3.1.46 \(\int \frac {x^3 (2+3 x^2)}{(5+x^4)^{3/2}} \, dx\) [46]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1266, 792, 221} \begin {gather*} \frac {3}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {3 x^2+2}{2 \sqrt {x^4+5}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(2 + 3*x^2)/Sqrt[5 + x^4] + (3*ArcSinh[x^2/Sqrt[5]])/2

Rule 221

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]

Rule 792

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 1266

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]

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} \text {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} \text {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*}

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Mathematica [A]
time = 0.11, size = 39, normalized size = 1.11 \begin {gather*} \frac {-2-3 x^2}{2 \sqrt {5+x^4}}+\frac {3}{2} \tanh ^{-1}\left (\frac {x^2}{\sqrt {5+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.15, size = 34, normalized size = 0.97

method	result	size
risch	\(-\frac {3 x^{2}+2}{2 \sqrt {x^{4}+5}}+\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}\)	\(29\)
trager	\(-\frac {3 x^{2}+2}{2 \sqrt {x^{4}+5}}+\frac {3 \ln \left (x^{2}+\sqrt {x^{4}+5}\right )}{2}\)	\(32\)
default	\(-\frac {3 x^{2}}{2 \sqrt {x^{4}+5}}+\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}-\frac {1}{\sqrt {x^{4}+5}}\)	\(34\)
elliptic	\(-\frac {3 x^{2}}{2 \sqrt {x^{4}+5}}+\frac {3 \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}-\frac {1}{\sqrt {x^{4}+5}}\)	\(34\)
meijerg	\(\frac {-\frac {3 \sqrt {\pi }\, x^{2} \sqrt {5}}{10 \sqrt {1+\frac {x^{4}}{5}}}+\frac {3 \sqrt {\pi }\, \arcsinh \left (\frac {x^{2} \sqrt {5}}{5}\right )}{2}}{\sqrt {\pi }}+\frac {\sqrt {5}\, \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {1+\frac {x^{4}}{5}}}\right )}{5 \sqrt {\pi }}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]
time = 0.51, size = 54, normalized size = 1.54 \begin {gather*} -\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 {gather*}

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)

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Fricas [A]
time = 0.38, size = 52, normalized size = 1.49 \begin {gather*} -\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 {gather*}

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)

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Sympy [A]
time = 4.42, size = 39, normalized size = 1.11 \begin {gather*} - \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 {gather*}

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)

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Giac [A]
time = 3.32, size = 33, normalized size = 0.94 \begin {gather*} -\frac {3 \, x^{2} + 2}{2 \, \sqrt {x^{4} + 5}} - \frac {3}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.84, size = 82, normalized size = 2.34 \begin {gather*} \frac {3\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{2}-\frac {\sqrt {5}\,\left (2+\sqrt {5}\,3{}\mathrm {i}\right )\,\sqrt {x^4+5}\,1{}\mathrm {i}}{20\,\left (-x^2+\sqrt {5}\,1{}\mathrm {i}\right )}+\frac {\sqrt {5}\,\left (-2+\sqrt {5}\,3{}\mathrm {i}\right )\,\sqrt {x^4+5}\,1{}\mathrm {i}}{20\,\left (x^2+\sqrt {5}\,1{}\mathrm {i}\right )} \end {gather*}

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*asinh((5^(1/2)*x^2)/5))/2 - (5^(1/2)*(5^(1/2)*3i + 2)*(x^4 + 5)^(1/2)*1i)/(20*(5^(1/2)*1i - x^2)) + (5^(1/2
)*(5^(1/2)*3i - 2)*(x^4 + 5)^(1/2)*1i)/(20*(5^(1/2)*1i + x^2))

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