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

Optimal. Leaf size=99 \[ \frac{3}{10} \left (x^4+5 x^2+3\right )^{5/2}-\frac{11}{32} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{429}{256} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}-\frac{5577}{512} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

[Out]

(429*(5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/256 - (11*(5 + 2*x^2)*(3 + 5*x^2 + x^4)^
(3/2))/32 + (3*(3 + 5*x^2 + x^4)^(5/2))/10 - (5577*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3
+ 5*x^2 + x^4])])/512

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Rubi [A]  time = 0.127127, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{3}{10} \left (x^4+5 x^2+3\right )^{5/2}-\frac{11}{32} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}+\frac{429}{256} \left (2 x^2+5\right ) \sqrt{x^4+5 x^2+3}-\frac{5577}{512} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(429*(5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/256 - (11*(5 + 2*x^2)*(3 + 5*x^2 + x^4)^
(3/2))/32 + (3*(3 + 5*x^2 + x^4)^(5/2))/10 - (5577*ArcTanh[(5 + 2*x^2)/(2*Sqrt[3
+ 5*x^2 + x^4])])/512

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Rubi in Sympy [A]  time = 12.8707, size = 90, normalized size = 0.91 \[ - \frac{11 \left (2 x^{2} + 5\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{32} + \frac{429 \left (2 x^{2} + 5\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{256} + \frac{3 \left (x^{4} + 5 x^{2} + 3\right )^{\frac{5}{2}}}{10} - \frac{5577 \operatorname{atanh}{\left (\frac{2 x^{2} + 5}{2 \sqrt{x^{4} + 5 x^{2} + 3}} \right )}}{512} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-11*(2*x**2 + 5)*(x**4 + 5*x**2 + 3)**(3/2)/32 + 429*(2*x**2 + 5)*sqrt(x**4 + 5*
x**2 + 3)/256 + 3*(x**4 + 5*x**2 + 3)**(5/2)/10 - 5577*atanh((2*x**2 + 5)/(2*sqr
t(x**4 + 5*x**2 + 3)))/512

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Mathematica [A]  time = 0.0455454, size = 69, normalized size = 0.7 \[ \frac{2 \sqrt{x^4+5 x^2+3} \left (384 x^8+2960 x^6+5304 x^4+2170 x^2+7581\right )-27885 \log \left (2 x^2+2 \sqrt{x^4+5 x^2+3}+5\right )}{2560} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[3 + 5*x^2 + x^4]*(7581 + 2170*x^2 + 5304*x^4 + 2960*x^6 + 384*x^8) - 278
85*Log[5 + 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/2560

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Maple [A]  time = 0.019, size = 104, normalized size = 1.1 \[{\frac{37\,{x}^{6}}{16}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{663\,{x}^{4}}{160}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{217\,{x}^{2}}{128}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{7581}{1280}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{5577}{512}\ln \left ({x}^{2}+{\frac{5}{2}}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }+{\frac{3\,{x}^{8}}{10}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

37/16*x^6*(x^4+5*x^2+3)^(1/2)+663/160*x^4*(x^4+5*x^2+3)^(1/2)+217/128*x^2*(x^4+5
*x^2+3)^(1/2)+7581/1280*(x^4+5*x^2+3)^(1/2)-5577/512*ln(x^2+5/2+(x^4+5*x^2+3)^(1
/2))+3/10*x^8*(x^4+5*x^2+3)^(1/2)

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Maxima [A]  time = 0.722598, size = 136, normalized size = 1.37 \[ -\frac{11}{16} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} x^{2} + \frac{3}{10} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{5}{2}} + \frac{429}{128} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} - \frac{55}{32} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} + \frac{2145}{256} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{5577}{512} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

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]

-11/16*(x^4 + 5*x^2 + 3)^(3/2)*x^2 + 3/10*(x^4 + 5*x^2 + 3)^(5/2) + 429/128*sqrt
(x^4 + 5*x^2 + 3)*x^2 - 55/32*(x^4 + 5*x^2 + 3)^(3/2) + 2145/256*sqrt(x^4 + 5*x^
2 + 3) - 5577/512*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)

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Fricas [A]  time = 0.282536, size = 360, normalized size = 3.64 \[ -\frac{3145728 \, x^{20} + 71434240 \, x^{18} + 684195840 \, x^{16} + 3609026560 \, x^{14} + 11552849920 \, x^{12} + 23642158080 \, x^{10} + 32610490880 \, x^{8} + 32442765280 \, x^{6} + 23399554320 \, x^{4} + 10176374950 \, x^{2} - 223080 \,{\left (512 \, x^{10} + 6400 \, x^{8} + 29920 \, x^{6} + 64400 \, x^{4} + 62690 \, x^{2} - 2 \,{\left (256 \, x^{8} + 2560 \, x^{6} + 8976 \, x^{4} + 12880 \, x^{2} + 6269\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 21725\right )} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 2 \,{\left (1572864 \, x^{18} + 31784960 \, x^{16} + 265191424 \, x^{14} + 1186795520 \, x^{12} + 3129296896 \, x^{10} + 5158310400 \, x^{8} + 5776010240 \, x^{6} + 4755043600 \, x^{4} + 2513463120 \, x^{2} + 506839375\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 1752842119}{20480 \,{\left (512 \, x^{10} + 6400 \, x^{8} + 29920 \, x^{6} + 64400 \, x^{4} + 62690 \, x^{2} - 2 \,{\left (256 \, x^{8} + 2560 \, x^{6} + 8976 \, x^{4} + 12880 \, x^{2} + 6269\right )} \sqrt{x^{4} + 5 \, x^{2} + 3} + 21725\right )}} \]

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]

-1/20480*(3145728*x^20 + 71434240*x^18 + 684195840*x^16 + 3609026560*x^14 + 1155
2849920*x^12 + 23642158080*x^10 + 32610490880*x^8 + 32442765280*x^6 + 2339955432
0*x^4 + 10176374950*x^2 - 223080*(512*x^10 + 6400*x^8 + 29920*x^6 + 64400*x^4 +
62690*x^2 - 2*(256*x^8 + 2560*x^6 + 8976*x^4 + 12880*x^2 + 6269)*sqrt(x^4 + 5*x^
2 + 3) + 21725)*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5) - 2*(1572864*x^18 + 31
784960*x^16 + 265191424*x^14 + 1186795520*x^12 + 3129296896*x^10 + 5158310400*x^
8 + 5776010240*x^6 + 4755043600*x^4 + 2513463120*x^2 + 506839375)*sqrt(x^4 + 5*x
^2 + 3) + 1752842119)/(512*x^10 + 6400*x^8 + 29920*x^6 + 64400*x^4 + 62690*x^2 -
2*(256*x^8 + 2560*x^6 + 8976*x^4 + 12880*x^2 + 6269)*sqrt(x^4 + 5*x^2 + 3) + 21
725)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.276172, size = 90, normalized size = 0.91 \[ \frac{1}{1280} \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (2 \,{\left (4 \,{\left (2 \,{\left (24 \, x^{2} + 185\right )} x^{2} + 663\right )} x^{2} + 1085\right )} x^{2} + 7581\right )} + \frac{5577}{512} \,{\rm ln}\left (2 \, x^{2} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \]

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,x, algorithm="giac")

[Out]

1/1280*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(2*(24*x^2 + 185)*x^2 + 663)*x^2 + 1085)*x^2
+ 7581) + 5577/512*ln(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)