3.1364 \(\int \frac{1}{(3-2 x)^{5/2} \sqrt{1-3 x+x^2}} \, dx\)

Optimal. Leaf size=79 \[ -\frac{4 \sqrt{x^2-3 x+1}}{15 (3-2 x)^{3/2}}-\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{15 \sqrt [4]{5} \sqrt{x^2-3 x+1}} \]

[Out]

(-4*Sqrt[1 - 3*x + x^2])/(15*(3 - 2*x)^(3/2)) - (2*Sqrt[-1 + 3*x - x^2]*Elliptic
F[ArcSin[Sqrt[3 - 2*x]/5^(1/4)], -1])/(15*5^(1/4)*Sqrt[1 - 3*x + x^2])

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Rubi [A]  time = 0.103523, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{4 \sqrt{x^2-3 x+1}}{15 (3-2 x)^{3/2}}-\frac{2 \sqrt{-x^2+3 x-1} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{3-2 x}}{\sqrt [4]{5}}\right )\right |-1\right )}{15 \sqrt [4]{5} \sqrt{x^2-3 x+1}} \]

Antiderivative was successfully verified.

[In]  Int[1/((3 - 2*x)^(5/2)*Sqrt[1 - 3*x + x^2]),x]

[Out]

(-4*Sqrt[1 - 3*x + x^2])/(15*(3 - 2*x)^(3/2)) - (2*Sqrt[-1 + 3*x - x^2]*Elliptic
F[ArcSin[Sqrt[3 - 2*x]/5^(1/4)], -1])/(15*5^(1/4)*Sqrt[1 - 3*x + x^2])

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Rubi in Sympy [A]  time = 13.302, size = 82, normalized size = 1.04 \[ - \frac{2 \sqrt [4]{5} \sqrt{- \frac{x^{2}}{5} + \frac{3 x}{5} - \frac{1}{5}} F\left (\operatorname{asin}{\left (\frac{5^{\frac{3}{4}} \sqrt{- 2 x + 3}}{5} \right )}\middle | -1\right )}{15 \sqrt{x^{2} - 3 x + 1}} - \frac{4 \sqrt{x^{2} - 3 x + 1}}{15 \left (- 2 x + 3\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-2*x)**(5/2)/(x**2-3*x+1)**(1/2),x)

[Out]

-2*5**(1/4)*sqrt(-x**2/5 + 3*x/5 - 1/5)*elliptic_f(asin(5**(3/4)*sqrt(-2*x + 3)/
5), -1)/(15*sqrt(x**2 - 3*x + 1)) - 4*sqrt(x**2 - 3*x + 1)/(15*(-2*x + 3)**(3/2)
)

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Mathematica [A]  time = 0.177516, size = 79, normalized size = 1. \[ \frac{2}{75} \sqrt{x^2-3 x+1} \left (\frac{5^{3/4} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{5}}{\sqrt{3-2 x}}\right )\right |-1\right )}{(3-2 x) \sqrt{\frac{x^2-3 x+1}{(3-2 x)^2}}}-\frac{10}{(3-2 x)^{3/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/((3 - 2*x)^(5/2)*Sqrt[1 - 3*x + x^2]),x]

[Out]

(2*Sqrt[1 - 3*x + x^2]*(-10/(3 - 2*x)^(3/2) + (5^(3/4)*EllipticF[ArcSin[5^(1/4)/
Sqrt[3 - 2*x]], -1])/((3 - 2*x)*Sqrt[(1 - 3*x + x^2)/(3 - 2*x)^2])))/75

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Maple [B]  time = 0.056, size = 172, normalized size = 2.2 \[{\frac{1}{75\, \left ( -3+2\,x \right ) ^{2}} \left ( 2\,\sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticF} \left ( 1/10\,\sqrt{2}\sqrt{5}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}},\sqrt{2} \right ) \sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}x-3\,\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}}\sqrt{ \left ( -3+2\,x \right ) \sqrt{5}}\sqrt{ \left ( 2\,x-3+\sqrt{5} \right ) \sqrt{5}}{\it EllipticF} \left ( 1/10\,\sqrt{2}\sqrt{5}\sqrt{ \left ( -2\,x+3+\sqrt{5} \right ) \sqrt{5}},\sqrt{2} \right ) -20\,{x}^{2}+60\,x-20 \right ) \sqrt{3-2\,x}{\frac{1}{\sqrt{{x}^{2}-3\,x+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-2*x)^(5/2)/(x^2-3*x+1)^(1/2),x)

[Out]

1/75*(2*((-3+2*x)*5^(1/2))^(1/2)*((2*x-3+5^(1/2))*5^(1/2))^(1/2)*EllipticF(1/10*
2^(1/2)*5^(1/2)*((-2*x+3+5^(1/2))*5^(1/2))^(1/2),2^(1/2))*((-2*x+3+5^(1/2))*5^(1
/2))^(1/2)*x-3*((-2*x+3+5^(1/2))*5^(1/2))^(1/2)*((-3+2*x)*5^(1/2))^(1/2)*((2*x-3
+5^(1/2))*5^(1/2))^(1/2)*EllipticF(1/10*2^(1/2)*5^(1/2)*((-2*x+3+5^(1/2))*5^(1/2
))^(1/2),2^(1/2))-20*x^2+60*x-20)/(x^2-3*x+1)^(1/2)*(3-2*x)^(1/2)/(-3+2*x)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} - 3 \, x + 1}{\left (-2 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 3*x + 1)*(-2*x + 3)^(5/2)),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(x^2 - 3*x + 1)*(-2*x + 3)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (4 \, x^{2} - 12 \, x + 9\right )} \sqrt{x^{2} - 3 \, x + 1} \sqrt{-2 \, x + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 3*x + 1)*(-2*x + 3)^(5/2)),x, algorithm="fricas")

[Out]

integral(1/((4*x^2 - 12*x + 9)*sqrt(x^2 - 3*x + 1)*sqrt(-2*x + 3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- 2 x + 3\right )^{\frac{5}{2}} \sqrt{x^{2} - 3 x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3-2*x)**(5/2)/(x**2-3*x+1)**(1/2),x)

[Out]

Integral(1/((-2*x + 3)**(5/2)*sqrt(x**2 - 3*x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{2} - 3 \, x + 1}{\left (-2 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^2 - 3*x + 1)*(-2*x + 3)^(5/2)),x, algorithm="giac")

[Out]

integrate(1/(sqrt(x^2 - 3*x + 1)*(-2*x + 3)^(5/2)), x)