3.2598 \(\int \frac{(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{5/2}} \, dx\)

Optimal. Leaf size=205 \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{5/2}}{9 (2 x+3)^{3/2}}+\frac{5 (121 x+745) \left (3 x^2+5 x+2\right )^{3/2}}{126 \sqrt{2 x+3}}+\frac{5}{756} (326-6957 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}+\frac{306175 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1512 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{33335 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{216 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(5*(326 - 6957*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/756 + (5*(745 + 121*x)*(2
+ 5*x + 3*x^2)^(3/2))/(126*Sqrt[3 + 2*x]) - ((21 + x)*(2 + 5*x + 3*x^2)^(5/2))/
(9*(3 + 2*x)^(3/2)) - (33335*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqr
t[1 + x]], -2/3])/(216*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (306175*Sqrt[-2 - 5*x -
3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1512*Sqrt[3]*Sqrt[2 + 5*x
+ 3*x^2])

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Rubi [A]  time = 0.412931, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{(x+21) \left (3 x^2+5 x+2\right )^{5/2}}{9 (2 x+3)^{3/2}}+\frac{5 (121 x+745) \left (3 x^2+5 x+2\right )^{3/2}}{126 \sqrt{2 x+3}}+\frac{5}{756} (326-6957 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}+\frac{306175 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{1512 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{33335 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{216 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(5*(326 - 6957*x)*Sqrt[3 + 2*x]*Sqrt[2 + 5*x + 3*x^2])/756 + (5*(745 + 121*x)*(2
+ 5*x + 3*x^2)^(3/2))/(126*Sqrt[3 + 2*x]) - ((21 + x)*(2 + 5*x + 3*x^2)^(5/2))/
(9*(3 + 2*x)^(3/2)) - (33335*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqr
t[1 + x]], -2/3])/(216*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) + (306175*Sqrt[-2 - 5*x -
3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(1512*Sqrt[3]*Sqrt[2 + 5*x
+ 3*x^2])

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Rubi in Sympy [A]  time = 59.1892, size = 196, normalized size = 0.96 \[ \frac{\left (- 104355 x + 4890\right ) \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}{2268} - \frac{33335 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{648 \sqrt{3 x^{2} + 5 x + 2}} + \frac{306175 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{4536 \sqrt{3 x^{2} + 5 x + 2}} + \frac{5 \left (363 x + 2235\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}}{378 \sqrt{2 x + 3}} - \frac{\left (3 x + 63\right ) \left (3 x^{2} + 5 x + 2\right )^{\frac{5}{2}}}{27 \left (2 x + 3\right )^{\frac{3}{2}}} \]

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)**(5/2),x)

[Out]

(-104355*x + 4890)*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)/2268 - 33335*sqrt(-9*x**
2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(648*sqrt(3*x**2 +
5*x + 2)) + 306175*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(2)*sqrt(6*x +
6)/2), -2/3)/(4536*sqrt(3*x**2 + 5*x + 2)) + 5*(363*x + 2235)*(3*x**2 + 5*x + 2)
**(3/2)/(378*sqrt(2*x + 3)) - (3*x + 63)*(3*x**2 + 5*x + 2)**(5/2)/(27*(2*x + 3)
**(3/2))

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Mathematica [A]  time = 0.504449, size = 202, normalized size = 0.99 \[ -\frac{13608 x^7-38232 x^6-234684 x^5-561564 x^4+120594 x^3+2607724 x^2+3207982 x-49640 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+233345 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} (2 x+3)^{5/2} \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+1099572}{4536 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(1099572 + 3207982*x + 2607724*x^2 + 120594*x^3 - 561564*x^4 - 234684*x^5 - 382
32*x^6 + 13608*x^7 + 233345*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(5/2)*Sqrt
[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 49640*Sq
rt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(5/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*Elliptic
F[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/(4536*(3 + 2*x)^(3/2)*Sqrt[2 + 5*x + 3*
x^2])

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Maple [A]  time = 0.026, size = 235, normalized size = 1.2 \[{\frac{1}{9072} \left ( -27216\,{x}^{7}+29132\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+93338\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+76464\,{x}^{6}+43698\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +140007\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +469368\,{x}^{5}+1123128\,{x}^{4}+5359092\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+3401136 \right ) \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

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)^(5/2),x)

[Out]

1/9072*(-27216*x^7+29132*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)+93338*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)+76464*x^6+43698*(3+2*x)^(1/2)*15^(1/2)*(-2-2*x)^(1/2)*(-30*x-20)^(1/2)*El
lipticF(1/5*15^(1/2)*(3+2*x)^(1/2),1/3*15^(1/2))+140007*(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)
)+469368*x^5+1123128*x^4+5359092*x^3+12518772*x^2+11318256*x+3401136)/(3*x^2+5*x
+2)^(1/2)/(3+2*x)^(3/2)

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

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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (9 \, x^{5} - 15 \, x^{4} - 113 \, x^{3} - 165 \, x^{2} - 96 \, x - 20\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left (4 \, x^{2} + 12 \, x + 9\right )} \sqrt{2 \, x + 3}}, x\right ) \]

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)^(5/2),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)
/((4*x^2 + 12*x + 9)*sqrt(2*x + 3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{20 \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{96 x \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{165 x^{2} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{113 x^{3} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \left (- \frac{15 x^{4} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\right )\, dx - \int \frac{9 x^{5} \sqrt{3 x^{2} + 5 x + 2}}{4 x^{2} \sqrt{2 x + 3} + 12 x \sqrt{2 x + 3} + 9 \sqrt{2 x + 3}}\, dx \]

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)**(5/2),x)

[Out]

-Integral(-20*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3)
+ 9*sqrt(2*x + 3)), x) - Integral(-96*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x
+ 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x) - Integral(-165*x**2*sqrt(3*x**
2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x) -
Integral(-113*x**3*sqrt(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x
+ 3) + 9*sqrt(2*x + 3)), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2)/(4*x**2*
sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3)), x) - Integral(9*x**5*sqrt
(3*x**2 + 5*x + 2)/(4*x**2*sqrt(2*x + 3) + 12*x*sqrt(2*x + 3) + 9*sqrt(2*x + 3))
, x)

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

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)^(5/2),x, algorithm="giac")

[Out]

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