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]

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

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Rubi [A]  time = 0.32, antiderivative size = 24, normalized size of antiderivative = 1.00, 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 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]

Rule 6688

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

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*}

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Mathematica [A]  time = 0.17, size = 24, normalized size = 1.00 \[ \sqrt {x^5-2 x+1}-\sqrt {x^3+1} \]

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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fricas [A]  time = 0.41, size = 20, normalized size = 0.83 \[ \sqrt {x^{5} - 2 \, x + 1} - \sqrt {x^{3} + 1} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {5 \, \sqrt {x^{3} + 1} x^{4} - 3 \, \sqrt {x^{5} - 2 \, x + 1} x^{2} - 2 \, \sqrt {x^{3} + 1}}{2 \, \sqrt {x^{5} - 2 \, x + 1} \sqrt {x^{3} + 1}}\,{d x} \]

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)

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maple [A]  time = 0.00, size = 21, normalized size = 0.88 \[ -\sqrt {x^{3}+1}+\sqrt {x^{5}-2 x +1} \]

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 = 1.19, size = 30, normalized size = 1.25 \[ \sqrt {x^{4} + x^{3} + x^{2} + x - 1} \sqrt {x - 1} - \sqrt {x^{3} + 1} \]

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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mupad [B]  time = 0.31, size = 20, normalized size = 0.83 \[ \sqrt {x^5-2\,x+1}-\sqrt {x^3+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.34, size = 19, normalized size = 0.79 \[ - \sqrt {x^{3} + 1} + \sqrt {x^{5} - 2 x + 1} \]

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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