3.2596 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt{3+2 x}} \, dx\)

Optimal. Leaf size=207 \[ \frac{1}{429} (224-33 x) \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac{5 \sqrt{2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}}{18018}+\frac{(34372-676791 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{324324}+\frac{5983645 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{648648 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{651617 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{92664 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

((34372 - 676791*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/324324 - (5*Sqrt[3 + 2*
x]*(563 + 4669*x)*(2 + 5*x + 3*x^2)^(3/2))/18018 + ((224 - 33*x)*Sqrt[3 + 2*x]*(
2 + 5*x + 3*x^2)^(5/2))/429 - (651617*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sq
rt[3]*Sqrt[1 + x]], -2/3])/(92664*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5983645*Sqrt
[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(648648*Sqrt[3]
*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.411534, antiderivative size = 207, 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 \[ \frac{1}{429} (224-33 x) \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{5/2}-\frac{5 \sqrt{2 x+3} (4669 x+563) \left (3 x^2+5 x+2\right )^{3/2}}{18018}+\frac{(34372-676791 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{324324}+\frac{5983645 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{648648 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{651617 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{92664 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

((34372 - 676791*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/324324 - (5*Sqrt[3 + 2*
x]*(563 + 4669*x)*(2 + 5*x + 3*x^2)^(3/2))/18018 + ((224 - 33*x)*Sqrt[3 + 2*x]*(
2 + 5*x + 3*x^2)^(5/2))/429 - (651617*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sq
rt[3]*Sqrt[1 + x]], -2/3])/(92664*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (5983645*Sqrt
[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(648648*Sqrt[3]
*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 55.9338, size = 196, normalized size = 0.95 \[ \frac{\left (- 2030373 x + 103116\right ) \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}{972972} + \frac{\left (- 33 x + 224\right ) \sqrt{2 x + 3} \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{429} - \frac{5 \sqrt{2 x + 3} \left (42021 x + 5067\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{162162} - \frac{651617 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{277992 \sqrt{3 x^{2} + 5 x + 2}} + \frac{5983645 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{1945944 \sqrt{3 x^{2} + 5 x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2030373*x + 103116)*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)/972972 + (-33*x + 224
)*sqrt(2*x + 3)*(3*x**2 + 5*x + 2)**(5/2)/429 - 5*sqrt(2*x + 3)*(42021*x + 5067)
*(3*x**2 + 5*x + 2)**(3/2)/162162 - 651617*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(a
sin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(277992*sqrt(3*x**2 + 5*x + 2)) + 5983645*sq
rt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(1945944*
sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.565497, size = 213, normalized size = 1.03 \[ -\frac{2 \left (4041576 x^8-1163484 x^7-83553120 x^6-268524558 x^5-406647648 x^4-349849791 x^3-170798082 x^2-39284147 x-1864706\right ) \sqrt{2 x+3}-971132 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+4561319 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{1945944 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*Sqrt[3 + 2*x]*(-1864706 - 39284147*x - 170798082*x^2 - 349849791*x^3 - 40664
7648*x^4 - 268524558*x^5 - 83553120*x^6 - 1163484*x^7 + 4041576*x^8) + 4561319*S
qrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[A
rcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 971132*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])/(1945944*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.016, size = 167, normalized size = 0.8 \[{\frac{1}{116756640\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+116756640}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -80831520\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}+1422326\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +4561319\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+219746880 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/19459440*(3*x^2+5*x+2)^(1/2)*(3+2*x)^(1/2)*(-80831520*x^8+23269680*x^7+1671062
400*x^6+5370491160*x^5+1422326*(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))+4561319*(3+2*x)^(1/2)*1
5^(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))[phone]*x^4+6996995820*x^3+3689640780*x^2+1241814840*x+219746880)/
(6*x^3+19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{\sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(9*x^5 - 15*x^4 - 113*x^3 - 165*x^2 - 96*x - 20)*sqrt(3*x^2 + 5*x + 2)
/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)**(5/2)/(3+2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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