3.2612 \(\int \frac{(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=197 \[ -\frac{2 (139 x+121) (2 x+3)^{5/2}}{3 \sqrt{3 x^2+5 x+2}}+\frac{308}{5} \sqrt{3 x^2+5 x+2} (2 x+3)^{3/2}+\frac{12068}{135} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}-\frac{6034 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{34174 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{135 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*(3 + 2*x)^(5/2)*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (12068*Sqrt[3 + 2
*x]*Sqrt[2 + 5*x + 3*x^2])/135 + (308*(3 + 2*x)^(3/2)*Sqrt[2 + 5*x + 3*x^2])/5 +
(34174*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(13
5*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (6034*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(27*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.408446, 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 \[ -\frac{2 (139 x+121) (2 x+3)^{5/2}}{3 \sqrt{3 x^2+5 x+2}}+\frac{308}{5} \sqrt{3 x^2+5 x+2} (2 x+3)^{3/2}+\frac{12068}{135} \sqrt{3 x^2+5 x+2} \sqrt{2 x+3}-\frac{6034 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}}+\frac{34174 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{135 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3 + 2*x)^(5/2)*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (12068*Sqrt[3 + 2
*x]*Sqrt[2 + 5*x + 3*x^2])/135 + (308*(3 + 2*x)^(3/2)*Sqrt[2 + 5*x + 3*x^2])/5 +
(34174*Sqrt[-2 - 5*x - 3*x^2]*EllipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(13
5*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2]) - (6034*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin
[Sqrt[3]*Sqrt[1 + x]], -2/3])/(27*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 55.1346, size = 189, normalized size = 0.96 \[ - \frac{2 \left (2 x + 3\right )^{\frac{5}{2}} \left (139 x + 121\right )}{3 \sqrt{3 x^{2} + 5 x + 2}} + \frac{308 \left (2 x + 3\right )^{\frac{3}{2}} \sqrt{3 x^{2} + 5 x + 2}}{5} + \frac{12068 \sqrt{2 x + 3} \sqrt{3 x^{2} + 5 x + 2}}{135} + \frac{34174 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{405 \sqrt{3 x^{2} + 5 x + 2}} - \frac{6034 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(3/2),x)

[Out]

-2*(2*x + 3)**(5/2)*(139*x + 121)/(3*sqrt(3*x**2 + 5*x + 2)) + 308*(2*x + 3)**(3
/2)*sqrt(3*x**2 + 5*x + 2)/5 + 12068*sqrt(2*x + 3)*sqrt(3*x**2 + 5*x + 2)/135 +
34174*sqrt(-9*x**2 - 15*x - 6)*elliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(
405*sqrt(3*x**2 + 5*x + 2)) - 6034*sqrt(-9*x**2 - 15*x - 6)*elliptic_f(asin(sqrt
(2)*sqrt(6*x + 6)/2), -2/3)/(81*sqrt(3*x**2 + 5*x + 2))

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Mathematica [A]  time = 0.511766, size = 192, normalized size = 0.97 \[ -\frac{2 \left (\left (432 x^4-936 x^3-30120 x^2+13231 x+39553\right ) \sqrt{2 x+3}+8036 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )-17087 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )\right )}{405 (2 x+3) \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(Sqrt[3 + 2*x]*(39553 + 13231*x - 30120*x^2 - 936*x^3 + 432*x^4) - 17087*Sqr
t[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticE[Arc
Sin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5] + 8036*Sqrt[5]*Sqrt[(1 + x)/(3 + 2*x)]*(3 + 2
*x)^2*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])
)/(405*(3 + 2*x)*Sqrt[2 + 5*x + 3*x^2])

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Maple [A]  time = 0.068, size = 147, normalized size = 0.8 \[{\frac{1}{12150\,{x}^{3}+38475\,{x}^{2}+38475\,x+12150}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 2002\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -17087\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) -4320\,{x}^{4}+9360\,{x}^{3}-724020\,{x}^{2}-1841010\,x-1079010 \right ) } \]

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)^(3/2),x)

[Out]

1/2025*(3+2*x)^(1/2)*(3*x^2+5*x+2)^(1/2)*(2002*(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))-17087*(
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))-4320*x^4+9360*x^3-724020*x^2-1841010*x-1079010)/(6*x^3+
19*x^2+19*x+6)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (8 \, x^{4} - 4 \, x^{3} - 126 \, x^{2} - 243 \, x - 135\right )} \sqrt{2 \, x + 3}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(8*x^4 - 4*x^3 - 126*x^2 - 243*x - 135)*sqrt(2*x + 3)/(3*x^2 + 5*x + 2
)^(3/2), 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)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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