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

Optimal. Leaf size=67 \[ \frac{3}{8} \sqrt{x^4+5} x^6+\frac{1}{3} \sqrt{x^4+5} x^4-\frac{5}{48} \left (27 x^2+32\right ) \sqrt{x^4+5}+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

[Out]

(x^4*Sqrt[5 + x^4])/3 + (3*x^6*Sqrt[5 + x^4])/8 - (5*(32 + 27*x^2)*Sqrt[5 + x^4])/48 + (225*ArcSinh[x^2/Sqrt[5
]])/16

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Rubi [A]  time = 0.0575545, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1252, 833, 780, 215} \[ \frac{3}{8} \sqrt{x^4+5} x^6+\frac{1}{3} \sqrt{x^4+5} x^4-\frac{5}{48} \left (27 x^2+32\right ) \sqrt{x^4+5}+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^4*Sqrt[5 + x^4])/3 + (3*x^6*Sqrt[5 + x^4])/8 - (5*(32 + 27*x^2)*Sqrt[5 + x^4])/48 + (225*ArcSinh[x^2/Sqrt[5
]])/16

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 833

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(g*(d + e*x)
^m*(a + 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 + c*x^2)^p*
Simp[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p
}, x] && NeQ[c*d^2 + 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 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[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^7 \left (2+3 x^2\right )}{\sqrt{5+x^4}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3 (2+3 x)}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=\frac{3}{8} x^6 \sqrt{5+x^4}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{x^2 (-45+8 x)}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^4 \sqrt{5+x^4}+\frac{3}{8} x^6 \sqrt{5+x^4}+\frac{1}{24} \operatorname{Subst}\left (\int \frac{(-80-135 x) x}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^4 \sqrt{5+x^4}+\frac{3}{8} x^6 \sqrt{5+x^4}-\frac{5}{48} \left (32+27 x^2\right ) \sqrt{5+x^4}+\frac{225}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{5+x^2}} \, dx,x,x^2\right )\\ &=\frac{1}{3} x^4 \sqrt{5+x^4}+\frac{3}{8} x^6 \sqrt{5+x^4}-\frac{5}{48} \left (32+27 x^2\right ) \sqrt{5+x^4}+\frac{225}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0348459, size = 44, normalized size = 0.66 \[ \frac{1}{48} \left (\sqrt{x^4+5} \left (18 x^6+16 x^4-135 x^2-160\right )+675 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[5 + x^4]*(-160 - 135*x^2 + 16*x^4 + 18*x^6) + 675*ArcSinh[x^2/Sqrt[5]])/48

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Maple [A]  time = 0.017, size = 51, normalized size = 0.8 \begin{align*}{\frac{3\,{x}^{6}}{8}\sqrt{{x}^{4}+5}}-{\frac{45\,{x}^{2}}{16}\sqrt{{x}^{4}+5}}+{\frac{225}{16}{\it Arcsinh} \left ({\frac{{x}^{2}\sqrt{5}}{5}} \right ) }+{\frac{{x}^{4}-10}{3}\sqrt{{x}^{4}+5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(3*x^2+2)/(x^4+5)^(1/2),x)

[Out]

3/8*x^6*(x^4+5)^(1/2)-45/16*x^2*(x^4+5)^(1/2)+225/16*arcsinh(1/5*x^2*5^(1/2))+1/3*(x^4+5)^(1/2)*(x^4-10)

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Maxima [A]  time = 1.45685, size = 140, normalized size = 2.09 \begin{align*} \frac{1}{3} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{4} + 5} - \frac{75 \,{\left (\frac{5 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{3 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{16 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac{225}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{225}{32} \, \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^7*(3*x^2+2)/(x^4+5)^(1/2),x, algorithm="maxima")

[Out]

1/3*(x^4 + 5)^(3/2) - 5*sqrt(x^4 + 5) - 75/16*(5*sqrt(x^4 + 5)/x^2 - 3*(x^4 + 5)^(3/2)/x^6)/(2*(x^4 + 5)/x^4 -
(x^4 + 5)^2/x^8 - 1) + 225/32*log(sqrt(x^4 + 5)/x^2 + 1) - 225/32*log(sqrt(x^4 + 5)/x^2 - 1)

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Fricas [A]  time = 1.54282, size = 120, normalized size = 1.79 \begin{align*} \frac{1}{48} \,{\left (18 \, x^{6} + 16 \, x^{4} - 135 \, x^{2} - 160\right )} \sqrt{x^{4} + 5} - \frac{225}{16} \, \log \left (-x^{2} + \sqrt{x^{4} + 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)^(1/2),x, algorithm="fricas")

[Out]

1/48*(18*x^6 + 16*x^4 - 135*x^2 - 160)*sqrt(x^4 + 5) - 225/16*log(-x^2 + sqrt(x^4 + 5))

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Sympy [A]  time = 6.5133, size = 85, normalized size = 1.27 \begin{align*} \frac{3 x^{10}}{8 \sqrt{x^{4} + 5}} - \frac{15 x^{6}}{16 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{3} - \frac{225 x^{2}}{16 \sqrt{x^{4} + 5}} - \frac{10 \sqrt{x^{4} + 5}}{3} + \frac{225 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(3*x**2+2)/(x**4+5)**(1/2),x)

[Out]

3*x**10/(8*sqrt(x**4 + 5)) - 15*x**6/(16*sqrt(x**4 + 5)) + x**4*sqrt(x**4 + 5)/3 - 225*x**2/(16*sqrt(x**4 + 5)
) - 10*sqrt(x**4 + 5)/3 + 225*asinh(sqrt(5)*x**2/5)/16

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Giac [A]  time = 1.13552, size = 62, normalized size = 0.93 \begin{align*} \frac{1}{48} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left (9 \, x^{2} + 8\right )} x^{2} - 135\right )} x^{2} - 160\right )} - \frac{225}{16} \, \log \left (-x^{2} + \sqrt{x^{4} + 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)^(1/2),x, algorithm="giac")

[Out]

1/48*sqrt(x^4 + 5)*((2*(9*x^2 + 8)*x^2 - 135)*x^2 - 160) - 225/16*log(-x^2 + sqrt(x^4 + 5))