3.272 \(\int (1+x+x^2)^{3/2} \, dx\)

Optimal. Leaf size=55 \[ \frac {1}{8} (2 x+1) \left (x^2+x+1\right )^{3/2}+\frac {9}{64} (2 x+1) \sqrt {x^2+x+1}+\frac {27}{128} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {612, 619, 215} \[ \frac {1}{8} (2 x+1) \left (x^2+x+1\right )^{3/2}+\frac {9}{64} (2 x+1) \sqrt {x^2+x+1}+\frac {27}{128} \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x + x^2)^(3/2),x]

[Out]

(9*(1 + 2*x)*Sqrt[1 + x + x^2])/64 + ((1 + 2*x)*(1 + x + x^2)^(3/2))/8 + (27*ArcSinh[(1 + 2*x)/Sqrt[3]])/128

Rule 215

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

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 619

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

Rubi steps

\begin {align*} \int \left (1+x+x^2\right )^{3/2} \, dx &=\frac {1}{8} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac {9}{16} \int \sqrt {1+x+x^2} \, dx\\ &=\frac {9}{64} (1+2 x) \sqrt {1+x+x^2}+\frac {1}{8} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac {27}{128} \int \frac {1}{\sqrt {1+x+x^2}} \, dx\\ &=\frac {9}{64} (1+2 x) \sqrt {1+x+x^2}+\frac {1}{8} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac {1}{128} \left (9 \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{3}}} \, dx,x,1+2 x\right )\\ &=\frac {9}{64} (1+2 x) \sqrt {1+x+x^2}+\frac {1}{8} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac {27}{128} \sinh ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 46, normalized size = 0.84 \[ \frac {1}{128} \left (2 \sqrt {x^2+x+1} \left (16 x^3+24 x^2+42 x+17\right )+27 \sinh ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x + x^2)^(3/2),x]

[Out]

(2*Sqrt[1 + x + x^2]*(17 + 42*x + 24*x^2 + 16*x^3) + 27*ArcSinh[(1 + 2*x)/Sqrt[3]])/128

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.15, size = 52, normalized size = 0.95 \[ \frac {1}{64} \sqrt {x^2+x+1} \left (16 x^3+24 x^2+42 x+17\right )-\frac {27}{128} \log \left (2 \sqrt {x^2+x+1}-2 x-1\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x + x^2)^(3/2),x]

[Out]

(Sqrt[1 + x + x^2]*(17 + 42*x + 24*x^2 + 16*x^3))/64 - (27*Log[-1 - 2*x + 2*Sqrt[1 + x + x^2]])/128

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 44, normalized size = 0.80 \[ \frac {1}{64} \, {\left (16 \, x^{3} + 24 \, x^{2} + 42 \, x + 17\right )} \sqrt {x^{2} + x + 1} - \frac {27}{128} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(16*x^3 + 24*x^2 + 42*x + 17)*sqrt(x^2 + x + 1) - 27/128*log(-2*x + 2*sqrt(x^2 + x + 1) - 1)

________________________________________________________________________________________

giac [A]  time = 0.63, size = 44, normalized size = 0.80 \[ \frac {1}{64} \, {\left (2 \, {\left (4 \, {\left (2 \, x + 3\right )} x + 21\right )} x + 17\right )} \sqrt {x^{2} + x + 1} - \frac {27}{128} \, \log \left (-2 \, x + 2 \, \sqrt {x^{2} + x + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(2*(4*(2*x + 3)*x + 21)*x + 17)*sqrt(x^2 + x + 1) - 27/128*log(-2*x + 2*sqrt(x^2 + x + 1) - 1)

________________________________________________________________________________________

maple [A]  time = 0.35, size = 38, normalized size = 0.69

method	result	size
risch	\(\frac {\left (16 x^{3}+24 x^{2}+42 x +17\right ) \sqrt {x^{2}+x +1}}{64}+\frac {27 \arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{128}\)	\(38\)
default	\(\frac {\left (1+2 x \right ) \left (x^{2}+x +1\right )^{\frac {3}{2}}}{8}+\frac {9 \left (1+2 x \right ) \sqrt {x^{2}+x +1}}{64}+\frac {27 \arcsinh \left (\frac {2 \left (\frac {1}{2}+x \right ) \sqrt {3}}{3}\right )}{128}\)	\(43\)
trager	\(\left (\frac {1}{4} x^{3}+\frac {3}{8} x^{2}+\frac {21}{32} x +\frac {17}{64}\right ) \sqrt {x^{2}+x +1}+\frac {27 \ln \left (1+2 x +2 \sqrt {x^{2}+x +1}\right )}{128}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+x+1)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/64*(16*x^3+24*x^2+42*x+17)*(x^2+x+1)^(1/2)+27/128*arcsinh(2/3*(1/2+x)*3^(1/2))

________________________________________________________________________________________

maxima [A]  time = 1.29, size = 56, normalized size = 1.02 \[ \frac {1}{4} \, {\left (x^{2} + x + 1\right )}^{\frac {3}{2}} x + \frac {1}{8} \, {\left (x^{2} + x + 1\right )}^{\frac {3}{2}} + \frac {9}{32} \, \sqrt {x^{2} + x + 1} x + \frac {9}{64} \, \sqrt {x^{2} + x + 1} + \frac {27}{128} \, \operatorname {arsinh}\left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(x^2 + x + 1)^(3/2)*x + 1/8*(x^2 + x + 1)^(3/2) + 9/32*sqrt(x^2 + x + 1)*x + 9/64*sqrt(x^2 + x + 1) + 27/1
28*arcsinh(1/3*sqrt(3)*(2*x + 1))

________________________________________________________________________________________

mupad [B]  time = 0.21, size = 43, normalized size = 0.78 \[ \frac {27\,\ln \left (x+\sqrt {x^2+x+1}+\frac {1}{2}\right )}{128}+\frac {\left (x+\frac {1}{2}\right )\,{\left (x^2+x+1\right )}^{3/2}}{4}+\frac {9\,\left (\frac {x}{2}+\frac {1}{4}\right )\,\sqrt {x^2+x+1}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x^2 + 1)^(3/2),x)

[Out]

(27*log(x + (x + x^2 + 1)^(1/2) + 1/2))/128 + ((x + 1/2)*(x + x^2 + 1)^(3/2))/4 + (9*(x/2 + 1/4)*(x + x^2 + 1)
^(1/2))/16

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (x^{2} + x + 1\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+x+1)**(3/2),x)

[Out]

Integral((x**2 + x + 1)**(3/2), x)

________________________________________________________________________________________