3.201 \(\int \frac{-2 \sqrt{1+x^3}+5 x^4 \sqrt{1+x^3}-3 x^2 \sqrt{1-2 x+x^5}}{2 \sqrt{1+x^3} \sqrt{1-2 x+x^5}} \, dx\)

Optimal. Leaf size=24 \[ \sqrt{x^5-2 x+1}-\sqrt{x^3+1} \]

[Out]

-Sqrt[1 + x^3] + Sqrt[1 - 2*x + x^5]

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Rubi [A]  time = 0.323708, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 68, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {12, 6688, 261, 2099} \[ \sqrt{x^5-2 x+1}-\sqrt{x^3+1} \]

Antiderivative was successfully verified.

[In]

Int[(-2*Sqrt[1 + x^3] + 5*x^4*Sqrt[1 + x^3] - 3*x^2*Sqrt[1 - 2*x + x^5])/(2*Sqrt[1 + x^3]*Sqrt[1 - 2*x + x^5])
,x]

[Out]

-Sqrt[1 + x^3] + Sqrt[1 - 2*x + x^5]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 2099

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1)
)/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[Simplify[Pm - (Coeff[Pm, x, m]*D[Qn, x])/(n*Coeff[Qn, x, n])], Int[Qn
^p, x], x] /; EqQ[m, n - 1] && EqQ[D[Simplify[Pm - (Coeff[Pm, x, m]*D[Qn, x])/(n*Coeff[Qn, x, n])], x], 0]] /;
FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{-2 \sqrt{1+x^3}+5 x^4 \sqrt{1+x^3}-3 x^2 \sqrt{1-2 x+x^5}}{2 \sqrt{1+x^3} \sqrt{1-2 x+x^5}} \, dx &=\frac{1}{2} \int \frac{-2 \sqrt{1+x^3}+5 x^4 \sqrt{1+x^3}-3 x^2 \sqrt{1-2 x+x^5}}{\sqrt{1+x^3} \sqrt{1-2 x+x^5}} \, dx\\ &=\frac{1}{2} \int \left (-\frac{3 x^2}{\sqrt{1+x^3}}-\frac{2}{\sqrt{1-2 x+x^5}}+\frac{5 x^4}{\sqrt{1-2 x+x^5}}\right ) \, dx\\ &=-\left (\frac{3}{2} \int \frac{x^2}{\sqrt{1+x^3}} \, dx\right )+\frac{5}{2} \int \frac{x^4}{\sqrt{1-2 x+x^5}} \, dx-\int \frac{1}{\sqrt{1-2 x+x^5}} \, dx\\ &=-\sqrt{1+x^3}+\sqrt{1-2 x+x^5}\\ \end{align*}

Mathematica [A]  time = 0.16667, size = 24, normalized size = 1. \[ \sqrt{x^5-2 x+1}-\sqrt{x^3+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(-2*Sqrt[1 + x^3] + 5*x^4*Sqrt[1 + x^3] - 3*x^2*Sqrt[1 - 2*x + x^5])/(2*Sqrt[1 + x^3]*Sqrt[1 - 2*x +
x^5]),x]

[Out]

-Sqrt[1 + x^3] + Sqrt[1 - 2*x + x^5]

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Maple [A]  time = 0.005, size = 21, normalized size = 0.9 \begin{align*} -\sqrt{{x}^{3}+1}+\sqrt{{x}^{5}-2\,x+1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-2*(x^3+1)^(1/2)+5*x^4*(x^3+1)^(1/2)-3*x^2*(x^5-2*x+1)^(1/2))/(x^3+1)^(1/2)/(x^5-2*x+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.60047, size = 41, normalized size = 1.71 \begin{align*} \sqrt{x^{4} + x^{3} + x^{2} + x - 1} \sqrt{x - 1} - \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*(x^3+1)^(1/2)+5*x^4*(x^3+1)^(1/2)-3*x^2*(x^5-2*x+1)^(1/2))/(x^3+1)^(1/2)/(x^5-2*x+1)^(1/2),x
, algorithm="maxima")

[Out]

sqrt(x^4 + x^3 + x^2 + x - 1)*sqrt(x - 1) - sqrt(x^3 + 1)

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Fricas [A]  time = 1.6198, size = 50, normalized size = 2.08 \begin{align*} \sqrt{x^{5} - 2 \, x + 1} - \sqrt{x^{3} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*(x^3+1)^(1/2)+5*x^4*(x^3+1)^(1/2)-3*x^2*(x^5-2*x+1)^(1/2))/(x^3+1)^(1/2)/(x^5-2*x+1)^(1/2),x
, algorithm="fricas")

[Out]

sqrt(x^5 - 2*x + 1) - sqrt(x^3 + 1)

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Sympy [A]  time = 1.6297, size = 19, normalized size = 0.79 \begin{align*} - \sqrt{x^{3} + 1} + \sqrt{x^{5} - 2 x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*(x**3+1)**(1/2)+5*x**4*(x**3+1)**(1/2)-3*x**2*(x**5-2*x+1)**(1/2))/(x**3+1)**(1/2)/(x**5-2*x
+1)**(1/2),x)

[Out]

-sqrt(x**3 + 1) + sqrt(x**5 - 2*x + 1)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \, \sqrt{x^{3} + 1} x^{4} - 3 \, \sqrt{x^{5} - 2 \, x + 1} x^{2} - 2 \, \sqrt{x^{3} + 1}}{2 \, \sqrt{x^{5} - 2 \, x + 1} \sqrt{x^{3} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*(x^3+1)^(1/2)+5*x^4*(x^3+1)^(1/2)-3*x^2*(x^5-2*x+1)^(1/2))/(x^3+1)^(1/2)/(x^5-2*x+1)^(1/2),x
, algorithm="giac")

[Out]

integrate(1/2*(5*sqrt(x^3 + 1)*x^4 - 3*sqrt(x^5 - 2*x + 1)*x^2 - 2*sqrt(x^3 + 1))/(sqrt(x^5 - 2*x + 1)*sqrt(x^
3 + 1)), x)