3.579 \(\int \frac{\sqrt{-1+x} x^3}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{4} (x-1)^{3/2} \sqrt{x+1} x^2+\frac{1}{24} (7-2 x) (x-1)^{3/2} \sqrt{x+1}-\frac{3}{8} \sqrt{x-1} \sqrt{x+1}+\frac{3}{8} \cosh ^{-1}(x) \]

[Out]

(-3*Sqrt[-1 + x]*Sqrt[1 + x])/8 + ((7 - 2*x)*(-1 + x)^(3/2)*Sqrt[1 + x])/24 + ((
-1 + x)^(3/2)*x^2*Sqrt[1 + x])/4 + (3*ArcCosh[x])/8

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Rubi [A]  time = 0.0708545, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{4} (x-1)^{3/2} \sqrt{x+1} x^2+\frac{1}{24} (7-2 x) (x-1)^{3/2} \sqrt{x+1}-\frac{3}{8} \sqrt{x-1} \sqrt{x+1}+\frac{3}{8} \cosh ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

(-3*Sqrt[-1 + x]*Sqrt[1 + x])/8 + ((7 - 2*x)*(-1 + x)^(3/2)*Sqrt[1 + x])/24 + ((
-1 + x)^(3/2)*x^2*Sqrt[1 + x])/4 + (3*ArcCosh[x])/8

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Rubi in Sympy [A]  time = 3.89663, size = 61, normalized size = 0.88 \[ \frac{x^{2} \left (x - 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{4} + \frac{\left (- 2 x + 7\right ) \left (x - 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{24} - \frac{3 \sqrt{x - 1} \sqrt{x + 1}}{8} + \frac{3 \operatorname{acosh}{\left (x \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(x - 1)**(3/2)*sqrt(x + 1)/4 + (-2*x + 7)*(x - 1)**(3/2)*sqrt(x + 1)/24 - 3
*sqrt(x - 1)*sqrt(x + 1)/8 + 3*acosh(x)/8

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Mathematica [A]  time = 0.0658282, size = 74, normalized size = 1.07 \[ \frac{\sqrt{\frac{x-1}{x+1}} \left (\sqrt{x-1} \left (6 x^4-2 x^3+x^2-7 x-16\right )+18 \sqrt{x+1} \sinh ^{-1}\left (\frac{\sqrt{x-1}}{\sqrt{2}}\right )\right )}{24 \sqrt{x-1}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(-1 + x)/(1 + x)]*(Sqrt[-1 + x]*(-16 - 7*x + x^2 - 2*x^3 + 6*x^4) + 18*Sqr
t[1 + x]*ArcSinh[Sqrt[-1 + x]/Sqrt[2]]))/(24*Sqrt[-1 + x])

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Maple [A]  time = 0.014, size = 76, normalized size = 1.1 \[{\frac{1}{24}\sqrt{-1+x}\sqrt{1+x} \left ( 6\,{x}^{3}\sqrt{{x}^{2}-1}-8\,{x}^{2}\sqrt{{x}^{2}-1}+9\,x\sqrt{{x}^{2}-1}+9\,\ln \left ( x+\sqrt{{x}^{2}-1} \right ) -16\,\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(-1+x)^(1/2)*(1+x)^(1/2)*(6*x^3*(x^2-1)^(1/2)-8*x^2*(x^2-1)^(1/2)+9*x*(x^2-
1)^(1/2)+9*ln(x+(x^2-1)^(1/2))-16*(x^2-1)^(1/2))/(x^2-1)^(1/2)

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Maxima [A]  time = 0.719345, size = 74, normalized size = 1.07 \[ \frac{1}{4} \,{\left (x^{2} - 1\right )}^{\frac{3}{2}} x - \frac{1}{3} \,{\left (x^{2} - 1\right )}^{\frac{3}{2}} + \frac{5}{8} \, \sqrt{x^{2} - 1} x - \sqrt{x^{2} - 1} + \frac{3}{8} \, \log \left (2 \, x + 2 \, \sqrt{x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(x^2 - 1)^(3/2)*x - 1/3*(x^2 - 1)^(3/2) + 5/8*sqrt(x^2 - 1)*x - sqrt(x^2 - 1
) + 3/8*log(2*x + 2*sqrt(x^2 - 1))

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Fricas [A]  time = 0.276165, size = 228, normalized size = 3.3 \[ -\frac{48 \, x^{8} - 64 \, x^{7} - 32 \, x^{5} - 84 \, x^{4} + 160 \, x^{3} -{\left (48 \, x^{7} - 64 \, x^{6} + 24 \, x^{5} - 64 \, x^{4} - 66 \, x^{3} + 120 \, x^{2} + 9 \, x - 16\right )} \sqrt{x + 1} \sqrt{x - 1} + 36 \, x^{2} + 9 \,{\left (8 \, x^{4} - 4 \,{\left (2 \, x^{3} - x\right )} \sqrt{x + 1} \sqrt{x - 1} - 8 \, x^{2} + 1\right )} \log \left (\sqrt{x + 1} \sqrt{x - 1} - x\right ) - 64 \, x}{24 \,{\left (8 \, x^{4} - 4 \,{\left (2 \, x^{3} - x\right )} \sqrt{x + 1} \sqrt{x - 1} - 8 \, x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24*(48*x^8 - 64*x^7 - 32*x^5 - 84*x^4 + 160*x^3 - (48*x^7 - 64*x^6 + 24*x^5 -
64*x^4 - 66*x^3 + 120*x^2 + 9*x - 16)*sqrt(x + 1)*sqrt(x - 1) + 36*x^2 + 9*(8*x
^4 - 4*(2*x^3 - x)*sqrt(x + 1)*sqrt(x - 1) - 8*x^2 + 1)*log(sqrt(x + 1)*sqrt(x -
1) - x) - 64*x)/(8*x^4 - 4*(2*x^3 - x)*sqrt(x + 1)*sqrt(x - 1) - 8*x^2 + 1)

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Sympy [A]  time = 42.1345, size = 83, normalized size = 1.2 \[ \frac{\left (x - 1\right )^{\frac{7}{2}} \sqrt{x + 1}}{4} + \frac{5 \left (x - 1\right )^{\frac{5}{2}} \sqrt{x + 1}}{12} + \frac{11 \left (x - 1\right )^{\frac{3}{2}} \sqrt{x + 1}}{24} - \frac{3 \sqrt{x - 1} \sqrt{x + 1}}{8} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{x - 1}}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x - 1)**(7/2)*sqrt(x + 1)/4 + 5*(x - 1)**(5/2)*sqrt(x + 1)/12 + 11*(x - 1)**(3/
2)*sqrt(x + 1)/24 - 3*sqrt(x - 1)*sqrt(x + 1)/8 + 3*asinh(sqrt(2)*sqrt(x - 1)/2)
/4

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GIAC/XCAS [A]  time = 0.292087, size = 65, normalized size = 0.94 \[ \frac{1}{24} \,{\left ({\left (2 \,{\left (3 \, x - 10\right )}{\left (x + 1\right )} + 43\right )}{\left (x + 1\right )} - 39\right )} \sqrt{x + 1} \sqrt{x - 1} - \frac{3}{4} \,{\rm ln}\left ({\left | -\sqrt{x + 1} + \sqrt{x - 1} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*((2*(3*x - 10)*(x + 1) + 43)*(x + 1) - 39)*sqrt(x + 1)*sqrt(x - 1) - 3/4*ln
(abs(-sqrt(x + 1) + sqrt(x - 1)))