3.2628 \(\int \frac{5-x}{(3+2 x)^{7/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=256 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{215096 \sqrt{3 x^2+5 x+2}}{15625 \sqrt{2 x+3}}+\frac{258536 \sqrt{3 x^2+5 x+2}}{3125 (2 x+3)^{3/2}}+\frac{87144 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)^{5/2}}+\frac{4 (2013 x+1858)}{25 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}}+\frac{129268 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3125 \sqrt{3 x^2+5 x+2}}-\frac{107548 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{15625 \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(3/2)) + (4*(1858 + 2013*x
))/(25*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2]) + (87144*Sqrt[2 + 5*x + 3*x^2])/(6
25*(3 + 2*x)^(5/2)) + (258536*Sqrt[2 + 5*x + 3*x^2])/(3125*(3 + 2*x)^(3/2)) + (2
15096*Sqrt[2 + 5*x + 3*x^2])/(15625*Sqrt[3 + 2*x]) - (107548*Sqrt[3]*Sqrt[-2 - 5
*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(15625*Sqrt[2 + 5*x +
3*x^2]) + (129268*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1
+ x]], -2/3])/(3125*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.591847, antiderivative size = 256, 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 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{215096 \sqrt{3 x^2+5 x+2}}{15625 \sqrt{2 x+3}}+\frac{258536 \sqrt{3 x^2+5 x+2}}{3125 (2 x+3)^{3/2}}+\frac{87144 \sqrt{3 x^2+5 x+2}}{625 (2 x+3)^{5/2}}+\frac{4 (2013 x+1858)}{25 (2 x+3)^{5/2} \sqrt{3 x^2+5 x+2}}+\frac{129268 \sqrt{3} \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{3125 \sqrt{3 x^2+5 x+2}}-\frac{107548 \sqrt{3} \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{15625 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

[In]  Int[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^(5/2)),x]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(3/2)) + (4*(1858 + 2013*x
))/(25*(3 + 2*x)^(5/2)*Sqrt[2 + 5*x + 3*x^2]) + (87144*Sqrt[2 + 5*x + 3*x^2])/(6
25*(3 + 2*x)^(5/2)) + (258536*Sqrt[2 + 5*x + 3*x^2])/(3125*(3 + 2*x)^(3/2)) + (2
15096*Sqrt[2 + 5*x + 3*x^2])/(15625*Sqrt[3 + 2*x]) - (107548*Sqrt[3]*Sqrt[-2 - 5
*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(15625*Sqrt[2 + 5*x +
3*x^2]) + (129268*Sqrt[3]*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1
+ x]], -2/3])/(3125*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 74.4348, size = 241, normalized size = 0.94 \[ - \frac{107548 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{15625 \sqrt{3 x^{2} + 5 x + 2}} + \frac{129268 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{3125 \sqrt{3 x^{2} + 5 x + 2}} + \frac{215096 \sqrt{3 x^{2} + 5 x + 2}}{15625 \sqrt{2 x + 3}} + \frac{258536 \sqrt{3 x^{2} + 5 x + 2}}{3125 \left (2 x + 3\right )^{\frac{3}{2}}} - \frac{2 \left (141 x + 111\right )}{15 \left (2 x + 3\right )^{\frac{5}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (6039 x + 5574\right )}{75 \left (2 x + 3\right )^{\frac{5}{2}} \sqrt{3 x^{2} + 5 x + 2}} + \frac{87144 \sqrt{3 x^{2} + 5 x + 2}}{625 \left (2 x + 3\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5-x)/(3+2*x)**(7/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

-107548*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)
/(15625*sqrt(3*x**2 + 5*x + 2)) + 129268*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asi
n(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(3125*sqrt(3*x**2 + 5*x + 2)) + 215096*sqrt(3*
x**2 + 5*x + 2)/(15625*sqrt(2*x + 3)) + 258536*sqrt(3*x**2 + 5*x + 2)/(3125*(2*x
+ 3)**(3/2)) - 2*(141*x + 111)/(15*(2*x + 3)**(5/2)*(3*x**2 + 5*x + 2)**(3/2))
+ 4*(6039*x + 5574)/(75*(2*x + 3)**(5/2)*sqrt(3*x**2 + 5*x + 2)) + 87144*sqrt(3*
x**2 + 5*x + 2)/(625*(2*x + 3)**(5/2))

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Mathematica [A]  time = 0.851403, size = 229, normalized size = 0.89 \[ \frac{2 \left (3871728 x^6+36155064 x^5+129381052 x^4+231620622 x^3+220795962 x^2-2 (2 x+3)^2 \left (3 x^2+5 x+2\right ) \left (53774 \left (3 x^2+5 x+2\right )+70064 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+26887 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )[phone] x+20514383\right )}{15625 (2 x+3)^{5/2} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(5 - x)/((3 + 2*x)^(7/2)*(2 + 5*x + 3*x^2)^(5/2)),x]

[Out]

(2*(20514383 + 106756189*x + 220795962*x^2 + 231620622*x^3 + 129381052*x^4 + 361
55064*x^5 + 3871728*x^6 - 2*(3 + 2*x)^2*(2 + 5*x + 3*x^2)*(53774*(2 + 5*x + 3*x^
2) + 26887*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] + 70064*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])))/(15625*(3 + 2*x)^(5/2)*(2 + 5*x + 3*x^2)^(3/2))

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Maple [B]  time = 0.037, size = 524, normalized size = 2.1 \[{\frac{2}{78125\, \left ( 1+x \right ) ^{2} \left ( 2+3\,x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 322644\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+1616376\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{4}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+1505672\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+7543088\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+2554265\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}[phone]\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+1855203\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+9294162\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}[phone]\,{x}^{6}+483966\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +2424564\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) [phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+102571915 \right ) \left ( 3+2\,x \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5-x)/(3+2*x)^(7/2)/(3*x^2+5*x+2)^(5/2),x)

[Out]

2/78125*(3*x^2+5*x+2)^(1/2)*(322644*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)+1616376*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)+1505672*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)+7543088*15^(1/2)*E
llipticF(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)+2554265*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)*Elli
pticF(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)+1855203*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)+9294162*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^6+483966*(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))+2424564*(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*1
5^(1/2))[phone]*x^5+646905260*x^4+1158103110*x^3+1103979810*x^2+533780945*x+1
02571915)/(3+2*x)^(5/2)/(1+x)^2/(2+3*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (2 \, x + 3\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^(7/2)),x, algorithm="maxima")

[Out]

-integrate((x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^(7/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x - 5}{{\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^(7/2)),x, algorithm="fricas")

[Out]

integral(-(x - 5)/((72*x^7 + 564*x^6 + 1862*x^5 + 3355*x^4 + 3560*x^3 + 2223*x^2
+ 756*x + 108)*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+2*x)**(7/2)/(3*x**2+5*x+2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (2 \, x + 3\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^(7/2)),x, algorithm="giac")

[Out]

integrate(-(x - 5)/((3*x^2 + 5*x + 2)^(5/2)*(2*x + 3)^(7/2)), x)