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

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]]

________________________________________________________________________________________

Rubi [A]  time = 0.0532846, 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, Rules used = {1802, 827, 1162, 617, 204} \[ \frac{2 x^{3/2}}{3}+\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-\sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right ) \]

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]]

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{-1+x^3}{\sqrt{x} \left (1+x^2\right )} \, dx &=\int \left (\sqrt{x}-\frac{1+x}{\sqrt{x} \left (1+x^2\right )}\right ) \, dx\\ &=\frac{2 x^{3/2}}{3}-\int \frac{1+x}{\sqrt{x} \left (1+x^2\right )} \, dx\\ &=\frac{2 x^{3/2}}{3}-2 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{3/2}}{3}-\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )-\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 x^{3/2}}{3}-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )+\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )\\ &=\frac{2 x^{3/2}}{3}+\sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )-\sqrt{2} \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.023999, 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]]

________________________________________________________________________________________

Maple [B]  time = 0.008, size = 97, normalized size = 1.9 \begin{align*}{\frac{2}{3}{x}^{{\frac{3}{2}}}}-\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) \sqrt{2}-\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) \sqrt{2}-{\frac{\sqrt{2}}{4}\ln \left ({ \left ( x+\sqrt{2}\sqrt{x}+1 \right ) \left ( x-\sqrt{2}\sqrt{x}+1 \right ) ^{-1}} \right ) }-{\frac{\sqrt{2}}{4}\ln \left ({ \left ( x-\sqrt{2}\sqrt{x}+1 \right ) \left ( x+\sqrt{2}\sqrt{x}+1 \right ) ^{-1}} \right ) } \end{align*}

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(-1+2^(1/2)*x^(1/2))*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))

________________________________________________________________________________________

Maxima [A]  time = 1.50326, size = 62, normalized size = 1.19 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)/(x^2+1)/x^(1/2),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*sqr
t(x)))

________________________________________________________________________________________

Fricas [A]  time = 1.47922, size = 80, normalized size = 1.54 \begin{align*} \frac{2}{3} \, x^{\frac{3}{2}} - \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (x - 1\right )}}{2 \, \sqrt{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)/(x^2+1)/x^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.896422, size = 44, normalized size = 0.85 \begin{align*} \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 )} \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.19831, size = 62, normalized size = 1.19 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)/(x^2+1)/x^(1/2),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*sqr
t(x)))