3.2582 \(\int \frac{(5-x) \sqrt{2+5 x+3 x^2}}{(3+2 x)^{9/2}} \, dx\)

Optimal. Leaf size=197 \[ \frac{\sqrt{3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}+\frac{159 \sqrt{3 x^2+5 x+2}}{625 \sqrt{2 x+3}}+\frac{183 \sqrt{3 x^2+5 x+2}}{875 (2 x+3)^{3/2}}+\frac{183 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1750 \sqrt{3 x^2+5 x+2}}-\frac{159 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1250 \sqrt{3 x^2+5 x+2}} \]

[Out]

(183*Sqrt[2 + 5*x + 3*x^2])/(875*(3 + 2*x)^(3/2)) + (159*Sqrt[2 + 5*x + 3*x^2])/
(625*Sqrt[3 + 2*x]) + ((46 + 139*x)*Sqrt[2 + 5*x + 3*x^2])/(175*(3 + 2*x)^(7/2))
- (159*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2
/3])/(1250*Sqrt[2 + 5*x + 3*x^2]) + (183*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*Elliptic
F[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1750*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.402205, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{\sqrt{3 x^2+5 x+2} (139 x+46)}{175 (2 x+3)^{7/2}}+\frac{159 \sqrt{3 x^2+5 x+2}}{625 \sqrt{2 x+3}}+\frac{183 \sqrt{3 x^2+5 x+2}}{875 (2 x+3)^{3/2}}+\frac{183 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1750 \sqrt{3 x^2+5 x+2}}-\frac{159 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1250 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(9/2),x]

[Out]

(183*Sqrt[2 + 5*x + 3*x^2])/(875*(3 + 2*x)^(3/2)) + (159*Sqrt[2 + 5*x + 3*x^2])/
(625*Sqrt[3 + 2*x]) + ((46 + 139*x)*Sqrt[2 + 5*x + 3*x^2])/(175*(3 + 2*x)^(7/2))
- (159*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2
/3])/(1250*Sqrt[2 + 5*x + 3*x^2]) + (183*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*Elliptic
F[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1750*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 56.3714, size = 187, normalized size = 0.95 \[ - \frac{159 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{1250 \sqrt{3 x^{2} + 5 x + 2}} + \frac{183 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{1750 \sqrt{3 x^{2} + 5 x + 2}} + \frac{159 \sqrt{3 x^{2} + 5 x + 2}}{625 \sqrt{2 x + 3}} + \frac{183 \sqrt{3 x^{2} + 5 x + 2}}{875 \left (2 x + 3\right )^{\frac{3}{2}}} + \frac{\left (139 x + 46\right ) \sqrt{3 x^{2} + 5 x + 2}}{175 \left (2 x + 3\right )^{\frac{7}{2}}} \]

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)**(9/2),x)

[Out]

-159*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(1
250*sqrt(3*x**2 + 5*x + 2)) + 183*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(
2)*sqrt(6*x + 6)/2), -2/3)/(1750*sqrt(3*x**2 + 5*x + 2)) + 159*sqrt(3*x**2 + 5*x
+ 2)/(625*sqrt(2*x + 3)) + 183*sqrt(3*x**2 + 5*x + 2)/(875*(2*x + 3)**(3/2)) +
(139*x + 46)*sqrt(3*x**2 + 5*x + 2)/(175*(2*x + 3)**(7/2))

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Mathematica [A]  time = 0.626963, size = 217, normalized size = 1.1 \[ -\frac{6 (2 x+3)^3 \left (742 \left (3 x^2+5 x+2\right )-188 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+371 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )-4 \left (3 x^2+5 x+2\right ) \left (8904 x^3+43728 x^2+74557 x+39436\right )}{17500 (2 x+3)^{7/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*Sqrt[2 + 5*x + 3*x^2])/(3 + 2*x)^(9/2),x]

[Out]

-(-4*(2 + 5*x + 3*x^2)*(39436 + 74557*x + 43728*x^2 + 8904*x^3) + 6*(3 + 2*x)^3*
(742*(2 + 5*x + 3*x^2) + 371*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqr
t[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 188*Sqr
t[5]*Sqrt[(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]))/(17500*(3 + 2*x)^(7/2)*Sqrt[2 + 5*x + 3
*x^2])

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Maple [B]  time = 0.048, size = 413, normalized size = 2.1 \[ -{\frac{1}{87500} \left ( 1584\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}-8904\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+7128\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}-40068\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+10692\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}-60102\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+5346\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -30051\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -534240\,{x}^{5}-3514080\,{x}^{4}-9202380\,{x}^{3}-11570980\,{x}^{2}-6925880\,x-1577440 \right ){\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}} \left ( 3+2\,x \right ) ^{-{\frac{7}{2}}}} \]

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)^(9/2),x)

[Out]

-1/87500*(1584*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^3*(
3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)-8904*15^(1/2)*EllipticE(1/5*15^(1/2
)*(3+2*x)^(1/2),1/3*15^(1/2))*x^3*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)+
7128*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^2*(-30*x-20)^
(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)-40068*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)
^(1/2),1/3*15^(1/2))*x^2*(-30*x-20)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)+10692*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)-60102*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)+5346*(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))-30051*(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))-534240*x^5-3514080*x^4-9202380*x^3-115
70980*x^2-6925880*x-1577440)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(7/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^(9/2),x, algorithm="maxima")

[Out]

-integrate(sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{{\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^(9/2),x, algorithm="fricas")

[Out]

integral(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/((16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81
)*sqrt(2*x + 3)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{3 \, x^{2} + 5 \, x + 2}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{9}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^(9/2),x, algorithm="giac")

[Out]

integrate(-sqrt(3*x^2 + 5*x + 2)*(x - 5)/(2*x + 3)^(9/2), x)