3.273 \(\int (1+x+x^2)^{5/2} \, dx\)

Optimal. Leaf size=74 \[ \frac{1}{12} (2 x+1) \left (x^2+x+1\right )^{5/2}+\frac{5}{64} (2 x+1) \left (x^2+x+1\right )^{3/2}+\frac{45}{512} (2 x+1) \sqrt{x^2+x+1}+\frac{135 \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{1024} \]

[Out]

(45*(1 + 2*x)*Sqrt[1 + x + x^2])/512 + (5*(1 + 2*x)*(1 + x + x^2)^(3/2))/64 + ((1 + 2*x)*(1 + x + x^2)^(5/2))/
12 + (135*ArcSinh[(1 + 2*x)/Sqrt[3]])/1024

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Rubi [A]  time = 0.0190233, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {612, 619, 215} \[ \frac{1}{12} (2 x+1) \left (x^2+x+1\right )^{5/2}+\frac{5}{64} (2 x+1) \left (x^2+x+1\right )^{3/2}+\frac{45}{512} (2 x+1) \sqrt{x^2+x+1}+\frac{135 \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{1024} \]

Antiderivative was successfully verified.

[In]

Int[(1 + x + x^2)^(5/2),x]

[Out]

(45*(1 + 2*x)*Sqrt[1 + x + x^2])/512 + (5*(1 + 2*x)*(1 + x + x^2)^(3/2))/64 + ((1 + 2*x)*(1 + x + x^2)^(5/2))/
12 + (135*ArcSinh[(1 + 2*x)/Sqrt[3]])/1024

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]

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]

Rubi steps

\begin{align*} \int \left (1+x+x^2\right )^{5/2} \, dx &=\frac{1}{12} (1+2 x) \left (1+x+x^2\right )^{5/2}+\frac{5}{8} \int \left (1+x+x^2\right )^{3/2} \, dx\\ &=\frac{5}{64} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac{1}{12} (1+2 x) \left (1+x+x^2\right )^{5/2}+\frac{45}{128} \int \sqrt{1+x+x^2} \, dx\\ &=\frac{45}{512} (1+2 x) \sqrt{1+x+x^2}+\frac{5}{64} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac{1}{12} (1+2 x) \left (1+x+x^2\right )^{5/2}+\frac{135 \int \frac{1}{\sqrt{1+x+x^2}} \, dx}{1024}\\ &=\frac{45}{512} (1+2 x) \sqrt{1+x+x^2}+\frac{5}{64} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac{1}{12} (1+2 x) \left (1+x+x^2\right )^{5/2}+\frac{\left (45 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{3}}} \, dx,x,1+2 x\right )}{1024}\\ &=\frac{45}{512} (1+2 x) \sqrt{1+x+x^2}+\frac{5}{64} (1+2 x) \left (1+x+x^2\right )^{3/2}+\frac{1}{12} (1+2 x) \left (1+x+x^2\right )^{5/2}+\frac{135 \sinh ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{1024}\\ \end{align*}

Mathematica [A]  time = 0.0206426, size = 56, normalized size = 0.76 \[ \frac{2 \sqrt{x^2+x+1} \left (256 x^5+640 x^4+1264 x^3+1256 x^2+1142 x+383\right )+405 \sinh ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3072} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x + x^2)^(5/2),x]

[Out]

(2*Sqrt[1 + x + x^2]*(383 + 1142*x + 1256*x^2 + 1264*x^3 + 640*x^4 + 256*x^5) + 405*ArcSinh[(1 + 2*x)/Sqrt[3]]
)/3072

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Maple [A]  time = 0.003, size = 58, normalized size = 0.8 \begin{align*}{\frac{1+2\,x}{12} \left ({x}^{2}+x+1 \right ) ^{{\frac{5}{2}}}}+{\frac{5+10\,x}{64} \left ({x}^{2}+x+1 \right ) ^{{\frac{3}{2}}}}+{\frac{45+90\,x}{512}\sqrt{{x}^{2}+x+1}}+{\frac{135}{1024}{\it Arcsinh} \left ({\frac{2\,\sqrt{3}}{3} \left ( x+{\frac{1}{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+x+1)^(5/2),x)

[Out]

1/12*(1+2*x)*(x^2+x+1)^(5/2)+5/64*(1+2*x)*(x^2+x+1)^(3/2)+45/512*(1+2*x)*(x^2+x+1)^(1/2)+135/1024*arcsinh(2/3*
3^(1/2)*(x+1/2))

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Maxima [A]  time = 1.44686, size = 104, normalized size = 1.41 \begin{align*} \frac{1}{6} \,{\left (x^{2} + x + 1\right )}^{\frac{5}{2}} x + \frac{1}{12} \,{\left (x^{2} + x + 1\right )}^{\frac{5}{2}} + \frac{5}{32} \,{\left (x^{2} + x + 1\right )}^{\frac{3}{2}} x + \frac{5}{64} \,{\left (x^{2} + x + 1\right )}^{\frac{3}{2}} + \frac{45}{256} \, \sqrt{x^{2} + x + 1} x + \frac{45}{512} \, \sqrt{x^{2} + x + 1} + \frac{135}{1024} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)^(5/2),x, algorithm="maxima")

[Out]

1/6*(x^2 + x + 1)^(5/2)*x + 1/12*(x^2 + x + 1)^(5/2) + 5/32*(x^2 + x + 1)^(3/2)*x + 5/64*(x^2 + x + 1)^(3/2) +
45/256*sqrt(x^2 + x + 1)*x + 45/512*sqrt(x^2 + x + 1) + 135/1024*arcsinh(1/3*sqrt(3)*(2*x + 1))

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Fricas [A]  time = 2.18232, size = 176, normalized size = 2.38 \begin{align*} \frac{1}{1536} \,{\left (256 \, x^{5} + 640 \, x^{4} + 1264 \, x^{3} + 1256 \, x^{2} + 1142 \, x + 383\right )} \sqrt{x^{2} + x + 1} - \frac{135}{1024} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)^(5/2),x, algorithm="fricas")

[Out]

1/1536*(256*x^5 + 640*x^4 + 1264*x^3 + 1256*x^2 + 1142*x + 383)*sqrt(x^2 + x + 1) - 135/1024*log(-2*x + 2*sqrt
(x^2 + x + 1) - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (x^{2} + x + 1\right )^{\frac{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+x+1)**(5/2),x)

[Out]

Integral((x**2 + x + 1)**(5/2), x)

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Giac [A]  time = 1.06681, size = 73, normalized size = 0.99 \begin{align*} \frac{1}{1536} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (2 \, x + 5\right )} x + 79\right )} x + 157\right )} x + 571\right )} x + 383\right )} \sqrt{x^{2} + x + 1} - \frac{135}{1024} \, \log \left (-2 \, x + 2 \, \sqrt{x^{2} + x + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)^(5/2),x, algorithm="giac")

[Out]

1/1536*(2*(4*(2*(8*(2*x + 5)*x + 79)*x + 157)*x + 571)*x + 383)*sqrt(x^2 + x + 1) - 135/1024*log(-2*x + 2*sqrt
(x^2 + x + 1) - 1)