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

Optimal. Leaf size=127 \[ \frac{3}{14} \left (x^4+5 x^2+3\right )^{5/2} x^4+\frac{\left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}}{1680}-\frac{2183}{768} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{28379 \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}}{2048}-\frac{368927 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )}{4096} \]

[Out]

(28379*(5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/2048 - (2183*(5 + 2*x^2)*(3 + 5*x^2 +
x^4)^(3/2))/768 + (3*x^4*(3 + 5*x^2 + x^4)^(5/2))/14 + ((3313 - 1070*x^2)*(3 + 5
*x^2 + x^4)^(5/2))/1680 - (368927*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])]
)/4096

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Rubi [A]  time = 0.229955, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{3}{14} \left (x^4+5 x^2+3\right )^{5/2} x^4+\frac{\left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}}{1680}-\frac{2183}{768} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{28379 \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}}{2048}-\frac{368927 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )}{4096} \]

Antiderivative was successfully verified.

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

[Out]

(28379*(5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/2048 - (2183*(5 + 2*x^2)*(3 + 5*x^2 +
x^4)^(3/2))/768 + (3*x^4*(3 + 5*x^2 + x^4)^(5/2))/14 + ((3313 - 1070*x^2)*(3 + 5
*x^2 + x^4)^(5/2))/1680 - (368927*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])]
)/4096

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Rubi in Sympy [A]  time = 21.1937, size = 119, normalized size = 0.94 \[ \frac{3 x^{4} \left (x^{4} + 5 x^{2} + 3\right )^{\frac{5}{2}}}{14} + \frac{\left (- \frac{535 x^{2}}{2} + \frac{3313}{4}\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{5}{2}}}{420} - \frac{2183 \left (2 x^{2} + 5\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{768} + \frac{28379 \left (2 x^{2} + 5\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{2048} - \frac{368927 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{4096} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x**4*(x**4 + 5*x**2 + 3)**(5/2)/14 + (-535*x**2/2 + 3313/4)*(x**4 + 5*x**2 + 3
)**(5/2)/420 - 2183*(2*x**2 + 5)*(x**4 + 5*x**2 + 3)**(3/2)/768 + 28379*(2*x**2
+ 5)*sqrt(x**4 + 5*x**2 + 3)/2048 - 368927*atanh((2*x**2 + 5)/(2*sqrt(x**4 + 5*x
**2 + 3)))/4096

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Mathematica [A]  time = 0.0672841, size = 79, normalized size = 0.62 \[ \frac{2 \sqrt{x^4+5 x^2+3} \left (46080 x^{12}+323840 x^{10}+482944 x^8+154800 x^6+283304 x^4-1499570 x^2+9546951\right )-38737335 \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )}{430080} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x^2 + x^4]*(9546951 - 1499570*x^2 + 283304*x^4 + 154800*x^6 + 4829
44*x^8 + 323840*x^10 + 46080*x^12) - 38737335*Log[5 + 2*x^2 + 2*Sqrt[3 + 5*x^2 +
x^4]])/430080

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Maple [A]  time = 0.047, size = 138, normalized size = 1.1 \[{\frac{253\,{x}^{10}}{168}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{539\,{x}^{8}}{240}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{645\,{x}^{6}}{896}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{5059\,{x}^{4}}{3840}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{149957\,{x}^{2}}{21504}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{3182317}{71680}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{368927}{4096}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }+{\frac{3\,{x}^{12}}{14}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

253/168*x^10*(x^4+5*x^2+3)^(1/2)+539/240*x^8*(x^4+5*x^2+3)^(1/2)+645/896*x^6*(x^
4+5*x^2+3)^(1/2)+5059/3840*x^4*(x^4+5*x^2+3)^(1/2)-149957/21504*x^2*(x^4+5*x^2+3
)^(1/2)+3182317/71680*(x^4+5*x^2+3)^(1/2)-368927/4096*ln(x^2+5/2+(x^4+5*x^2+3)^(
1/2))+3/14*x^12*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.740455, size = 182, normalized size = 1.43 \[ \frac{3}{14} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{5}{2}} x^{4} - \frac{107}{168} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{5}{2}} x^{2} - \frac{2183}{384} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} x^{2} + \frac{3313}{1680} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{5}{2}} + \frac{28379}{1024} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \frac{10915}{768} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} + \frac{141895}{2048} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{368927}{4096} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/14*(x^4 + 5*x^2 + 3)^(5/2)*x^4 - 107/168*(x^4 + 5*x^2 + 3)^(5/2)*x^2 - 2183/38
4*(x^4 + 5*x^2 + 3)^(3/2)*x^2 + 3313/1680*(x^4 + 5*x^2 + 3)^(5/2) + 28379/1024*s
qrt(x^4 + 5*x^2 + 3)*x^2 - 10915/768*(x^4 + 5*x^2 + 3)^(3/2) + 141895/2048*sqrt(
x^4 + 5*x^2 + 3) - 368927/4096*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 0.261605, size = 468, normalized size = 3.69 \[ -\frac{6039797760 \, x^{28} + 163242311680 \, x^{26} + 1925026152448 \, x^{24} + 13026354790400 \, x^{22} + 56020526170112 \, x^{20} + 160351875891200 \, x^{18} + 311486603624448 \, x^{16} + 417859472998400 \, x^{14} + 446544716914688 \, x^{12} + 647316389099520 \, x^{10} + 1305695545011200 \, x^{8} + 1940788986577280 \, x^{6} + 1629316481183552 \, x^{4} + 665049507193190 \, x^{2} - 309898680 \,{\left (8192 \, x^{14} + 143360 \, x^{12} + 1028608 \, x^{10} + 3897600 \, x^{8} + 8363712 \, x^{6} + 10087840 \, x^{4} + 6288842 \, x^{2} - 2 \,{\left (4096 \, x^{12} + 61440 \, x^{10} + 367360 \, x^{8} + 1113600 \, x^{6} + 1792224 \, x^{4} + 1441120 \, x^{2} + 449203\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 1556105\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 2 \,{\left (3019898880 \, x^{26} + 74071408640 \, x^{24} + 782241890304 \, x^{22} + 4665670369280 \, x^{20} + 17350973194240 \, x^{18} + 41945865584640 \, x^{16} + 66830623850496 \, x^{14} + 72809996984320 \, x^{12} + 79101766213632 \, x^{10} + 159399745182720 \, x^{8} + 307775900170240 \, x^{6} + 334552245101760 \, x^{4} + 168959327923136 \, x^{2} + 27616762406365\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 95657842018463}{3440640 \,{\left (8192 \, x^{14} + 143360 \, x^{12} + 1028608 \, x^{10} + 3897600 \, x^{8} + 8363712 \, x^{6} + 10087840 \, x^{4} + 6288842 \, x^{2} - 2 \,{\left (4096 \, x^{12} + 61440 \, x^{10} + 367360 \, x^{8} + 1113600 \, x^{6} + 1792224 \, x^{4} + 1441120 \, x^{2} + 449203\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 1556105\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3440640*(6039797760*x^28 + 163242311680*x^26 + 1925026152448*x^24 + 130263547
90400*x^22 + 56020526170112*x^20 + 160351875891200*x^18 + 311486603624448*x^16 +
417859472998400*x^14 + 446544716914688*x^12 + 647316389099520*x^10 + 1305695545
011200*x^8 + 1940788986577280*x^6 + 1629316481183552*x^4 + 665049507193190*x^2 -
309898680*(8192*x^14 + 143360*x^12 + 1028608*x^10 + 3897600*x^8 + 8363712*x^6 +
10087840*x^4 + 6288842*x^2 - 2*(4096*x^12 + 61440*x^10 + 367360*x^8 + 1113600*x
^6 + 1792224*x^4 + 1441120*x^2 + 449203)*sqrt(x^4 + 5*x^2 + 3) + 1556105)*log(-2
*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) - 2*(3019898880*x^26 + 74071408640*x^24 + 78
2241890304*x^22 + 4665670369280*x^20 + 17350973194240*x^18 + 41945865584640*x^16
+ 66830623850496*x^14 + 72809996984320*x^12 + 79101766213632*x^10 + 15939974518
2720*x^8 + 307775900170240*x^6 + 334552245101760*x^4 + 168959327923136*x^2 + 276
16762406365)*sqrt(x^4 + 5*x^2 + 3) + 95657842018463)/(8192*x^14 + 143360*x^12 +
1028608*x^10 + 3897600*x^8 + 8363712*x^6 + 10087840*x^4 + 6288842*x^2 - 2*(4096*
x^12 + 61440*x^10 + 367360*x^8 + 1113600*x^6 + 1792224*x^4 + 1441120*x^2 + 44920
3)*sqrt(x^4 + 5*x^2 + 3) + 1556105)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{5} \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.282346, size = 109, normalized size = 0.86 \[ \frac{1}{215040} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \,{\left (36 \, x^{2} + 253\right )} x^{2} + 3773\right )} x^{2} + 9675\right )} x^{2} + 35413\right )} x^{2} - 749785\right )} x^{2} + 9546951\right )} + \frac{368927}{4096} \,{\rm ln}\left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/215040*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(2*(8*(10*(36*x^2 + 253)*x^2 + 3773)*x^2 +
9675)*x^2 + 35413)*x^2 - 749785)*x^2 + 9546951) + 368927/4096*ln(2*x^2 - 2*sqrt(
x^4 + 5*x^2 + 3) + 5)