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

Optimal. Leaf size=51 \[ -\frac{75}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{15}{16} \sqrt{x^4+5} x^2+\frac{1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2} \]

[Out]

(-15*x^2*Sqrt[5 + x^4])/16 + ((8 + 9*x^2)*(5 + x^4)^(3/2))/24 - (75*ArcSinh[x^2/
Sqrt[5]])/16

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Rubi [A]  time = 0.0932686, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{75}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{15}{16} \sqrt{x^4+5} x^2+\frac{1}{24} \left (9 x^2+8\right ) \left (x^4+5\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-15*x^2*Sqrt[5 + x^4])/16 + ((8 + 9*x^2)*(5 + x^4)^(3/2))/24 - (75*ArcSinh[x^2/
Sqrt[5]])/16

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Rubi in Sympy [A]  time = 8.58032, size = 46, normalized size = 0.9 \[ - \frac{15 x^{2} \sqrt{x^{4} + 5}}{16} + \frac{\left (9 x^{2} + 8\right ) \left (x^{4} + 5\right )^{\frac{3}{2}}}{24} - \frac{75 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(3*x**2+2)*(x**4+5)**(1/2),x)

[Out]

-15*x**2*sqrt(x**4 + 5)/16 + (9*x**2 + 8)*(x**4 + 5)**(3/2)/24 - 75*asinh(sqrt(5
)*x**2/5)/16

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Mathematica [A]  time = 0.0343582, size = 44, normalized size = 0.86 \[ \frac{1}{48} \left (\sqrt{x^4+5} \left (18 x^6+16 x^4+45 x^2+80\right )-225 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[5 + x^4]*(80 + 45*x^2 + 16*x^4 + 18*x^6) - 225*ArcSinh[x^2/Sqrt[5]])/48

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Maple [A]  time = 0.009, size = 46, normalized size = 0.9 \[{\frac{1}{3} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{x}^{2}}{8} \left ({x}^{4}+5 \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{x}^{2}}{16}\sqrt{{x}^{4}+5}}-{\frac{75}{16}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(3*x^2+2)*(x^4+5)^(1/2),x)

[Out]

1/3*(x^4+5)^(3/2)+3/8*x^2*(x^4+5)^(3/2)-15/16*x^2*(x^4+5)^(1/2)-75/16*arcsinh(1/
5*5^(1/2)*x^2)

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Maxima [A]  time = 0.784207, size = 126, normalized size = 2.47 \[ \frac{1}{3} \,{\left (x^{4} + 5\right )}^{\frac{3}{2}} - \frac{75 \,{\left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + \frac{{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}}\right )}}{16 \,{\left (\frac{2 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{{\left (x^{4} + 5\right )}^{2}}{x^{8}} - 1\right )}} - \frac{75}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{75}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.270577, size = 231, normalized size = 4.53 \[ -\frac{144 \, x^{16} + 128 \, x^{14} + 1440 \, x^{12} + 1600 \, x^{10} + 4500 \, x^{8} + 6400 \, x^{6} + 4500 \, x^{4} + 8000 \, x^{2} - 225 \,{\left (8 \, x^{8} + 40 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (144 \, x^{14} + 128 \, x^{12} + 1080 \, x^{10} + 1280 \, x^{8} + 2250 \, x^{6} + 3600 \, x^{4} + 1125 \, x^{2} + 2000\right )} \sqrt{x^{4} + 5}}{48 \,{\left (8 \, x^{8} + 40 \, x^{4} - 4 \,{\left (2 \, x^{6} + 5 \, x^{2}\right )} \sqrt{x^{4} + 5} + 25\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(144*x^16 + 128*x^14 + 1440*x^12 + 1600*x^10 + 4500*x^8 + 6400*x^6 + 4500*
x^4 + 8000*x^2 - 225*(8*x^8 + 40*x^4 - 4*(2*x^6 + 5*x^2)*sqrt(x^4 + 5) + 25)*log
(-x^2 + sqrt(x^4 + 5)) - (144*x^14 + 128*x^12 + 1080*x^10 + 1280*x^8 + 2250*x^6
+ 3600*x^4 + 1125*x^2 + 2000)*sqrt(x^4 + 5))/(8*x^8 + 40*x^4 - 4*(2*x^6 + 5*x^2)
*sqrt(x^4 + 5) + 25)

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Sympy [A]  time = 10.5978, size = 70, normalized size = 1.37 \[ \frac{3 x^{10}}{8 \sqrt{x^{4} + 5}} + \frac{45 x^{6}}{16 \sqrt{x^{4} + 5}} + \frac{75 x^{2}}{16 \sqrt{x^{4} + 5}} + \frac{\left (x^{4} + 5\right )^{\frac{3}{2}}}{3} - \frac{75 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(3*x**2+2)*(x**4+5)**(1/2),x)

[Out]

3*x**10/(8*sqrt(x**4 + 5)) + 45*x**6/(16*sqrt(x**4 + 5)) + 75*x**2/(16*sqrt(x**4
+ 5)) + (x**4 + 5)**(3/2)/3 - 75*asinh(sqrt(5)*x**2/5)/16

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GIAC/XCAS [A]  time = 0.263236, size = 62, normalized size = 1.22 \[ \frac{1}{48} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left (9 \, x^{2} + 8\right )} x^{2} + 45\right )} x^{2} + 80\right )} + \frac{75}{16} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*sqrt(x^4 + 5)*((2*(9*x^2 + 8)*x^2 + 45)*x^2 + 80) + 75/16*ln(-x^2 + sqrt(x^
4 + 5))