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

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

[Out]

(15*x^2*Sqrt[5 + x^4])/8 + (x^2*(5 + x^4)^(3/2))/4 + (3*(5 + x^4)^(5/2))/10 + (7
5*ArcSinh[x^2/Sqrt[5]])/8

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Rubi [A]  time = 0.0720775, 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 \[ \frac{3}{10} \left (x^4+5\right )^{5/2}+\frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{4} x^2 \left (x^4+5\right )^{3/2}+\frac{15}{8} x^2 \sqrt{x^4+5} \]

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 + (7
5*ArcSinh[x^2/Sqrt[5]])/8

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Rubi in Sympy [A]  time = 7.308, size = 54, normalized size = 0.9 \[ \frac{x^{2} \left (x^{4} + 5\right )^{\frac{3}{2}}}{4} + \frac{15 x^{2} \sqrt{x^{4} + 5}}{8} + \frac{3 \left (x^{4} + 5\right )^{\frac{5}{2}}}{10} + \frac{75 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(3*x**2+2)*(x**4+5)**(3/2),x)

[Out]

x**2*(x**4 + 5)**(3/2)/4 + 15*x**2*sqrt(x**4 + 5)/8 + 3*(x**4 + 5)**(5/2)/10 + 7
5*asinh(sqrt(5)*x**2/5)/8

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Mathematica [A]  time = 0.031343, size = 56, normalized size = 0.93 \[ \frac{75}{8} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )+\frac{1}{2} \sqrt{x^4+5} \left (\frac{3 x^8}{5}+\frac{x^6}{2}+6 x^4+\frac{25 x^2}{4}+15\right ) \]

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 = 0.019, size = 46, normalized size = 0.8 \[{\frac{{x}^{6}}{4}\sqrt{{x}^{4}+5}}+{\frac{25\,{x}^{2}}{8}\sqrt{{x}^{4}+5}}+{\frac{75}{8}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }+{\frac{3}{10} \left ({x}^{4}+5 \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.776508, size = 128, normalized size = 2.13 \[ \frac{3}{10} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} + \frac{25 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{5 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{8 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} + \frac{75}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) - \frac{75}{16} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x,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 = 0.28594, size = 271, normalized size = 4.52 \[ -\frac{192 \, x^{20} + 160 \, x^{18} + 3600 \, x^{16} + 3400 \, x^{14} + 25500 \, x^{12} + 20750 \, x^{10} + 82500 \, x^{8} + 41875 \, x^{6} + 112500 \, x^{4} + 15625 \, x^{2} + 375 \,{\left (16 \, x^{10} + 100 \, x^{6} + 125 \, x^{2} -{\left (16 \, x^{8} + 60 \, x^{4} + 25\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (192 \, x^{18} + 160 \, x^{16} + 3120 \, x^{14} + 3000 \, x^{12} + 18300 \, x^{10} + 13750 \, x^{8} + 45000 \, x^{6} + 15625 \, x^{4} + 37500 \, x^{2}\right )} \sqrt{x^{4} + 5} + 37500}{40 \,{\left (16 \, x^{10} + 100 \, x^{6} + 125 \, x^{2} -{\left (16 \, x^{8} + 60 \, x^{4} + 25\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/40*(192*x^20 + 160*x^18 + 3600*x^16 + 3400*x^14 + 25500*x^12 + 20750*x^10 + 8
2500*x^8 + 41875*x^6 + 112500*x^4 + 15625*x^2 + 375*(16*x^10 + 100*x^6 + 125*x^2
- (16*x^8 + 60*x^4 + 25)*sqrt(x^4 + 5))*log(-x^2 + sqrt(x^4 + 5)) - (192*x^18 +
160*x^16 + 3120*x^14 + 3000*x^12 + 18300*x^10 + 13750*x^8 + 45000*x^6 + 15625*x
^4 + 37500*x^2)*sqrt(x^4 + 5) + 37500)/(16*x^10 + 100*x^6 + 125*x^2 - (16*x^8 +
60*x^4 + 25)*sqrt(x^4 + 5))

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Sympy [A]  time = 17.1906, size = 109, normalized size = 1.82 \[ \frac{x^{10}}{4 \sqrt{x^{4} + 5}} + \frac{3 x^{8} \sqrt{x^{4} + 5}}{10} + \frac{35 x^{6}}{8 \sqrt{x^{4} + 5}} + \frac{x^{4} \sqrt{x^{4} + 5}}{2} + \frac{125 x^{2}}{8 \sqrt{x^{4} + 5}} + \frac{5 \left (x^{4} + 5\right )^{\frac{3}{2}}}{2} - 5 \sqrt{x^{4} + 5} + \frac{75 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{8} \]

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/XCAS [A]  time = 0.267085, size = 70, normalized size = 1.17 \[ \frac{1}{40} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (6 \, x^{2} + 5\right )} x^{2} + 60\right )} x^{2} + 125\right )} x^{2} + 300\right )} - \frac{75}{8} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 5)^(3/2)*(3*x^2 + 2)*x,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*ln(-x
^2 + sqrt(x^4 + 5))