189 + (10*(3 + 2*x)^(3/2)*Sqrt[2 + 5*
x + 3*x^2])/7 - (2*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2])/21 + (865*Sqrt[-2 - 5*
x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(27*Sqrt[3]*Sqrt[2 + 5*
x + 3*x^2]) - (2525*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]]
, -2/3])/(189*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 56.7841, size = 184, normalized size = 0.96 \[ - \frac{2 \left (2 x + 3\right )^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}}{21} + \frac{10 \left (2 x + 3\right )^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}}{7} + \frac{1010 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}{189} + \frac{865 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2525 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{567 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(2*x + 3)**(5/2)*sqrt(3*x**2 + 5*x + 2)/21 + 10*(2*x + 3)**(3/2)*sqrt(3*x**2
+ 5*x + 2)/7 + 1010*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)/189 + 865*sqrt(-9*x**2
- 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(81*sqrt(3*x**2 + 5*
x + 2)) - 2525*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2)
, -2/3)/(567*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.5223, size = 198, normalized size = 1.03 \[ -\frac{8 \left (162 x^5-216 x^4-5526 x^3-18501 x^2-20111 x-6758\right ) \sqrt{2 x+3}+4540 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-6055 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{567 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(8*Sqrt[3 + 2*x]*(-6758 - 20111*x - 18501*x^2 - 5526*x^3 - 216*x^4 + 162*x^5) -
6055*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*Elli
pticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 4540*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])/(567*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.043, size = 152, normalized size = 0.8 \[{\frac{1}{6804\,{x}^{3}+21546\,{x}^{2}+21546\,x+6804}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -2592\,{x}^{5}+706\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -1211\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +3456\,{x}^{4}+88416\,{x}^{3}+223356\,{x}^{2}+200676\,x+59688 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1134*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(-2592*x^5+706*(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)
)-1211*(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))+3456*x^4+88416*x^3+223356*x^2+200676*x+59688)/(
6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x - 45\right )} \sqrt{2 \, x + 3}}{\sqrt{3 \, x^{2} + 5 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(4*x^3 - 8*x^2 - 51*x - 45)*sqrt(2*x + 3)/sqrt(3*x^2 + 5*x + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{45 \sqrt{2 x + 3}}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{51 x \sqrt{2 x + 3}}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac{8 x^{2} \sqrt{2 x + 3}}{\sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{4 x^{3} \sqrt{2 x + 3}}{\sqrt{3 x^{2} + 5 x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-45*sqrt(2*x + 3)/sqrt(3*x**2 + 5*x + 2), x) - Integral(-51*x*sqrt(2*x
+ 3)/sqrt(3*x**2 + 5*x + 2), x) - Integral(-8*x**2*sqrt(2*x + 3)/sqrt(3*x**2 +
5*x + 2), x) - Integral(4*x**3*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. \[ \int -\frac{{\left (2 \, x + 3\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{\sqrt{3 \, x^{2} + 5 \, x + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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