3.2604 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{17/2}} \, dx\)

Optimal. Leaf size=261 \[ \frac{(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}-\frac{(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{64350 (2 x+3)^{11/2}}-\frac{(328339 x+386846) \sqrt{3 x^2+5 x+2}}{7507500 (2 x+3)^{7/2}}+\frac{335723 \sqrt{3 x^2+5 x+2}}{80437500 \sqrt{2 x+3}}+\frac{594851 \sqrt{3 x^2+5 x+2}}{112612500 (2 x+3)^{3/2}}+\frac{594851 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{75075000 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{335723 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{53625000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(594851*Sqrt[2 + 5*x + 3*x^2])/(112612500*(3 + 2*x)^(3/2)) + (335723*Sqrt[2 + 5*
x + 3*x^2])/(80437500*Sqrt[3 + 2*x]) - ((386846 + 328339*x)*Sqrt[2 + 5*x + 3*x^2
])/(7507500*(3 + 2*x)^(7/2)) - ((8901 + 8399*x)*(2 + 5*x + 3*x^2)^(3/2))/(64350*
(3 + 2*x)^(11/2)) + ((94 + 119*x)*(2 + 5*x + 3*x^2)^(5/2))/(195*(3 + 2*x)^(15/2)
) - (335723*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])
/(53625000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (594851*Sqrt[-2 - 5*x - 3*x^2]*Ellip
ticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(75075000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]
)

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Rubi [A]  time = 0.579442, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{(119 x+94) \left (3 x^2+5 x+2\right )^{5/2}}{195 (2 x+3)^{15/2}}-\frac{(8399 x+8901) \left (3 x^2+5 x+2\right )^{3/2}}{64350 (2 x+3)^{11/2}}-\frac{(328339 x+386846) \sqrt{3 x^2+5 x+2}}{7507500 (2 x+3)^{7/2}}+\frac{335723 \sqrt{3 x^2+5 x+2}}{80437500 \sqrt{2 x+3}}+\frac{594851 \sqrt{3 x^2+5 x+2}}{112612500 (2 x+3)^{3/2}}+\frac{594851 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{75075000 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{335723 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{53625000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(17/2),x]

[Out]

(594851*Sqrt[2 + 5*x + 3*x^2])/(112612500*(3 + 2*x)^(3/2)) + (335723*Sqrt[2 + 5*
x + 3*x^2])/(80437500*Sqrt[3 + 2*x]) - ((386846 + 328339*x)*Sqrt[2 + 5*x + 3*x^2
])/(7507500*(3 + 2*x)^(7/2)) - ((8901 + 8399*x)*(2 + 5*x + 3*x^2)^(3/2))/(64350*
(3 + 2*x)^(11/2)) + ((94 + 119*x)*(2 + 5*x + 3*x^2)^(5/2))/(195*(3 + 2*x)^(15/2)
) - (335723*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])
/(53625000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (594851*Sqrt[-2 - 5*x - 3*x^2]*Ellip
ticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(75075000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]
)

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Rubi in Sympy [A]  time = 75.5675, size = 245, normalized size = 0.94 \[ - \frac{335723 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{160875000 \sqrt{3 x^{2} + 5 x + 2}} + \frac{594851 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{225225000 \sqrt{3 x^{2} + 5 x + 2}} + \frac{335723 \sqrt{3 x^{2} + 5 x + 2}}{80437500 \sqrt{2 x + 3}} + \frac{594851 \sqrt{3 x^{2} + 5 x + 2}}{112612500 \left (2 x + 3\right )^{\frac{3}{2}}} - \frac{\left (985017 x + 1160538\right ) \sqrt{3 x^{2} + 5 x + 2}}{22522500 \left (2 x + 3\right )^{\frac{7}{2}}} - \frac{\left (25197 x + 26703\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{193050 \left (2 x + 3\right )^{\frac{11}{2}}} + \frac{\left (595 x + 470\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{975 \left (2 x + 3\right )^{\frac{15}{2}}} \]

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)**(17/2),x)

[Out]

-335723*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)
/(160875000*sqrt(3*x**2 + 5*x + 2)) + 594851*sqrt(-9*x**2 - 15*x - 6)*elliptic_f
(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(225225000*sqrt(3*x**2 + 5*x + 2)) + 33572
3*sqrt(3*x**2 + 5*x + 2)/(80437500*sqrt(2*x + 3)) + 594851*sqrt(3*x**2 + 5*x + 2
)/(112612500*(2*x + 3)**(3/2)) - (985017*x + 1160538)*sqrt(3*x**2 + 5*x + 2)/(22
522500*(2*x + 3)**(7/2)) - (25197*x + 26703)*(3*x**2 + 5*x + 2)**(3/2)/(193050*(
2*x + 3)**(11/2)) + (595*x + 470)*(3*x**2 + 5*x + 2)**(5/2)/(975*(2*x + 3)**(15/
2))

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Mathematica [A]  time = 0.846446, size = 237, normalized size = 0.91 \[ -\frac{2 (2 x+3)^7 \left (9400244 \left (3 x^2+5 x+2\right )-1131016 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+4700122 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )-8 \left (3 x^2+5 x+2\right ) \left (300807808 x^7+3348834304 x^6+17742950508 x^5+46830142120 x^4+67557035830 x^3+55283449932 x^2+24502214271 x+4641518352\right )}{4504500000 (2 x+3)^{15/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(17/2),x]

[Out]

-(-8*(2 + 5*x + 3*x^2)*(4641518352 + 24502214271*x + 55283449932*x^2 + 675570358
30*x^3 + 46830142120*x^4 + 17742950508*x^5 + 3348834304*x^6 + 300807808*x^7) + 2
*(3 + 2*x)^7*(9400244*(2 + 5*x + 3*x^2) + 4700122*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)
]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 +
2*x]], 3/5] - 1131016*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]))/(4504500000*(3
+ 2*x)^(15/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.055, size = 809, normalized size = 3.1 \[ \text{result too large to display} \]

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)^(17/2),x)

[Out]

1/11261250000*(1444240406040*x+838916736*15^(1/2)*EllipticF(1/5*15^(1/2)*(3+2*x)
^(1/2),1/3*15^(1/2))*x^6*(-30*x-20)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)[phone]
4*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^6*(-30*x-20)^(1/
2)*(3+2*x)^(1/2)*(-2-2*x)^(1/2)[phone]*15^(1/2)*EllipticE(1/5*15^(1/2)*(3+2*x
)^(1/2),1/3*15^(1/2))*x^7*(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^7*(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^5*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)[phone]
920*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^3*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)+9437
813280*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^4*(3+2*x)^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)+1
4213168928*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)[phone]*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)*EllipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))*x^2*(-3
0*x-20)^(1/2)*(3+2*x)^(1/2)*(-2-2*x)^(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*(-
2-2*x)^(1/2)*(-30*x-20)^(1/2)*(3+2*x)^(1/2)[phone]*x^2+9446154382120*x^5+
4718056950160*x^6+1411492773200*x^7+11945916263720*x^4+9700759282660*x^3+2310108
39040*x^8+18048468480*x^9+1365112278*(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]*(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*x^2+5*x+2)^(1/2)/(3+2*x)^(15/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{17}{2}}}\,{d x} \]

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)^(17/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(17/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )} \sqrt{2 \, x + 3}}, x\right ) \]

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)^(17/2),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)
/((256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x
^2 + 34992*x + 6561)*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)**(17/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{17}{2}}}\,{d x} \]

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)^(17/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(17/2), x)