3.2624 \(\int \frac{(5-x) \sqrt{3+2 x}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{2 \sqrt{2 x+3} (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 \sqrt{2 x+3} (2139 x+1759)}{15 \sqrt{3 x^2+5 x+2}}+\frac{748 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2852 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*Sqrt[3 + 2*x]*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (4*Sqrt[3 + 2*x]*(1
759 + 2139*x))/(15*Sqrt[2 + 5*x + 3*x^2]) - (2852*Sqrt[-2 - 5*x - 3*x^2]*Ellipti
cE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (748*
Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sq
rt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.343448, antiderivative size = 173, 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 \[ -\frac{2 \sqrt{2 x+3} (35 x+29)}{3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{4 \sqrt{2 x+3} (2139 x+1759)}{15 \sqrt{3 x^2+5 x+2}}+\frac{748 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{\sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2852 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{5 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[3 + 2*x]*(29 + 35*x))/(3*(2 + 5*x + 3*x^2)^(3/2)) + (4*Sqrt[3 + 2*x]*(1
759 + 2139*x))/(15*Sqrt[2 + 5*x + 3*x^2]) - (2852*Sqrt[-2 - 5*x - 3*x^2]*Ellipti
cE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(5*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (748*
Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(Sqrt[3]*Sq
rt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 47.2894, size = 168, normalized size = 0.97 \[ - \frac{2 \sqrt{2 x + 3} \left (35 x + 29\right )}{3 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \sqrt{2 x + 3} \left (2139 x + 1759\right )}{15 \sqrt{3 x^{2} + 5 x + 2}} - \frac{2852 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{15 \sqrt{3 x^{2} + 5 x + 2}} + \frac{748 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(2*x + 3)*(35*x + 29)/(3*(3*x**2 + 5*x + 2)**(3/2)) + 4*sqrt(2*x + 3)*(21
39*x + 1759)/(15*sqrt(3*x**2 + 5*x + 2)) - 2852*sqrt(-9*x**2 - 15*x - 6)*ellipti
c_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(15*sqrt(3*x**2 + 5*x + 2)) + 748*sqrt(
-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(3*sqrt(3*x*
*2 + 5*x + 2))

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Mathematica [A]  time = 0.533483, size = 196, normalized size = 1.13 \[ -\frac{\frac{5704 \left (3 x^2+5 x+2\right )}{\sqrt{2 x+3}}-\frac{6 \sqrt{2 x+3} \left (4278 x^3+10648 x^2+8657 x+2297\right )}{3 x^2+5 x+2}-\frac{608 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}+\frac{2852 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}}{15 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-((5704*(2 + 5*x + 3*x^2))/Sqrt[3 + 2*x] - (6*Sqrt[3 + 2*x]*(2297 + 8657*x + 106
48*x^2 + 4278*x^3))/(2 + 5*x + 3*x^2) + (2852*(1 + x)*Sqrt[(2 + 3*x)/(3 + 2*x)]*
EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/Sqrt[(1 + x)/(15 + 10*x)] - (60
8*(1 + x)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3
/5])/Sqrt[(1 + x)/(15 + 10*x)])/(15*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.03, size = 326, normalized size = 1.9 \[{\frac{2}{75\, \left ( 1+x \right ) ^{2} \left ( 2+3\,x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 2139\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+666\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+3565\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+1110\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+1426\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +444\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +128340\,{x}^{4}+511950\,{x}^{3}+738870\,{x}^{2}+458475\,x+103365 \right ){\frac{1}{\sqrt{3+2\,x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/75*(3*x^2+5*x+2)^(1/2)*(2139*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)+666*15^(1/2)*Ellipt
icF(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)+3565*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)+1110*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)
+1426*(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))+444*(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))+128340*x^4+511950*
x^3+738870*x^2+458475*x+103365)/(1+x)^2/(2+3*x)^2/(3+2*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{2 \, x + 3}{\left (x - 5\right )}}{{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-sqrt(2*x + 3)*(x - 5)/((9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*sqrt(3*x^2
+ 5*x + 2)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{5 \sqrt{2 x + 3}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac{x \sqrt{2 x + 3}}{9 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 20 x \sqrt{3 x^{2} + 5 x + 2} + 4 \sqrt{3 x^{2} + 5 x + 2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-5*sqrt(2*x + 3)/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2
+ 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sq
rt(3*x**2 + 5*x + 2)), x) - Integral(x*sqrt(2*x + 3)/(9*x**4*sqrt(3*x**2 + 5*x +
2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqr
t(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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