3.2602 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{13/2}} \, dx\)

Optimal. Leaf size=207 \[ \frac{(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}+\frac{(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{34650 (2 x+3)^{7/2}}+\frac{(948443 x+1301762) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{3/2}}+\frac{198109 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{46200 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{107857 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{33000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

((1301762 + 948443*x)*Sqrt[2 + 5*x + 3*x^2])/(346500*(3 + 2*x)^(3/2)) + ((24161
+ 18699*x)*(2 + 5*x + 3*x^2)^(3/2))/(34650*(3 + 2*x)^(7/2)) + ((114 + 115*x)*(2
+ 5*x + 3*x^2)^(5/2))/(99*(3 + 2*x)^(11/2)) - (107857*Sqrt[-2 - 5*x - 3*x^2]*Ell
ipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(33000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])
+ (198109*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/
(46200*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.40977, 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 \[ \frac{(115 x+114) \left (3 x^2+5 x+2\right )^{5/2}}{99 (2 x+3)^{11/2}}+\frac{(18699 x+24161) \left (3 x^2+5 x+2\right )^{3/2}}{34650 (2 x+3)^{7/2}}+\frac{(948443 x+1301762) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{3/2}}+\frac{198109 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{46200 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{107857 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{33000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(13/2),x]

[Out]

((1301762 + 948443*x)*Sqrt[2 + 5*x + 3*x^2])/(346500*(3 + 2*x)^(3/2)) + ((24161
+ 18699*x)*(2 + 5*x + 3*x^2)^(3/2))/(34650*(3 + 2*x)^(7/2)) + ((114 + 115*x)*(2
+ 5*x + 3*x^2)^(5/2))/(99*(3 + 2*x)^(11/2)) - (107857*Sqrt[-2 - 5*x - 3*x^2]*Ell
ipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(33000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])
+ (198109*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/
(46200*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 58.9022, size = 194, normalized size = 0.94 \[ - \frac{107857 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{99000 \sqrt{3 x^{2} + 5 x + 2}} + \frac{198109 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{138600 \sqrt{3 x^{2} + 5 x + 2}} + \frac{\left (948443 x + 1301762\right ) \sqrt{3 x^{2} + 5 x + 2}}{346500 \left (2 x + 3\right )^{\frac{3}{2}}} + \frac{\left (18699 x + 24161\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{34650 \left (2 x + 3\right )^{\frac{7}{2}}} + \frac{\left (575 x + 570\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{495 \left (2 x + 3\right )^{\frac{11}{2}}} \]

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)**(13/2),x)

[Out]

-107857*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)
/(99000*sqrt(3*x**2 + 5*x + 2)) + 198109*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asi
n(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(138600*sqrt(3*x**2 + 5*x + 2)) + (948443*x +
1301762)*sqrt(3*x**2 + 5*x + 2)/(346500*(2*x + 3)**(3/2)) + (18699*x + 24161)*(3
*x**2 + 5*x + 2)**(3/2)/(34650*(2*x + 3)**(7/2)) + (575*x + 570)*(3*x**2 + 5*x +
2)**(5/2)/(495*(2*x + 3)**(11/2))

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Mathematica [A]  time = 0.730466, size = 227, normalized size = 1.1 \[ -\frac{4 (2 x+3)^5 \left (1509998 \left (3 x^2+5 x+2\right )-160672 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+754999 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )-8 \left (3 x^2+5 x+2\right ) \left (21041468 x^5+140915480 x^4+387989550 x^3+544712540 x^2+387631385 x+111387702\right )}{2772000 (2 x+3)^{11/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(13/2),x]

[Out]

-(-8*(2 + 5*x + 3*x^2)*(111387702 + 387631385*x + 544712540*x^2 + 387989550*x^3
+ 140915480*x^4 + 21041468*x^5) + 4*(3 + 2*x)^5*(1509998*(2 + 5*x + 3*x^2) + 754
999*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*El
lipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 160672*Sqrt[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]))/(2772000*(3 + 2*x)^(11/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.027, size = 611, normalized size = 3. \[ \text{result too large to display} \]

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)^(13/2),x)

[Out]

1/6930000*(24159968*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*
x^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)+7537472*15^(1/2)*EllipticF(1/5
*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20
)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^
4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)[phone]*15^(1/2)*EllipticF(1/5*
15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)
^(1/2)[phone]*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)[phone]*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)[phone]*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)[phone]*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)[phone]*x^7+611549190*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)[phone]*15^(1/2)*E
llipticF(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)[phone]*x^6+183464757*(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))[phone]*(
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))[phone]*x^5+77118326600*x^4+93248719100*x^3+6723490
2220*x^2+26644025600*x+4455508080)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(11/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2),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)
/((64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729)*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)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(13/2), x)