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

Optimal. Leaf size=35 \[ \frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3 x^2+2}{2 \sqrt{x^4+5}} \]

[Out]

-(2 + 3*x^2)/(2*Sqrt[5 + x^4]) + (3*ArcSinh[x^2/Sqrt[5]])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0265763, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {1252, 778, 215} \[ \frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{3 x^2+2}{2 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2 + 3*x^2)/(2*Sqrt[5 + x^4]) + (3*ArcSinh[x^2/Sqrt[5]])/2

Rule 1252

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

Rule 778

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

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 \frac{x^3 \left (2+3 x^2\right )}{\left (5+x^4\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (2+3 x)}{\left (5+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{2+3 x^2}{2 \sqrt{5+x^4}}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=-\frac{2+3 x^2}{2 \sqrt{5+x^4}}+\frac{3}{2} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0187678, size = 41, normalized size = 1.17 \[ \frac{-3 x^2+3 \sqrt{x^4+5} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-2}{2 \sqrt{x^4+5}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2 - 3*x^2 + 3*Sqrt[5 + x^4]*ArcSinh[x^2/Sqrt[5]])/(2*Sqrt[5 + x^4])

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 34, normalized size = 1. \begin{align*} -{\frac{3\,{x}^{2}}{2}{\frac{1}{\sqrt{{x}^{4}+5}}}}+{\frac{3}{2}{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{5}}{5}} \right ) }-{\frac{1}{\sqrt{{x}^{4}+5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x^2/(x^4+5)^(1/2)+3/2*arcsinh(1/5*x^2*5^(1/2))-1/(x^4+5)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.41979, size = 73, normalized size = 2.09 \begin{align*} -\frac{3 \, x^{2}}{2 \, \sqrt{x^{4} + 5}} - \frac{1}{\sqrt{x^{4} + 5}} + \frac{3}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{3}{4} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*x^2/sqrt(x^4 + 5) - 1/sqrt(x^4 + 5) + 3/4*log(sqrt(x^4 + 5)/x^2 + 1) - 3/4*log(sqrt(x^4 + 5)/x^2 - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.51848, size = 131, normalized size = 3.74 \begin{align*} -\frac{3 \, x^{4} + 3 \,{\left (x^{4} + 5\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) + \sqrt{x^{4} + 5}{\left (3 \, x^{2} + 2\right )} + 15}{2 \,{\left (x^{4} + 5\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*x^4 + 3*(x^4 + 5)*log(-x^2 + sqrt(x^4 + 5)) + sqrt(x^4 + 5)*(3*x^2 + 2) + 15)/(x^4 + 5)

________________________________________________________________________________________

Sympy [A]  time = 7.08634, size = 39, normalized size = 1.11 \begin{align*} - \frac{3 x^{2}}{2 \sqrt{x^{4} + 5}} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{2} - \frac{1}{\sqrt{x^{4} + 5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x**2/(2*sqrt(x**4 + 5)) + 3*asinh(sqrt(5)*x**2/5)/2 - 1/sqrt(x**4 + 5)

________________________________________________________________________________________

Giac [A]  time = 1.18008, size = 45, normalized size = 1.29 \begin{align*} -\frac{3 \, x^{2} + 2}{2 \, \sqrt{x^{4} + 5}} - \frac{3}{2} \, \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(3*x^2 + 2)/sqrt(x^4 + 5) - 3/2*log(-x^2 + sqrt(x^4 + 5))