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

Optimal. Leaf size=60 \[ \frac{3}{10} \left (x^4+5\right )^{5/2}+\frac{1}{4} x^2 \left (x^4+5\right )^{3/2}+\frac{15}{8} x^2 \sqrt{x^4+5}+\frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

[Out]

(15*x^2*Sqrt[5 + x^4])/8 + (x^2*(5 + x^4)^(3/2))/4 + (3*(5 + x^4)^(5/2))/10 + (75*ArcSinh[x^2/Sqrt[5]])/8

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Rubi [A]  time = 0.030477, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1248, 641, 195, 215} \[ \frac{3}{10} \left (x^4+5\right )^{5/2}+\frac{1}{4} x^2 \left (x^4+5\right )^{3/2}+\frac{15}{8} x^2 \sqrt{x^4+5}+\frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*(2 + 3*x^2)*(5 + x^4)^(3/2),x]

[Out]

(15*x^2*Sqrt[5 + x^4])/8 + (x^2*(5 + x^4)^(3/2))/4 + (3*(5 + x^4)^(5/2))/10 + (75*ArcSinh[x^2/Sqrt[5]])/8

Rule 1248

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

Rule 641

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 x \left (2+3 x^2\right ) \left (5+x^4\right )^{3/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (2+3 x) \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{3}{10} \left (5+x^4\right )^{5/2}+\operatorname{Subst}\left (\int \left (5+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac{1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac{3}{10} \left (5+x^4\right )^{5/2}+\frac{15}{4} \operatorname{Subst}\left (\int \sqrt{5+x^2} \, dx,x,x^2\right )\\ &=\frac{15}{8} x^2 \sqrt{5+x^4}+\frac{1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac{3}{10} \left (5+x^4\right )^{5/2}+\frac{75}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=\frac{15}{8} x^2 \sqrt{5+x^4}+\frac{1}{4} x^2 \left (5+x^4\right )^{3/2}+\frac{3}{10} \left (5+x^4\right )^{5/2}+\frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0324412, size = 56, normalized size = 0.93 \[ \frac{1}{2} \sqrt{x^4+5} \left (\frac{3 x^8}{5}+\frac{x^6}{2}+6 x^4+\frac{25 x^2}{4}+15\right )+\frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[5 + x^4]*(15 + (25*x^2)/4 + 6*x^4 + x^6/2 + (3*x^8)/5))/2 + (75*ArcSinh[x^2/Sqrt[5]])/8

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Maple [A]  time = 0.012, size = 46, normalized size = 0.8 \begin{align*}{\frac{3}{10} \left ({x}^{4}+5 \right ) ^{{\frac{5}{2}}}}+{\frac{{x}^{6}}{4}\sqrt{{x}^{4}+5}}+{\frac{25\,{x}^{2}}{8}\sqrt{{x}^{4}+5}}+{\frac{75}{8}{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/10*(x^4+5)^(5/2)+1/4*x^6*(x^4+5)^(1/2)+25/8*x^2*(x^4+5)^(1/2)+75/8*arcsinh(1/5*x^2*5^(1/2))

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Maxima [B]  time = 1.4211, size = 128, normalized size = 2.13 \begin{align*} \frac{3}{10} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} + \frac{25 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{5 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{8 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac{75}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{75}{16} \, \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*x^2+2)*(x^4+5)^(3/2),x, algorithm="maxima")

[Out]

3/10*(x^4 + 5)^(5/2) + 25/8*(3*sqrt(x^4 + 5)/x^2 - 5*(x^4 + 5)^(3/2)/x^6)/(2*(x^4 + 5)/x^4 - (x^4 + 5)^2/x^8 -
1) + 75/16*log(sqrt(x^4 + 5)/x^2 + 1) - 75/16*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 1.53592, size = 131, normalized size = 2.18 \begin{align*} \frac{1}{40} \,{\left (12 \, x^{8} + 10 \, x^{6} + 120 \, x^{4} + 125 \, x^{2} + 300\right )} \sqrt{x^{4} + 5} - \frac{75}{8} \, \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*x^2+2)*(x^4+5)^(3/2),x, algorithm="fricas")

[Out]

1/40*(12*x^8 + 10*x^6 + 120*x^4 + 125*x^2 + 300)*sqrt(x^4 + 5) - 75/8*log(-x^2 + sqrt(x^4 + 5))

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Sympy [B]  time = 7.25373, size = 109, normalized size = 1.82 \begin{align*} \frac{x^{10}}{4 \sqrt{x^{4} + 5}} + \frac{3 x^{8} \sqrt{x^{4} + 5}}{10} + \frac{35 x^{6}}{8 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{2} + \frac{125 x^{2}}{8 \sqrt{x^{4} + 5}} + \frac{5 \left (x^{4} + 5\right )^{\frac{3}{2}}}{2} - 5 \sqrt{x^{4} + 5} + \frac{75 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**10/(4*sqrt(x**4 + 5)) + 3*x**8*sqrt(x**4 + 5)/10 + 35*x**6/(8*sqrt(x**4 + 5)) + x**4*sqrt(x**4 + 5)/2 + 125
*x**2/(8*sqrt(x**4 + 5)) + 5*(x**4 + 5)**(3/2)/2 - 5*sqrt(x**4 + 5) + 75*asinh(sqrt(5)*x**2/5)/8

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Giac [A]  time = 1.11493, size = 70, normalized size = 1.17 \begin{align*} \frac{1}{40} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (6 \, x^{2} + 5\right )} x^{2} + 60\right )} x^{2} + 125\right )} x^{2} + 300\right )} - \frac{75}{8} \, \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*x^2+2)*(x^4+5)^(3/2),x, algorithm="giac")

[Out]

1/40*sqrt(x^4 + 5)*((2*((6*x^2 + 5)*x^2 + 60)*x^2 + 125)*x^2 + 300) - 75/8*log(-x^2 + sqrt(x^4 + 5))