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

Optimal. Leaf size=102 \[ \frac{3}{10} \left (x^4+5 x^2+3\right )^{3/2} x^4+\frac{1}{480} \left (1837-510 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac{1633}{256} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}+\frac{21229}{512} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

(-1633*(5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/256 + (3*x^4*(3 + 5*x^2 + x^4)^(3/2))/10 + ((1837 - 510*x^2)*(3 + 5*
x^2 + x^4)^(3/2))/480 + (21229*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/512

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Rubi [A]  time = 0.079444, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1251, 832, 779, 612, 621, 206} \[ \frac{3}{10} \left (x^4+5 x^2+3\right )^{3/2} x^4+\frac{1}{480} \left (1837-510 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac{1633}{256} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}+\frac{21229}{512} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1633*(5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/256 + (3*x^4*(3 + 5*x^2 + x^4)^(3/2))/10 + ((1837 - 510*x^2)*(3 + 5*
x^2 + x^4)^(3/2))/480 + (21229*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/512

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 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 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 x^5 \left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (2+3 x) \sqrt{3+5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{3}{10} x^4 \left (3+5 x^2+x^4\right )^{3/2}+\frac{1}{10} \operatorname{Subst}\left (\int \left (-18-\frac{85 x}{2}\right ) x \sqrt{3+5 x+x^2} \, dx,x,x^2\right )\\ &=\frac{3}{10} x^4 \left (3+5 x^2+x^4\right )^{3/2}+\frac{1}{480} \left (1837-510 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac{1633}{64} \operatorname{Subst}\left (\int \sqrt{3+5 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{1633}{256} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{3}{10} x^4 \left (3+5 x^2+x^4\right )^{3/2}+\frac{1}{480} \left (1837-510 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{21229}{512} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{1633}{256} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{3}{10} x^4 \left (3+5 x^2+x^4\right )^{3/2}+\frac{1}{480} \left (1837-510 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{21229}{256} \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{1633}{256} \left (5+2 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{3}{10} x^4 \left (3+5 x^2+x^4\right )^{3/2}+\frac{1}{480} \left (1837-510 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac{21229}{512} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0306944, size = 71, normalized size = 0.7 \[ \frac{2 \sqrt{x^4+5 x^2+3} \left (1152 x^8+1680 x^6-2248 x^4+12250 x^2-78387\right )+318435 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )}{7680} \]

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]*(-78387 + 12250*x^2 - 2248*x^4 + 1680*x^6 + 1152*x^8) + 318435*ArcTanh[(5 + 2*x^2)/(2
*Sqrt[3 + 5*x^2 + x^4])])/7680

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Maple [A]  time = 0.023, size = 91, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{4}}{10} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{17\,{x}^{2}}{16} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{1837}{480} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{3266\,{x}^{2}+8165}{256}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{21229}{512}\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]

3/10*x^4*(x^4+5*x^2+3)^(3/2)-17/16*x^2*(x^4+5*x^2+3)^(3/2)+1837/480*(x^4+5*x^2+3)^(3/2)-1633/256*(2*x^2+5)*(x^
4+5*x^2+3)^(1/2)+21229/512*ln(5/2+x^2+(x^4+5*x^2+3)^(1/2))

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Maxima [A]  time = 0.982898, size = 140, normalized size = 1.37 \begin{align*} \frac{3}{10} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} x^{4} - \frac{17}{16} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} x^{2} - \frac{1633}{128} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{1837}{480} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} - \frac{8165}{256} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{21229}{512} \, \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]

3/10*(x^4 + 5*x^2 + 3)^(3/2)*x^4 - 17/16*(x^4 + 5*x^2 + 3)^(3/2)*x^2 - 1633/128*sqrt(x^4 + 5*x^2 + 3)*x^2 + 18
37/480*(x^4 + 5*x^2 + 3)^(3/2) - 8165/256*sqrt(x^4 + 5*x^2 + 3) + 21229/512*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3
) + 5)

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Fricas [A]  time = 1.5956, size = 185, normalized size = 1.81 \begin{align*} \frac{1}{3840} \,{\left (1152 \, x^{8} + 1680 \, x^{6} - 2248 \, x^{4} + 12250 \, x^{2} - 78387\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{21229}{512} \, \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]

1/3840*(1152*x^8 + 1680*x^6 - 2248*x^4 + 12250*x^2 - 78387)*sqrt(x^4 + 5*x^2 + 3) - 21229/512*log(-2*x^2 + 2*s
qrt(x^4 + 5*x^2 + 3) - 5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int 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 = 1.11535, size = 90, normalized size = 0.88 \begin{align*} \frac{1}{3840} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \,{\left (6 \,{\left (24 \, x^{2} + 35\right )} x^{2} - 281\right )} x^{2} + 6125\right )} x^{2} - 78387\right )} - \frac{21229}{512} \, \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]

1/3840*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(6*(24*x^2 + 35)*x^2 - 281)*x^2 + 6125)*x^2 - 78387) - 21229/512*log(2*x^2
- 2*sqrt(x^4 + 5*x^2 + 3) + 5)