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

Optimal. Leaf size=119 \[ \frac{1}{48} \left (18 x^2+61\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{1}{128} \left (199-74 x^2\right ) \sqrt{x^4+5 x^2+3}+\frac{2401}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-3 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

((199 - 74*x^2)*Sqrt[3 + 5*x^2 + x^4])/128 + ((61 + 18*x^2)*(3 + 5*x^2 + x^4)^(3
/2))/48 + (2401*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/256 - 3*Sqrt[3]*
ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]

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Rubi [A]  time = 0.256481, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{1}{48} \left (18 x^2+61\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{1}{128} \left (199-74 x^2\right ) \sqrt{x^4+5 x^2+3}+\frac{2401}{256} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-3 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((199 - 74*x^2)*Sqrt[3 + 5*x^2 + x^4])/128 + ((61 + 18*x^2)*(3 + 5*x^2 + x^4)^(3
/2))/48 + (2401*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/256 - 3*Sqrt[3]*
ArcTanh[(6 + 5*x^2)/(2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4])]

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Rubi in Sympy [A]  time = 27.0119, size = 110, normalized size = 0.92 \[ \frac{\left (- \frac{37 x^{2}}{2} + \frac{199}{4}\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{32} + \frac{\left (9 x^{2} + \frac{61}{2}\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{24} + \frac{2401 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{256} - 3 \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \left (5 x^{2} + 6\right )}{6 \sqrt{x^{4} + 5 x^{2} + 3}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-37*x**2/2 + 199/4)*sqrt(x**4 + 5*x**2 + 3)/32 + (9*x**2 + 61/2)*(x**4 + 5*x**2
+ 3)**(3/2)/24 + 2401*atanh((2*x**2 + 5)/(2*sqrt(x**4 + 5*x**2 + 3)))/256 - 3*s
qrt(3)*atanh(sqrt(3)*(5*x**2 + 6)/(6*sqrt(x**4 + 5*x**2 + 3)))

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Mathematica [A]  time = 0.139443, size = 107, normalized size = 0.9 \[ \frac{2401}{256} \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )+3 \sqrt{3} \left (2 \log (x)-\log \left (5 x^2+2 \sqrt{3} \sqrt{x^4+5 x^2+3}+6\right )\right )+\frac{1}{384} \sqrt{x^4+5 x^2+3} \left (144 x^6+1208 x^4+2650 x^2+2061\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[3 + 5*x^2 + x^4]*(2061 + 2650*x^2 + 1208*x^4 + 144*x^6))/384 + (2401*Log[5
+ 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/256 + 3*Sqrt[3]*(2*Log[x] - Log[6 + 5*x^2 +
2*Sqrt[3]*Sqrt[3 + 5*x^2 + x^4]])

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Maple [A]  time = 0.022, size = 117, normalized size = 1. \[{\frac{151\,{x}^{4}}{48}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{1325\,{x}^{2}}{192}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{687}{128}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{2401}{256}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }-3\,{\it Artanh} \left ( 1/6\,{\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}} \right ) \sqrt{3}+{\frac{3\,{x}^{6}}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

151/48*x^4*(x^4+5*x^2+3)^(1/2)+1325/192*x^2*(x^4+5*x^2+3)^(1/2)+687/128*(x^4+5*x
^2+3)^(1/2)+2401/256*ln(x^2+5/2+(x^4+5*x^2+3)^(1/2))-3*arctanh(1/6*(5*x^2+6)*3^(
1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)+3/8*x^6*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.818915, size = 162, normalized size = 1.36 \[ \frac{3}{8} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} x^{2} - \frac{37}{64} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{61}{48} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} - 3 \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{199}{128} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{2401}{256} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

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,x, algorithm="maxima")

[Out]

3/8*(x^4 + 5*x^2 + 3)^(3/2)*x^2 - 37/64*sqrt(x^4 + 5*x^2 + 3)*x^2 + 61/48*(x^4 +
5*x^2 + 3)^(3/2) - 3*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x^2 + 3)/x^2 + 6/x^2 +
5) + 199/128*sqrt(x^4 + 5*x^2 + 3) + 2401/256*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3
) + 5)

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Fricas [A]  time = 0.265556, size = 491, normalized size = 4.13 \[ -\frac{294912 \, x^{16} + 6160384 \, x^{14} + 53346304 \, x^{12} + 249921536 \, x^{10} + 691886464 \, x^{8} + 1153924864 \, x^{6} + 1117721440 \, x^{4} + 560180576 \, x^{2} + 57624 \,{\left (128 \, x^{8} + 1280 \, x^{6} + 4384 \, x^{4} + 5920 \, x^{2} - 8 \,{\left (16 \, x^{6} + 120 \, x^{4} + 274 \, x^{2} + 185\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 2569\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) + 18432 \,{\left (8 \, \sqrt{3}{\left (16 \, x^{6} + 120 \, x^{4} + 274 \, x^{2} + 185\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} - \sqrt{3}{\left (128 \, x^{8} + 1280 \, x^{6} + 4384 \, x^{4} + 5920 \, x^{2} + 2569\right )}\right )} \log \left (\frac{2 \, x^{4} + 2 \, \sqrt{3} x^{2} + 5 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (x^{2} + \sqrt{3}\right )} + 6}{2 \, x^{4} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + 5 \, x^{2}}\right ) - 8 \,{\left (36864 \, x^{14} + 677888 \, x^{12} + 5033472 \, x^{10} + 19608320 \, x^{8} + 43313552 \, x^{6} + 53850552 \, x^{4} + 33901090 \, x^{2} + 7809793\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 107814215}{6144 \,{\left (128 \, x^{8} + 1280 \, x^{6} + 4384 \, x^{4} + 5920 \, x^{2} - 8 \,{\left (16 \, x^{6} + 120 \, x^{4} + 274 \, x^{2} + 185\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 2569\right )}} \]

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,x, algorithm="fricas")

[Out]

-1/6144*(294912*x^16 + 6160384*x^14 + 53346304*x^12 + 249921536*x^10 + 691886464
*x^8 + 1153924864*x^6 + 1117721440*x^4 + 560180576*x^2 + 57624*(128*x^8 + 1280*x
^6 + 4384*x^4 + 5920*x^2 - 8*(16*x^6 + 120*x^4 + 274*x^2 + 185)*sqrt(x^4 + 5*x^2
+ 3) + 2569)*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) + 18432*(8*sqrt(3)*(16*x
^6 + 120*x^4 + 274*x^2 + 185)*sqrt(x^4 + 5*x^2 + 3) - sqrt(3)*(128*x^8 + 1280*x^
6 + 4384*x^4 + 5920*x^2 + 2569))*log((2*x^4 + 2*sqrt(3)*x^2 + 5*x^2 - 2*sqrt(x^4
+ 5*x^2 + 3)*(x^2 + sqrt(3)) + 6)/(2*x^4 - 2*sqrt(x^4 + 5*x^2 + 3)*x^2 + 5*x^2)
) - 8*(36864*x^14 + 677888*x^12 + 5033472*x^10 + 19608320*x^8 + 43313552*x^6 + 5
3850552*x^4 + 33901090*x^2 + 7809793)*sqrt(x^4 + 5*x^2 + 3) + 107814215)/(128*x^
8 + 1280*x^6 + 4384*x^4 + 5920*x^2 - 8*(16*x^6 + 120*x^4 + 274*x^2 + 185)*sqrt(x
^4 + 5*x^2 + 3) + 2569)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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,x, algorithm="giac")

[Out]

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