3.2627 \(\int \frac{5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=229 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{190792 \sqrt{3 x^2+5 x+2}}{1875 \sqrt{2 x+3}}+\frac{61672 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^{3/2}}+\frac{12 (737 x+652)}{25 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}}+\frac{30836 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{95396 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3/2)) + (12*(652 + 737*x)
)/(25*(3 + 2*x)^(3/2)*Sqrt[2 + 5*x + 3*x^2]) + (61672*Sqrt[2 + 5*x + 3*x^2])/(37
5*(3 + 2*x)^(3/2)) + (190792*Sqrt[2 + 5*x + 3*x^2])/(1875*Sqrt[3 + 2*x]) - (9539
6*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(625*Sqrt
[3]*Sqrt[2 + 5*x + 3*x^2]) + (30836*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt
[3]*Sqrt[1 + x]], -2/3])/(125*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.505582, antiderivative size = 229, 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 \[ -\frac{2 (47 x+37)}{5 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}}+\frac{190792 \sqrt{3 x^2+5 x+2}}{1875 \sqrt{2 x+3}}+\frac{61672 \sqrt{3 x^2+5 x+2}}{375 (2 x+3)^{3/2}}+\frac{12 (737 x+652)}{25 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}}+\frac{30836 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{95396 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{625 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(37 + 47*x))/(5*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3/2)) + (12*(652 + 737*x)
)/(25*(3 + 2*x)^(3/2)*Sqrt[2 + 5*x + 3*x^2]) + (61672*Sqrt[2 + 5*x + 3*x^2])/(37
5*(3 + 2*x)^(3/2)) + (190792*Sqrt[2 + 5*x + 3*x^2])/(1875*Sqrt[3 + 2*x]) - (9539
6*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(625*Sqrt
[3]*Sqrt[2 + 5*x + 3*x^2]) + (30836*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt
[3]*Sqrt[1 + x]], -2/3])/(125*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 64.6441, size = 216, normalized size = 0.94 \[ - \frac{95396 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{1875 \sqrt{3 x^{2} + 5 x + 2}} + \frac{30836 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{375 \sqrt{3 x^{2} + 5 x + 2}} + \frac{190792 \sqrt{3 x^{2} + 5 x + 2}}{1875 \sqrt{2 x + 3}} - \frac{2 \left (141 x + 111\right )}{15 \left (2 x + 3\right )^{\frac{3}{2}} \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \left (6633 x + 5868\right )}{75 \left (2 x + 3\right )^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}} + \frac{61672 \sqrt{3 x^{2} + 5 x + 2}}{375 \left (2 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-95396*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/
(1875*sqrt(3*x**2 + 5*x + 2)) + 30836*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(s
qrt(2)*sqrt(6*x + 6)/2), -2/3)/(375*sqrt(3*x**2 + 5*x + 2)) + 190792*sqrt(3*x**2
+ 5*x + 2)/(1875*sqrt(2*x + 3)) - 2*(141*x + 111)/(15*(2*x + 3)**(3/2)*(3*x**2
+ 5*x + 2)**(3/2)) + 4*(6633*x + 5868)/(75*(2*x + 3)**(3/2)*sqrt(3*x**2 + 5*x +
2)) + 61672*sqrt(3*x**2 + 5*x + 2)/(375*(2*x + 3)**(3/2))

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Mathematica [A]  time = 0.924573, size = 222, normalized size = 0.97 \[ \frac{2 \left (1717128 x^5+9687072 x^4+21265294 x^3+22647906 x^2-2 (2 x+3) \left (3 x^2+5 x+2\right ) \left (47698 \left (3 x^2+5 x+2\right )-722 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+23849 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{3/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )[phone] x+2334397\right )}{1875 (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(2334397 + 11683203*x + 22647906*x^2 + 21265294*x^3 + 9687072*x^4 + 1717128*x
^5 - 2*(3 + 2*x)*(2 + 5*x + 3*x^2)*(47698*(2 + 5*x + 3*x^2) + 23849*Sqrt[5]*Sqrt
[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(3/2)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[ArcSin[S
qrt[5/3]/Sqrt[3 + 2*x]], 3/5] - 722*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])))
/(1875*(3 + 2*x)^(3/2)*(2 + 5*x + 3*x^2)^(3/2))

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Maple [B]  time = 0.038, size = 425, normalized size = 1.9 \[{\frac{2}{9375\, \left ( 1+x \right ) ^{2} \left ( 2+3\,x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 143094\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+88176\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{3}\sqrt{3+2\,x}\sqrt{-2-2\,x}\sqrt{-30\,x-20}+453131\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+279224\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+453131\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+279224\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+143094\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +88176\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +8585640\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+11671985 \right ) \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9375*(3*x^2+5*x+2)^(1/2)*(143094*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)+88176*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)+453131*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)+279224*15^(1/2)*Ellipt
icF(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)+453131*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)+279224*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)+143094*(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))+88176*(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))+8585640*x^
5+48435360*x^4+106326470*x^3+113239530*x^2+58416015*x+11671985)/(3+2*x)^(3/2)/(1
+x)^2/(2+3*x)^2

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

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)^(5/2)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x - 5}{{\left (36 \, x^{6} + 228 \, x^{5} + 589 \, x^{4} + 794 \, x^{3} + 589 \, x^{2} + 228 \, x + 36\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}, x\right ) \]

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)^(5/2)),x, algorithm="fricas")

[Out]

integral(-(x - 5)/((36*x^6 + 228*x^5 + 589*x^4 + 794*x^3 + 589*x^2 + 228*x + 36)
*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)**(5/2)/(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. \[ \int -\frac{x - 5}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (2 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(5/2)),x, algorithm="giac")

[Out]

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