3.2618 \(\int \frac{5-x}{(3+2 x)^{5/2} \left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=197 \[ -\frac{6 (47 x+37)}{5 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}}-\frac{14876 \sqrt{3 x^2+5 x+2}}{375 \sqrt{2 x+3}}-\frac{2516 \sqrt{3 x^2+5 x+2}}{75 (2 x+3)^{3/2}}-\frac{1258 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{7438 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)^(3/2)*Sqrt[2 + 5*x + 3*x^2]) - (2516*Sqrt[2 + 5*x
+ 3*x^2])/(75*(3 + 2*x)^(3/2)) - (14876*Sqrt[2 + 5*x + 3*x^2])/(375*Sqrt[3 + 2*x
]) + (7438*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/
(125*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (1258*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[Arc
Sin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.423953, antiderivative size = 197, 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 \[ -\frac{6 (47 x+37)}{5 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}}-\frac{14876 \sqrt{3 x^2+5 x+2}}{375 \sqrt{2 x+3}}-\frac{2516 \sqrt{3 x^2+5 x+2}}{75 (2 x+3)^{3/2}}-\frac{1258 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{25 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{7438 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{125 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-6*(37 + 47*x))/(5*(3 + 2*x)^(3/2)*Sqrt[2 + 5*x + 3*x^2]) - (2516*Sqrt[2 + 5*x
+ 3*x^2])/(75*(3 + 2*x)^(3/2)) - (14876*Sqrt[2 + 5*x + 3*x^2])/(375*Sqrt[3 + 2*x
]) + (7438*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/
(125*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (1258*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[Arc
Sin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(25*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 54.4561, size = 187, normalized size = 0.95 \[ \frac{7438 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{375 \sqrt{3 x^{2} + 5 x + 2}} - \frac{1258 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{75 \sqrt{3 x^{2} + 5 x + 2}} - \frac{14876 \sqrt{3 x^{2} + 5 x + 2}}{375 \sqrt{2 x + 3}} - \frac{2 \left (141 x + 111\right )}{5 \left (2 x + 3\right )^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}} - \frac{2516 \sqrt{3 x^{2} + 5 x + 2}}{75 \left (2 x + 3\right )^{\frac{3}{2}}} \]

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)**(3/2),x)

[Out]

7438*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(3
75*sqrt(3*x**2 + 5*x + 2)) - 1258*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt(
2)*sqrt(6*x + 6)/2), -2/3)/(75*sqrt(3*x**2 + 5*x + 2)) - 14876*sqrt(3*x**2 + 5*x
+ 2)/(375*sqrt(2*x + 3)) - 2*(141*x + 111)/(5*(2*x + 3)**(3/2)*sqrt(3*x**2 + 5*
x + 2)) - 2516*sqrt(3*x**2 + 5*x + 2)/(75*(2*x + 3)**(3/2))

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Mathematica [A]  time = 0.555298, size = 177, normalized size = 0.9 \[ \frac{2 \left (-18870 x^2-42025 x-1832 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{5/2} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )+3719 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^{5/2} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-20905\right )}{375 (2 x+3)^{3/2} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-20905 - 42025*x - 18870*x^2 + 3719*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
] - 1832*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^(5/2)*Sqrt[(2 + 3*x)/(3 + 2*x
)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5]))/(375*(3 + 2*x)^(3/2)*Sqrt[2
+ 5*x + 3*x^2])

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Maple [A]  time = 0.033, size = 215, normalized size = 1.1 \[{\frac{1}{1875} \left ( 1148\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}-7438\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+1722\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -11157\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -446280\,{x}^{3}-1601920\,{x}^{2}-1833470\,x-655330 \right ) \left ( 3+2\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{3\,{x}^{2}+5\,x+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1875*(1148*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)-7438*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)+1722*(
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))-11157*(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))-446280*x^3-1601920*x^2-
1833470*x-655330)/(3+2*x)^(3/2)/(3*x^2+5*x+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x - 5}{{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(x - 5)/((12*x^4 + 56*x^3 + 95*x^2 + 69*x + 18)*sqrt(3*x^2 + 5*x + 2)*
sqrt(2*x + 3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x}{12 x^{4} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 56 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 95 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 69 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 18 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac{5}{12 x^{4} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 56 x^{3} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 95 x^{2} \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 69 x \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2} + 18 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}\right )\, dx \]

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)**(3/2),x)

[Out]

-Integral(x/(12*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 56*x**3*sqrt(2*x + 3
)*sqrt(3*x**2 + 5*x + 2) + 95*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 69*x*s
qrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 18*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)),
x) - Integral(-5/(12*x**4*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 56*x**3*sqrt(2*
x + 3)*sqrt(3*x**2 + 5*x + 2) + 95*x**2*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 6
9*x*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2) + 18*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x +
2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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