3.2621 \(\int \frac{(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=175 \[ -\frac{2 (139 x+121) (2 x+3)^{5/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{28 (1177 x+1018) \sqrt{2 x+3}}{27 \sqrt{3 x^2+5 x+2}}+\frac{41860 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{31892 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

[Out]

(-2*(3 + 2*x)^(5/2)*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (28*Sqrt[3 + 2*
x]*(1018 + 1177*x))/(27*Sqrt[2 + 5*x + 3*x^2]) - (31892*Sqrt[-2 - 5*x - 3*x^2]*E
llipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(27*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])
+ (41860*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2
7*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi [A]  time = 0.336706, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{2 (139 x+121) (2 x+3)^{5/2}}{9 \left (3 x^2+5 x+2\right )^{3/2}}+\frac{28 (1177 x+1018) \sqrt{2 x+3}}{27 \sqrt{3 x^2+5 x+2}}+\frac{41860 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{31892 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{27 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(3 + 2*x)^(5/2)*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) + (28*Sqrt[3 + 2*
x]*(1018 + 1177*x))/(27*Sqrt[2 + 5*x + 3*x^2]) - (31892*Sqrt[-2 - 5*x - 3*x^2]*E
llipticE[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(27*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])
+ (41860*Sqrt[-2 - 5*x - 3*x^2]*EllipticF[ArcSin[Sqrt[3]*Sqrt[1 + x]], -2/3])/(2
7*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])

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Rubi in Sympy [A]  time = 48.2359, size = 168, normalized size = 0.96 \[ - \frac{2 \left (2 x + 3\right )^{\frac{5}{2}} \left (139 x + 121\right )}{9 \left (3 x^{2} + 5 x + 2\right )^{\frac{3}{2}}} + \frac{4 \sqrt{2 x + 3} \left (8239 x + 7126\right )}{27 \sqrt{3 x^{2} + 5 x + 2}} - \frac{31892 \sqrt{- 9 x^{2} - 15 x - 6} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} + \frac{41860 \sqrt{- 9 x^{2} - 15 x - 6} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{6 x + 6}}{2} \right )}\middle | - \frac{2}{3}\right )}{81 \sqrt{3 x^{2} + 5 x + 2}} \]

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)**(5/2),x)

[Out]

-2*(2*x + 3)**(5/2)*(139*x + 121)/(9*(3*x**2 + 5*x + 2)**(3/2)) + 4*sqrt(2*x + 3
)*(8239*x + 7126)/(27*sqrt(3*x**2 + 5*x + 2)) - 31892*sqrt(-9*x**2 - 15*x - 6)*e
lliptic_e(asin(sqrt(2)*sqrt(6*x + 6)/2), -2/3)/(81*sqrt(3*x**2 + 5*x + 2)) + 418
60*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 = 0.544145, size = 196, normalized size = 1.12 \[ -\frac{\frac{63784 \left (3 x^2+5 x+2\right )}{\sqrt{2 x+3}}-\frac{6 \sqrt{2 x+3} \left (47766 x^3+118690 x^2+96107 x+25237\right )}{3 x^2+5 x+2}-\frac{6776 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}+\frac{31892 (x+1) \sqrt{\frac{3 x+2}{2 x+3}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{\sqrt{\frac{x+1}{10 x+15}}}}{81 \sqrt{3 x^2+5 x+2}} \]

Antiderivative was successfully verified.

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

[Out]

-((63784*(2 + 5*x + 3*x^2))/Sqrt[3 + 2*x] - (6*Sqrt[3 + 2*x]*(25237 + 96107*x +
118690*x^2 + 47766*x^3))/(2 + 5*x + 3*x^2) + (31892*(1 + x)*Sqrt[(2 + 3*x)/(3 +
2*x)]*EllipticE[ArcSin[Sqrt[5/3]/Sqrt[3 + 2*x]], 3/5])/Sqrt[(1 + x)/(15 + 10*x)]
- (6776*(1 + x)*Sqrt[(2 + 3*x)/(3 + 2*x)]*EllipticF[ArcSin[Sqrt[5/3]/Sqrt[3 + 2
*x]], 3/5])/Sqrt[(1 + x)/(15 + 10*x)])/(81*Sqrt[2 + 5*x + 3*x^2])

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Maple [B]  time = 0.033, size = 326, normalized size = 1.9 \[{\frac{2}{405\, \left ( 1+x \right ) ^{2} \left ( 2+3\,x \right ) ^{2}}\sqrt{3\,{x}^{2}+5\,x+2} \left ( 23919\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+7476\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ){x}^{2}\sqrt{-30\,x-20}\sqrt{3+2\,x}\sqrt{-2-2\,x}+39865\,\sqrt{15}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+12460\,\sqrt{15}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) x\sqrt{-2-2\,x}\sqrt{-30\,x-20}\sqrt{3+2\,x}+15946\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticE} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +4984\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-30\,x-20}{\it EllipticF} \left ( 1/5\,\sqrt{15}\sqrt{3+2\,x},1/3\,\sqrt{15} \right ) +1432980\,{x}^{4}+5710170\,{x}^{3}+8224260\,{x}^{2}+5081925\,x+1135665 \right ){\frac{1}{\sqrt{3+2\,x}}}} \]

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]

2/405*(3*x^2+5*x+2)^(1/2)*(23919*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)+7476*15^(1/2)*Ell
ipticF(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)+39865*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)+12460*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)+15946*(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))+4984*(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))+1432980*x^4
+5710170*x^3+8224260*x^2+5081925*x+1135665)/(1+x)^2/(2+3*x)^2/(3+2*x)^(1/2)

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

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)^(5/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (8 \, x^{4} - 4 \, x^{3} - 126 \, x^{2} - 243 \, x - 135\right )} \sqrt{2 \, x + 3}}{{\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}, x\right ) \]

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)^(5/2),x, algorithm="fricas")

[Out]

integral(-(8*x^4 - 4*x^3 - 126*x^2 - 243*x - 135)*sqrt(2*x + 3)/((9*x^4 + 30*x^3
+ 37*x^2 + 20*x + 4)*sqrt(3*x^2 + 5*x + 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)**(5/2),x)

[Out]

Timed out

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

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)^(5/2),x, algorithm="giac")

[Out]

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