3.2603 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{15/2}} \, dx\)

Optimal. Leaf size=234 \[ \frac{(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac{(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac{(17833 x+21492) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac{5083 \sqrt{3 x^2+5 x+2}}{247500 \sqrt{2 x+3}}-\frac{9421 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{231000 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{5083 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{165000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-5083*Sqrt[2 + 5*x + 3*x^2])/(247500*Sqrt[3 + 2*x]) + ((21492 + 17833*x)*Sqrt[2
+ 5*x + 3*x^2])/(346500*(3 + 2*x)^(5/2)) + ((73 - 33*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(6930*(3 + 2*x)^(9/2)) + ((8 + 9*x)*(2 + 5*x + 3*x^2)^(5/2))/(11*(3 + 2*x)^(13
/2)) + (5083*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3]
)/(165000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (9421*Sqrt[-2 - 5*x - 3*x^2]*Elliptic
F[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(231000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.485429, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{(9 x+8) \left (3 x^2+5 x+2\right )^{5/2}}{11 (2 x+3)^{13/2}}+\frac{(73-33 x) \left (3 x^2+5 x+2\right )^{3/2}}{6930 (2 x+3)^{9/2}}+\frac{(17833 x+21492) \sqrt{3 x^2+5 x+2}}{346500 (2 x+3)^{5/2}}-\frac{5083 \sqrt{3 x^2+5 x+2}}{247500 \sqrt{2 x+3}}-\frac{9421 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{231000 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{5083 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{165000 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]

[Out]

(-5083*Sqrt[2 + 5*x + 3*x^2])/(247500*Sqrt[3 + 2*x]) + ((21492 + 17833*x)*Sqrt[2
+ 5*x + 3*x^2])/(346500*(3 + 2*x)^(5/2)) + ((73 - 33*x)*(2 + 5*x + 3*x^2)^(3/2)
)/(6930*(3 + 2*x)^(9/2)) + ((8 + 9*x)*(2 + 5*x + 3*x^2)^(5/2))/(11*(3 + 2*x)^(13
/2)) + (5083*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3]
)/(165000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (9421*Sqrt[-2 - 5*x - 3*x^2]*Elliptic
F[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(231000*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 68.0175, size = 219, normalized size = 0.94 \[ \frac{\left (- 429 x + 949\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{90090 \left (2 x + 3\right )^{\frac{9}{2}}} + \frac{5083 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{495000 \sqrt{3 x^{2} + 5 x + 2}} - \frac{9421 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{693000 \sqrt{3 x^{2} + 5 x + 2}} - \frac{5083 \sqrt{3 x^{2} + 5 x + 2}}{247500 \sqrt{2 x + 3}} + \frac{\left (231829 x + 279396\right ) \sqrt{3 x^{2} + 5 x + 2}}{4504500 \left (2 x + 3\right )^{\frac{5}{2}}} + \frac{\left (585 x + 520\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{715 \left (2 x + 3\right )^{\frac{13}{2}}} \]

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)**(15/2),x)

[Out]

(-429*x + 949)*(3*x**2 + 5*x + 2)**(3/2)/(90090*(2*x + 3)**(9/2)) + 5083*sqrt(-9
*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(495000*sqrt(3
*x**2 + 5*x + 2)) - 9421*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6
*x + 6)/2), -2/3)/(693000*sqrt(3*x**2 + 5*x + 2)) - 5083*sqrt(3*x**2 + 5*x + 2)/
(247500*sqrt(2*x + 3)) + (231829*x + 279396)*sqrt(3*x**2 + 5*x + 2)/(4504500*(2*
x + 3)**(5/2)) + (585*x + 520)*(3*x**2 + 5*x + 2)**(5/2)/(715*(2*x + 3)**(13/2))

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Mathematica [A]  time = 0.801482, size = 232, normalized size = 0.99 \[ -\frac{8 \left (3 x^2+5 x+2\right ) \left (2277184 x^6+6409516 x^5+12953760 x^4+33648370 x^3+54318160 x^2+41339721 x+11865789\right )-4 (2 x+3)^6 \left (71162 \left (3 x^2+5 x+2\right )-7318 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+35581 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )}{13860000 (2 x+3)^{13/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((5 - x)*(2 + 5*x + 3*x^2)^(5/2))/(3 + 2*x)^(15/2),x]

[Out]

-(8*(2 + 5*x + 3*x^2)*(11865789 + 41339721*x + 54318160*x^2 + 33648370*x^3 + 129
53760*x^4 + 6409516*x^5 + 2277184*x^6) - 4*(3 + 2*x)^6*(71162*(2 + 5*x + 3*x^2)
+ 35581*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] - 7318*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]/Sq
rt[3 + 2*x]], 3/5]))/(13860000*(3 + 2*x)^(13/2)*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.054, size = 710, 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)^(15/2),x)

[Out]

-1/34650000*(737536*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)+2277184*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)+6637824*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^5*(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^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^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
)[phone]*x^8+56006640*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]*15^(1/2)*Ellipt
icE(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+33603984*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]*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]*x^6+8400996*(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))+2593854
9*(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]*x^5+7142077000*x^4+9258134060*x^3+7018645
840*x^2+2840167740*x+474631560)/(3*x^2+5*x+2)^(1/2)/(3+2*x)^(13/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{15}{2}}}\,{d x} \]

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)^(15/2),x, algorithm="maxima")

[Out]

-integrate((3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )} \sqrt{2 \, x + 3}}, x\right ) \]

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)^(15/2),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)
/((128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x +
2187)*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)**(15/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (x - 5\right )}}{{\left (2 \, x + 3\right )}^{\frac{15}{2}}}\,{d x} \]

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)^(15/2),x, algorithm="giac")

[Out]

integrate(-(3*x^2 + 5*x + 2)^(5/2)*(x - 5)/(2*x + 3)^(15/2), x)