3.553 \(\int \frac{-1+x^3}{\sqrt{x} \left (1+x^2\right )} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 x^{3/2}}{3}+\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right ) \]

[Out]

(2*x^(3/2))/3 + Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] - Sqrt[2]*ArcTan[1 + Sqrt[2]
*Sqrt[x]]

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Rubi [A]  time = 0.0959879, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ \frac{2 x^{3/2}}{3}+\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2))/3 + Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] - Sqrt[2]*ArcTan[1 + Sqrt[2]
*Sqrt[x]]

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Rubi in Sympy [A]  time = 7.70096, size = 44, normalized size = 0.85 \[ \frac{2 x^{\frac{3}{2}}}{3} - \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )} - \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/3 - sqrt(2)*atan(sqrt(2)*sqrt(x) - 1) - sqrt(2)*atan(sqrt(2)*sqrt(x)
+ 1)

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Mathematica [A]  time = 0.0323538, size = 52, normalized size = 1. \[ \frac{2 x^{3/2}}{3}+\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2))/3 + Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] - Sqrt[2]*ArcTan[1 + Sqrt[2]
*Sqrt[x]]

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Maple [B]  time = 0.013, size = 97, normalized size = 1.9 \[{\frac{2}{3}{x}^{{\frac{3}{2}}}}-\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) \sqrt{2}-\arctan \left ( \sqrt{2}\sqrt{x}-1 \right ) \sqrt{2}-{\frac{\sqrt{2}}{4}\ln \left ({1 \left ( x+\sqrt{2}\sqrt{x}+1 \right ) \left ( x-\sqrt{2}\sqrt{x}+1 \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4}\ln \left ({1 \left ( x-\sqrt{2}\sqrt{x}+1 \right ) \left ( x+\sqrt{2}\sqrt{x}+1 \right ) ^{-1}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3-1)/(x^2+1)/x^(1/2),x)

[Out]

2/3*x^(3/2)-arctan(1+2^(1/2)*x^(1/2))*2^(1/2)-arctan(2^(1/2)*x^(1/2)-1)*2^(1/2)-
1/4*2^(1/2)*ln((x+2^(1/2)*x^(1/2)+1)/(x-2^(1/2)*x^(1/2)+1))-1/4*2^(1/2)*ln((x-2^
(1/2)*x^(1/2)+1)/(x+2^(1/2)*x^(1/2)+1))

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Maxima [A]  time = 0.801757, size = 62, normalized size = 1.19 \[ \frac{2}{3} \, x^{\frac{3}{2}} - \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x^(3/2) - sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - sqrt(2)*arctan
(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x)))

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Fricas [A]  time = 0.266681, size = 31, normalized size = 0.6 \[ \frac{2}{3} \, x^{\frac{3}{2}} - \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x - 1\right )}}{2 \, \sqrt{x}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x^(3/2) - sqrt(2)*arctan(1/2*sqrt(2)*(x - 1)/sqrt(x))

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Sympy [A]  time = 3.3509, size = 44, normalized size = 0.85 \[ \frac{2 x^{\frac{3}{2}}}{3} - \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )} - \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(3/2)/3 - sqrt(2)*atan(sqrt(2)*sqrt(x) - 1) - sqrt(2)*atan(sqrt(2)*sqrt(x)
+ 1)

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GIAC/XCAS [A]  time = 0.261633, size = 62, normalized size = 1.19 \[ \frac{2}{3} \, x^{\frac{3}{2}} - \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) - \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*x^(3/2) - sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) - sqrt(2)*arctan
(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(x)))