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

Optimal. Leaf size=83 \[ \frac{3}{14} \left (x^4+5\right )^{5/2} x^4-\frac{125}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{5}{24} \left (x^4+5\right )^{3/2} x^2-\frac{25}{16} \sqrt{x^4+5} x^2-\frac{1}{42} \left (18-7 x^2\right ) \left (x^4+5\right )^{5/2} \]

[Out]

(-25*x^2*Sqrt[5 + x^4])/16 - (5*x^2*(5 + x^4)^(3/2))/24 + (3*x^4*(5 + x^4)^(5/2)
)/14 - ((18 - 7*x^2)*(5 + x^4)^(5/2))/42 - (125*ArcSinh[x^2/Sqrt[5]])/16

_______________________________________________________________________________________

Rubi [A]  time = 0.157682, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{3}{14} \left (x^4+5\right )^{5/2} x^4-\frac{125}{16} \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )-\frac{5}{24} \left (x^4+5\right )^{3/2} x^2-\frac{25}{16} \sqrt{x^4+5} x^2-\frac{1}{42} \left (18-7 x^2\right ) \left (x^4+5\right )^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(-25*x^2*Sqrt[5 + x^4])/16 - (5*x^2*(5 + x^4)^(3/2))/24 + (3*x^4*(5 + x^4)^(5/2)
)/14 - ((18 - 7*x^2)*(5 + x^4)^(5/2))/42 - (125*ArcSinh[x^2/Sqrt[5]])/16

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.7038, size = 76, normalized size = 0.92 \[ \frac{3 x^{4} \left (x^{4} + 5\right )^{\frac{5}{2}}}{14} - \frac{5 x^{2} \left (x^{4} + 5\right )^{\frac{3}{2}}}{24} - \frac{25 x^{2} \sqrt{x^{4} + 5}}{16} - \frac{\left (- 70 x^{2} + 180\right ) \left (x^{4} + 5\right )^{\frac{5}{2}}}{420} - \frac{125 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**4*(x**4 + 5)**(5/2)/14 - 5*x**2*(x**4 + 5)**(3/2)/24 - 25*x**2*sqrt(x**4 +
5)/16 - (-70*x**2 + 180)*(x**4 + 5)**(5/2)/420 - 125*asinh(sqrt(5)*x**2/5)/16

_______________________________________________________________________________________

Mathematica [A]  time = 0.0546489, size = 59, normalized size = 0.71 \[ \frac{1}{336} \left (\sqrt{x^4+5} \left (72 x^{12}+56 x^{10}+576 x^8+490 x^6+360 x^4+525 x^2-3600\right )-2625 \sinh ^{-1}\left (\frac{x^2}{\sqrt{5}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[5 + x^4]*(-3600 + 525*x^2 + 360*x^4 + 490*x^6 + 576*x^8 + 56*x^10 + 72*x^1
2) - 2625*ArcSinh[x^2/Sqrt[5]])/336

_______________________________________________________________________________________

Maple [A]  time = 0.032, size = 73, normalized size = 0.9 \[{\frac{{x}^{10}}{6}\sqrt{{x}^{4}+5}}+{\frac{35\,{x}^{6}}{24}\sqrt{{x}^{4}+5}}+{\frac{25\,{x}^{2}}{16}\sqrt{{x}^{4}+5}}-{\frac{125}{16}{\it Arcsinh} \left ({\frac{\sqrt{5}{x}^{2}}{5}} \right ) }+{\frac{ \left ( 3\,{x}^{4}-6 \right ) \left ({x}^{8}+10\,{x}^{4}+25 \right ) }{14}\sqrt{{x}^{4}+5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*x^10*(x^4+5)^(1/2)+35/24*x^6*(x^4+5)^(1/2)+25/16*x^2*(x^4+5)^(1/2)-125/16*ar
csinh(1/5*5^(1/2)*x^2)+3/14*(x^4+5)^(1/2)*(x^4-2)*(x^8+10*x^4+25)

_______________________________________________________________________________________

Maxima [A]  time = 0.776728, size = 171, normalized size = 2.06 \[ \frac{3}{14} \,{\left (x^{4} + 5\right )}^{\frac{7}{2}} - \frac{3}{2} \,{\left (x^{4} + 5\right )}^{\frac{5}{2}} - \frac{125 \,{\left (\frac{3 \, \sqrt{x^{4} + 5}}{x^{2}} - \frac{8 \,{\left (x^{4} + 5\right )}^{\frac{3}{2}}}{x^{6}} - \frac{3 \,{\left (x^{4} + 5\right )}^{\frac{5}{2}}}{x^{10}}\right )}}{48 \,{\left (\frac{3 \,{\left (x^{4} + 5\right )}}{x^{4}} - \frac{3 \,{\left (x^{4} + 5\right )}^{2}}{x^{8}} + \frac{{\left (x^{4} + 5\right )}^{3}}{x^{12}} - 1\right )}} - \frac{125}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} + 1\right ) + \frac{125}{32} \, \log \left (\frac{\sqrt{x^{4} + 5}}{x^{2}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*(x^4 + 5)^(7/2) - 3/2*(x^4 + 5)^(5/2) - 125/48*(3*sqrt(x^4 + 5)/x^2 - 8*(x^
4 + 5)^(3/2)/x^6 - 3*(x^4 + 5)^(5/2)/x^10)/(3*(x^4 + 5)/x^4 - 3*(x^4 + 5)^2/x^8
+ (x^4 + 5)^3/x^12 - 1) - 125/32*log(sqrt(x^4 + 5)/x^2 + 1) + 125/32*log(sqrt(x^
4 + 5)/x^2 - 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.264553, size = 352, normalized size = 4.24 \[ -\frac{4608 \, x^{28} + 3584 \, x^{26} + 88704 \, x^{24} + 71680 \, x^{22} + 624960 \, x^{20} + 532000 \, x^{18} + 1751400 \, x^{16} + 1827000 \, x^{14} + 189000 \, x^{12} + 2931250 \, x^{10} - 7875000 \, x^{8} + 1946875 \, x^{6} - 11025000 \, x^{4} + 328125 \, x^{2} - 2625 \,{\left (64 \, x^{14} + 560 \, x^{10} + 1400 \, x^{6} + 875 \, x^{2} -{\left (64 \, x^{12} + 400 \, x^{8} + 600 \, x^{4} + 125\right )} \sqrt{x^{4} + 5}\right )} \log \left (-x^{2} + \sqrt{x^{4} + 5}\right ) -{\left (4608 \, x^{26} + 3584 \, x^{24} + 77184 \, x^{22} + 62720 \, x^{20} + 446400 \, x^{18} + 386400 \, x^{16} + 840600 \, x^{14} + 1029000 \, x^{12} - 1008000 \, x^{10} + 1163750 \, x^{8} - 4725000 \, x^{6} + 459375 \, x^{4} - 3150000 \, x^{2}\right )} \sqrt{x^{4} + 5} - 2250000}{336 \,{\left (64 \, x^{14} + 560 \, x^{10} + 1400 \, x^{6} + 875 \, x^{2} -{\left (64 \, x^{12} + 400 \, x^{8} + 600 \, x^{4} + 125\right )} \sqrt{x^{4} + 5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/336*(4608*x^28 + 3584*x^26 + 88704*x^24 + 71680*x^22 + 624960*x^20 + 532000*x
^18 + 1751400*x^16 + 1827000*x^14 + 189000*x^12 + 2931250*x^10 - 7875000*x^8 + 1
946875*x^6 - 11025000*x^4 + 328125*x^2 - 2625*(64*x^14 + 560*x^10 + 1400*x^6 + 8
75*x^2 - (64*x^12 + 400*x^8 + 600*x^4 + 125)*sqrt(x^4 + 5))*log(-x^2 + sqrt(x^4
+ 5)) - (4608*x^26 + 3584*x^24 + 77184*x^22 + 62720*x^20 + 446400*x^18 + 386400*
x^16 + 840600*x^14 + 1029000*x^12 - 1008000*x^10 + 1163750*x^8 - 4725000*x^6 + 4
59375*x^4 - 3150000*x^2)*sqrt(x^4 + 5) - 2250000)/(64*x^14 + 560*x^10 + 1400*x^6
+ 875*x^2 - (64*x^12 + 400*x^8 + 600*x^4 + 125)*sqrt(x^4 + 5))

_______________________________________________________________________________________

Sympy [A]  time = 34.4574, size = 131, normalized size = 1.58 \[ \frac{x^{14}}{6 \sqrt{x^{4} + 5}} + \frac{3 x^{12} \sqrt{x^{4} + 5}}{14} + \frac{55 x^{10}}{24 \sqrt{x^{4} + 5}} + \frac{12 x^{8} \sqrt{x^{4} + 5}}{7} + \frac{425 x^{6}}{48 \sqrt{x^{4} + 5}} + \frac{15 x^{4} \sqrt{x^{4} + 5}}{14} + \frac{125 x^{2}}{16 \sqrt{x^{4} + 5}} - \frac{75 \sqrt{x^{4} + 5}}{7} - \frac{125 \operatorname{asinh}{\left (\frac{\sqrt{5} x^{2}}{5} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**14/(6*sqrt(x**4 + 5)) + 3*x**12*sqrt(x**4 + 5)/14 + 55*x**10/(24*sqrt(x**4 +
5)) + 12*x**8*sqrt(x**4 + 5)/7 + 425*x**6/(48*sqrt(x**4 + 5)) + 15*x**4*sqrt(x**
4 + 5)/14 + 125*x**2/(16*sqrt(x**4 + 5)) - 75*sqrt(x**4 + 5)/7 - 125*asinh(sqrt(
5)*x**2/5)/16

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.263861, size = 88, normalized size = 1.06 \[ \frac{1}{336} \, \sqrt{x^{4} + 5}{\left ({\left (2 \,{\left ({\left (4 \,{\left ({\left (9 \, x^{2} + 7\right )} x^{2} + 72\right )} x^{2} + 245\right )} x^{2} + 180\right )} x^{2} + 525\right )} x^{2} - 3600\right )} + \frac{125}{16} \,{\rm ln}\left (-x^{2} + \sqrt{x^{4} + 5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/336*sqrt(x^4 + 5)*((2*((4*((9*x^2 + 7)*x^2 + 72)*x^2 + 245)*x^2 + 180)*x^2 + 5
25)*x^2 - 3600) + 125/16*ln(-x^2 + sqrt(x^4 + 5))