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

Optimal. Leaf size=77 \[ \frac{1}{2} \sqrt{x^4+5 x^2+3} x^4+\frac{3}{16} \left (89-14 x^2\right ) \sqrt{x^4+5 x^2+3}-\frac{1083}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

(x^4*Sqrt[3 + 5*x^2 + x^4])/2 + (3*(89 - 14*x^2)*Sqrt[3 + 5*x^2 + x^4])/16 - (1083*ArcTanh[(5 + 2*x^2)/(2*Sqrt
[3 + 5*x^2 + x^4])])/32

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Rubi [A]  time = 0.0662238, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1251, 832, 779, 621, 206} \[ \frac{1}{2} \sqrt{x^4+5 x^2+3} x^4+\frac{3}{16} \left (89-14 x^2\right ) \sqrt{x^4+5 x^2+3}-\frac{1083}{32} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Sqrt[3 + 5*x^2 + x^4])/2 + (3*(89 - 14*x^2)*Sqrt[3 + 5*x^2 + x^4])/16 - (1083*ArcTanh[(5 + 2*x^2)/(2*Sqrt
[3 + 5*x^2 + x^4])])/32

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 832

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] + Dist[1/(c*(m + 2*p + 2)), Int[(d + e*x)^(m
- 1)*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m*(c*e*f + c*d*g - b*e*g) + e*(p
+ 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
&&  !(IGtQ[m, 0] && EqQ[f, 0])

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

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 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])

Rubi steps

\begin{align*} \int \frac{x^5 \left (2+3 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 (2+3 x)}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} x^4 \sqrt{3+5 x^2+x^4}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{\left (-18-\frac{63 x}{2}\right ) x}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} x^4 \sqrt{3+5 x^2+x^4}+\frac{3}{16} \left (89-14 x^2\right ) \sqrt{3+5 x^2+x^4}-\frac{1083}{32} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} x^4 \sqrt{3+5 x^2+x^4}+\frac{3}{16} \left (89-14 x^2\right ) \sqrt{3+5 x^2+x^4}-\frac{1083}{16} \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{5+2 x^2}{\sqrt{3+5 x^2+x^4}}\right )\\ &=\frac{1}{2} x^4 \sqrt{3+5 x^2+x^4}+\frac{3}{16} \left (89-14 x^2\right ) \sqrt{3+5 x^2+x^4}-\frac{1083}{32} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0215752, size = 61, normalized size = 0.79 \[ \frac{1}{32} \left (2 \sqrt{x^4+5 x^2+3} \left (8 x^4-42 x^2+267\right )-1083 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x^2 + x^4]*(267 - 42*x^2 + 8*x^4) - 1083*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/32

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Maple [A]  time = 0.012, size = 70, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{2}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{21\,{x}^{2}}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{267}{16}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{1083}{32}\ln \left ({\frac{5}{2}}+{x}^{2}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^4*(x^4+5*x^2+3)^(1/2)-21/8*x^2*(x^4+5*x^2+3)^(1/2)+267/16*(x^4+5*x^2+3)^(1/2)-1083/32*ln(5/2+x^2+(x^4+5*
x^2+3)^(1/2))

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Maxima [A]  time = 0.960459, size = 99, normalized size = 1.29 \begin{align*} \frac{1}{2} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{4} - \frac{21}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{267}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{1083}{32} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(x^4 + 5*x^2 + 3)*x^4 - 21/8*sqrt(x^4 + 5*x^2 + 3)*x^2 + 267/16*sqrt(x^4 + 5*x^2 + 3) - 1083/32*log(2*
x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 1.31517, size = 139, normalized size = 1.81 \begin{align*} \frac{1}{16} \,{\left (8 \, x^{4} - 42 \, x^{2} + 267\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{1083}{32} \, \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(8*x^4 - 42*x^2 + 267)*sqrt(x^4 + 5*x^2 + 3) + 1083/32*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \left (3 x^{2} + 2\right )}{\sqrt{x^{4} + 5 x^{2} + 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**5*(3*x**2 + 2)/sqrt(x**4 + 5*x**2 + 3), x)

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Giac [A]  time = 1.13247, size = 72, normalized size = 0.94 \begin{align*} \frac{1}{16} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \, x^{2} - 21\right )} x^{2} + 267\right )} + \frac{1083}{32} \, \log \left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(x^4 + 5*x^2 + 3)*(2*(4*x^2 - 21)*x^2 + 267) + 1083/32*log(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)