3.19.74 \(\int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx\)

Optimal. Leaf size=129 \[ -\frac {325}{128} \log \left (x^2+5 x+1\right )+\frac {\sqrt {x^6+9 x^5+16 x^4-27 x^3-36 x^2-11 x-1} \left (16 x^3+104 x^2-6 x-185\right )}{64 \left (x^2+5 x+1\right )}+\frac {325}{128} \log \left (-2 x^3-9 x^2+2 \sqrt {x^6+9 x^5+16 x^4-27 x^3-36 x^2-11 x-1}+3 x+1\right ) \]

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Rubi [A]  time = 0.08, antiderivative size = 209, normalized size of antiderivative = 1.62, number of steps used = 7, number of rules used = 7, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.219, Rules used = {6688, 6719, 1661, 640, 612, 621, 206} \begin {gather*} -\frac {65 \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )} (1-2 x)}{64 \left (x^2+5 x+1\right )}-\frac {x \left (-x^2+x+1\right ) \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )}}{4 \left (x^2+5 x+1\right )}-\frac {15 \left (-x^2+x+1\right ) \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )}}{8 \left (x^2+5 x+1\right )}+\frac {325 \sqrt {-\left (\left (-x^2+x+1\right ) \left (x^2+5 x+1\right )^2\right )} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )}{128 \sqrt {x^2-x-1} \left (x^2+5 x+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 - 11*x - 36*x^2 - 27*x^3 + 16*x^4 + 9*x^5 + x^6],x]

[Out]

(-65*(1 - 2*x)*Sqrt[-((1 + x - x^2)*(1 + 5*x + x^2)^2)])/(64*(1 + 5*x + x^2)) - (15*(1 + x - x^2)*Sqrt[-((1 +
x - x^2)*(1 + 5*x + x^2)^2)])/(8*(1 + 5*x + x^2)) - (x*(1 + x - x^2)*Sqrt[-((1 + x - x^2)*(1 + 5*x + x^2)^2)])
/(4*(1 + 5*x + x^2)) + (325*Sqrt[-((1 + x - x^2)*(1 + 5*x + x^2)^2)]*ArcTanh[(1 - 2*x)/(2*Sqrt[-1 - x + x^2])]
)/(128*Sqrt[-1 - x + x^2]*(1 + 5*x + x^2))

Rule 206

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 6688

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

Rule 6719

Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m*w^n)^FracPart[p])/(v^(m*F
racPart[p])*w^(n*FracPart[p])), Int[u*v^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] &&  !IntegerQ[p] &&  !
FreeQ[v, x] &&  !FreeQ[w, x]

Rubi steps

\begin {align*} \int \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6} \, dx &=\int \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2} \, dx\\ &=\frac {\sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2} \int \sqrt {-1-x+x^2} \left (1+5 x+x^2\right ) \, dx}{\sqrt {-1-x+x^2} \left (1+5 x+x^2\right )}\\ &=-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}+\frac {\sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2} \int \left (5+\frac {45 x}{2}\right ) \sqrt {-1-x+x^2} \, dx}{4 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )}\\ &=-\frac {15 \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{8 \left (1+5 x+x^2\right )}-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}+\frac {\left (65 \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2}\right ) \int \sqrt {-1-x+x^2} \, dx}{16 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )}\\ &=-\frac {65 (1-2 x) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{64 \left (1+5 x+x^2\right )}-\frac {15 \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{8 \left (1+5 x+x^2\right )}-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}-\frac {\left (325 \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2}\right ) \int \frac {1}{\sqrt {-1-x+x^2}} \, dx}{128 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )}\\ &=-\frac {65 (1-2 x) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{64 \left (1+5 x+x^2\right )}-\frac {15 \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{8 \left (1+5 x+x^2\right )}-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}-\frac {\left (325 \sqrt {\left (-1-x+x^2\right ) \left (1+5 x+x^2\right )^2}\right ) \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {-1+2 x}{\sqrt {-1-x+x^2}}\right )}{64 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )}\\ &=-\frac {65 (1-2 x) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{64 \left (1+5 x+x^2\right )}-\frac {15 \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{8 \left (1+5 x+x^2\right )}-\frac {x \left (1+x-x^2\right ) \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )}}{4 \left (1+5 x+x^2\right )}+\frac {325 \sqrt {-\left (\left (1+x-x^2\right ) \left (1+5 x+x^2\right )^2\right )} \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {-1-x+x^2}}\right )}{128 \sqrt {-1-x+x^2} \left (1+5 x+x^2\right )}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 101, normalized size = 0.78 \begin {gather*} \frac {\sqrt {x^2-x-1} \left (x^2+5 x+1\right ) \left (325 \tanh ^{-1}\left (\frac {1-2 x}{2 \sqrt {x^2-x-1}}\right )+2 \sqrt {x^2-x-1} \left (16 x^3+104 x^2-6 x-185\right )\right )}{128 \sqrt {\left (x^2-x-1\right ) \left (x^2+5 x+1\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 - 11*x - 36*x^2 - 27*x^3 + 16*x^4 + 9*x^5 + x^6],x]

[Out]

(Sqrt[-1 - x + x^2]*(1 + 5*x + x^2)*(2*Sqrt[-1 - x + x^2]*(-185 - 6*x + 104*x^2 + 16*x^3) + 325*ArcTanh[(1 - 2
*x)/(2*Sqrt[-1 - x + x^2])]))/(128*Sqrt[(-1 - x + x^2)*(1 + 5*x + x^2)^2])

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IntegrateAlgebraic [A]  time = 0.24, size = 129, normalized size = 1.00 \begin {gather*} \frac {\left (-185-6 x+104 x^2+16 x^3\right ) \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6}}{64 \left (1+5 x+x^2\right )}-\frac {325}{128} \log \left (1+5 x+x^2\right )+\frac {325}{128} \log \left (1+3 x-9 x^2-2 x^3+2 \sqrt {-1-11 x-36 x^2-27 x^3+16 x^4+9 x^5+x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-1 - 11*x - 36*x^2 - 27*x^3 + 16*x^4 + 9*x^5 + x^6],x]

[Out]

((-185 - 6*x + 104*x^2 + 16*x^3)*Sqrt[-1 - 11*x - 36*x^2 - 27*x^3 + 16*x^4 + 9*x^5 + x^6])/(64*(1 + 5*x + x^2)
) - (325*Log[1 + 5*x + x^2])/128 + (325*Log[1 + 3*x - 9*x^2 - 2*x^3 + 2*Sqrt[-1 - 11*x - 36*x^2 - 27*x^3 + 16*
x^4 + 9*x^5 + x^6]])/128

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fricas [A]  time = 0.52, size = 139, normalized size = 1.08 \begin {gather*} \frac {569 \, x^{2} + 2600 \, {\left (x^{2} + 5 \, x + 1\right )} \log \left (-\frac {2 \, x^{3} + 9 \, x^{2} - 3 \, x - 2 \, \sqrt {x^{6} + 9 \, x^{5} + 16 \, x^{4} - 27 \, x^{3} - 36 \, x^{2} - 11 \, x - 1} - 1}{x^{2} + 5 \, x + 1}\right ) + 16 \, \sqrt {x^{6} + 9 \, x^{5} + 16 \, x^{4} - 27 \, x^{3} - 36 \, x^{2} - 11 \, x - 1} {\left (16 \, x^{3} + 104 \, x^{2} - 6 \, x - 185\right )} + 2845 \, x + 569}{1024 \, {\left (x^{2} + 5 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+9*x^5+16*x^4-27*x^3-36*x^2-11*x-1)^(1/2),x, algorithm="fricas")

[Out]

1/1024*(569*x^2 + 2600*(x^2 + 5*x + 1)*log(-(2*x^3 + 9*x^2 - 3*x - 2*sqrt(x^6 + 9*x^5 + 16*x^4 - 27*x^3 - 36*x
^2 - 11*x - 1) - 1)/(x^2 + 5*x + 1)) + 16*sqrt(x^6 + 9*x^5 + 16*x^4 - 27*x^3 - 36*x^2 - 11*x - 1)*(16*x^3 + 10
4*x^2 - 6*x - 185) + 2845*x + 569)/(x^2 + 5*x + 1)

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giac [A]  time = 0.30, size = 97, normalized size = 0.75 \begin {gather*} \frac {325}{128} \, \log \left ({\left | -2 \, x + 2 \, \sqrt {x^{2} - x - 1} + 1 \right |}\right ) \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right ) + \frac {1}{64} \, {\left (2 \, {\left (4 \, {\left (2 \, x \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right ) + 13 \, \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right )\right )} x - 3 \, \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right )\right )} x - 185 \, \mathrm {sgn}\left (x^{2} + 5 \, x + 1\right )\right )} \sqrt {x^{2} - x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+9*x^5+16*x^4-27*x^3-36*x^2-11*x-1)^(1/2),x, algorithm="giac")

[Out]

325/128*log(abs(-2*x + 2*sqrt(x^2 - x - 1) + 1))*sgn(x^2 + 5*x + 1) + 1/64*(2*(4*(2*x*sgn(x^2 + 5*x + 1) + 13*
sgn(x^2 + 5*x + 1))*x - 3*sgn(x^2 + 5*x + 1))*x - 185*sgn(x^2 + 5*x + 1))*sqrt(x^2 - x - 1)

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maple [A]  time = 0.13, size = 107, normalized size = 0.83

method	result	size
risch	\(\frac {\left (16 x^{3}+104 x^{2}-6 x -185\right ) \sqrt {\left (x^{2}-x -1\right ) \left (x^{2}+5 x +1\right )^{2}}}{64 x^{2}+320 x +64}-\frac {325 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right ) \sqrt {\left (x^{2}-x -1\right ) \left (x^{2}+5 x +1\right )^{2}}}{128 \left (x^{2}+5 x +1\right ) \sqrt {x^{2}-x -1}}\)	\(107\)
default	\(\frac {\sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}\, \left (32 x \left (x^{2}-x -1\right )^{\frac {3}{2}}+240 \left (x^{2}-x -1\right )^{\frac {3}{2}}+260 x \sqrt {x^{2}-x -1}-130 \sqrt {x^{2}-x -1}-325 \ln \left (x -\frac {1}{2}+\sqrt {x^{2}-x -1}\right )\right )}{128 \left (x^{2}+5 x +1\right ) \sqrt {x^{2}-x -1}}\)	\(120\)
trager	\(\frac {\left (16 x^{3}+104 x^{2}-6 x -185\right ) \sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}}{64 x^{2}+320 x +64}-\frac {325 \ln \left (\frac {2 x^{3}+9 x^{2}+2 \sqrt {x^{6}+9 x^{5}+16 x^{4}-27 x^{3}-36 x^{2}-11 x -1}-3 x -1}{x^{2}+5 x +1}\right )}{128}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6+9*x^5+16*x^4-27*x^3-36*x^2-11*x-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/64*(16*x^3+104*x^2-6*x-185)*((x^2-x-1)*(x^2+5*x+1)^2)^(1/2)/(x^2+5*x+1)-325/128*ln(x-1/2+(x^2-x-1)^(1/2))*((
x^2-x-1)*(x^2+5*x+1)^2)^(1/2)/(x^2+5*x+1)/(x^2-x-1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{6} + 9 \, x^{5} + 16 \, x^{4} - 27 \, x^{3} - 36 \, x^{2} - 11 \, x - 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6+9*x^5+16*x^4-27*x^3-36*x^2-11*x-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(x^6 + 9*x^5 + 16*x^4 - 27*x^3 - 36*x^2 - 11*x - 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {x^6+9\,x^5+16\,x^4-27\,x^3-36\,x^2-11\,x-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^4 - 36*x^2 - 27*x^3 - 11*x + 9*x^5 + x^6 - 1)^(1/2),x)

[Out]

int((16*x^4 - 36*x^2 - 27*x^3 - 11*x + 9*x^5 + x^6 - 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x^{6} + 9 x^{5} + 16 x^{4} - 27 x^{3} - 36 x^{2} - 11 x - 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6+9*x**5+16*x**4-27*x**3-36*x**2-11*x-1)**(1/2),x)

[Out]

Integral(sqrt(x**6 + 9*x**5 + 16*x**4 - 27*x**3 - 36*x**2 - 11*x - 1), x)

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