3.181 \(\int \frac{x^7 (2+3 x^2)}{\sqrt{3+5 x^2+x^4}} \, dx\)

Optimal. Leaf size=98 \[ \frac{3}{8} \sqrt{x^4+5 x^2+3} x^6-\frac{89}{48} \sqrt{x^4+5 x^2+3} x^4-\frac{1}{384} \left (24243-3802 x^2\right ) \sqrt{x^4+5 x^2+3}+\frac{32801}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

(-89*x^4*Sqrt[3 + 5*x^2 + x^4])/48 + (3*x^6*Sqrt[3 + 5*x^2 + x^4])/8 - ((24243 - 3802*x^2)*Sqrt[3 + 5*x^2 + x^
4])/384 + (32801*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/256

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Rubi [A]  time = 0.086537, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{3}{8} \sqrt{x^4+5 x^2+3} x^6-\frac{89}{48} \sqrt{x^4+5 x^2+3} x^4-\frac{1}{384} \left (24243-3802 x^2\right ) \sqrt{x^4+5 x^2+3}+\frac{32801}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(x^7*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]

[Out]

(-89*x^4*Sqrt[3 + 5*x^2 + x^4])/48 + (3*x^6*Sqrt[3 + 5*x^2 + x^4])/8 - ((24243 - 3802*x^2)*Sqrt[3 + 5*x^2 + x^
4])/384 + (32801*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/256

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^7 \left (2+3 x^2\right )}{\sqrt{3+5 x^2+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3 (2+3 x)}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac{3}{8} x^6 \sqrt{3+5 x^2+x^4}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{\left (-27-\frac{89 x}{2}\right ) x^2}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{89}{48} x^4 \sqrt{3+5 x^2+x^4}+\frac{3}{8} x^6 \sqrt{3+5 x^2+x^4}+\frac{1}{24} \operatorname{Subst}\left (\int \frac{x \left (267+\frac{1901 x}{4}\right )}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{89}{48} x^4 \sqrt{3+5 x^2+x^4}+\frac{3}{8} x^6 \sqrt{3+5 x^2+x^4}-\frac{1}{384} \left (24243-3802 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{32801}{256} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{89}{48} x^4 \sqrt{3+5 x^2+x^4}+\frac{3}{8} x^6 \sqrt{3+5 x^2+x^4}-\frac{1}{384} \left (24243-3802 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{32801}{128} \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{89}{48} x^4 \sqrt{3+5 x^2+x^4}+\frac{3}{8} x^6 \sqrt{3+5 x^2+x^4}-\frac{1}{384} \left (24243-3802 x^2\right ) \sqrt{3+5 x^2+x^4}+\frac{32801}{256} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0312108, size = 66, normalized size = 0.67 \[ \frac{1}{768} \left (2 \sqrt{x^4+5 x^2+3} \left (144 x^6-712 x^4+3802 x^2-24243\right )+98403 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(x^7*(2 + 3*x^2))/Sqrt[3 + 5*x^2 + x^4],x]

[Out]

(2*Sqrt[3 + 5*x^2 + x^4]*(-24243 + 3802*x^2 - 712*x^4 + 144*x^6) + 98403*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2
+ x^4])])/768

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Maple [A]  time = 0.016, size = 87, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{6}}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{89\,{x}^{4}}{48}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{1901\,{x}^{2}}{192}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{8081}{128}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{32801}{256}\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^7*(3*x^2+2)/(x^4+5*x^2+3)^(1/2),x)

[Out]

3/8*x^6*(x^4+5*x^2+3)^(1/2)-89/48*x^4*(x^4+5*x^2+3)^(1/2)+1901/192*x^2*(x^4+5*x^2+3)^(1/2)-8081/128*(x^4+5*x^2
+3)^(1/2)+32801/256*ln(5/2+x^2+(x^4+5*x^2+3)^(1/2))

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Maxima [A]  time = 0.949552, size = 122, normalized size = 1.24 \begin{align*} \frac{3}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{6} - \frac{89}{48} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{4} + \frac{1901}{192} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \frac{8081}{128} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{32801}{256} \, \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^7*(3*x^2+2)/(x^4+5*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

3/8*sqrt(x^4 + 5*x^2 + 3)*x^6 - 89/48*sqrt(x^4 + 5*x^2 + 3)*x^4 + 1901/192*sqrt(x^4 + 5*x^2 + 3)*x^2 - 8081/12
8*sqrt(x^4 + 5*x^2 + 3) + 32801/256*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 1.37426, size = 165, normalized size = 1.68 \begin{align*} \frac{1}{384} \,{\left (144 \, x^{6} - 712 \, x^{4} + 3802 \, x^{2} - 24243\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{32801}{256} \, \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^7*(3*x^2+2)/(x^4+5*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

1/384*(144*x^6 - 712*x^4 + 3802*x^2 - 24243)*sqrt(x^4 + 5*x^2 + 3) - 32801/256*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^{7} \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**7*(3*x**2+2)/(x**4+5*x**2+3)**(1/2),x)

[Out]

Integral(x**7*(3*x**2 + 2)/sqrt(x**4 + 5*x**2 + 3), x)

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Giac [A]  time = 1.1422, size = 81, normalized size = 0.83 \begin{align*} \frac{1}{384} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \,{\left (18 \, x^{2} - 89\right )} x^{2} + 1901\right )} x^{2} - 24243\right )} - \frac{32801}{256} \, \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^7*(3*x^2+2)/(x^4+5*x^2+3)^(1/2),x, algorithm="giac")

[Out]

1/384*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(18*x^2 - 89)*x^2 + 1901)*x^2 - 24243) - 32801/256*log(2*x^2 - 2*sqrt(x^4 +
5*x^2 + 3) + 5)