3.376 \(\int x^5 \sqrt{1+x^2} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{7} \left (x^2+1\right )^{7/2}-\frac{2}{5} \left (x^2+1\right )^{5/2}+\frac{1}{3} \left (x^2+1\right )^{3/2} \]

[Out]

(1 + x^2)^(3/2)/3 - (2*(1 + x^2)^(5/2))/5 + (1 + x^2)^(7/2)/7

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Rubi [A]  time = 0.0336456, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{1}{7} \left (x^2+1\right )^{7/2}-\frac{2}{5} \left (x^2+1\right )^{5/2}+\frac{1}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[1 + x^2],x]

[Out]

(1 + x^2)^(3/2)/3 - (2*(1 + x^2)^(5/2))/5 + (1 + x^2)^(7/2)/7

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Rubi in Sympy [A]  time = 2.16245, size = 31, normalized size = 0.78 \[ \frac{\left (x^{2} + 1\right )^{\frac{7}{2}}}{7} - \frac{2 \left (x^{2} + 1\right )^{\frac{5}{2}}}{5} + \frac{\left (x^{2} + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**2 + 1)**(7/2)/7 - 2*(x**2 + 1)**(5/2)/5 + (x**2 + 1)**(3/2)/3

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Mathematica [A]  time = 0.0108458, size = 25, normalized size = 0.62 \[ \frac{1}{105} \left (x^2+1\right )^{3/2} \left (15 x^4-12 x^2+8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[1 + x^2],x]

[Out]

((1 + x^2)^(3/2)*(8 - 12*x^2 + 15*x^4))/105

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Maple [A]  time = 0.007, size = 22, normalized size = 0.6 \[{\frac{15\,{x}^{4}-12\,{x}^{2}+8}{105} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(x^2+1)^(1/2),x)

[Out]

1/105*(x^2+1)^(3/2)*(15*x^4-12*x^2+8)

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Maxima [A]  time = 1.4983, size = 46, normalized size = 1.15 \[ \frac{1}{7} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} x^{4} - \frac{4}{35} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} x^{2} + \frac{8}{105} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 1)*x^5,x, algorithm="maxima")

[Out]

1/7*(x^2 + 1)^(3/2)*x^4 - 4/35*(x^2 + 1)^(3/2)*x^2 + 8/105*(x^2 + 1)^(3/2)

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Fricas [A]  time = 0.205229, size = 174, normalized size = 4.35 \[ -\frac{960 \, x^{14} + 2352 \, x^{12} + 1736 \, x^{10} + 623 \, x^{8} + 826 \, x^{6} + 735 \, x^{4} + 196 \, x^{2} -{\left (960 \, x^{13} + 1872 \, x^{11} + 920 \, x^{9} + 337 \, x^{7} + 693 \, x^{5} + 420 \, x^{3} + 56 \, x\right )} \sqrt{x^{2} + 1} + 8}{105 \,{\left (64 \, x^{7} + 112 \, x^{5} + 56 \, x^{3} -{\left (64 \, x^{6} + 80 \, x^{4} + 24 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 7 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 1)*x^5,x, algorithm="fricas")

[Out]

-1/105*(960*x^14 + 2352*x^12 + 1736*x^10 + 623*x^8 + 826*x^6 + 735*x^4 + 196*x^2
- (960*x^13 + 1872*x^11 + 920*x^9 + 337*x^7 + 693*x^5 + 420*x^3 + 56*x)*sqrt(x^
2 + 1) + 8)/(64*x^7 + 112*x^5 + 56*x^3 - (64*x^6 + 80*x^4 + 24*x^2 + 1)*sqrt(x^2
+ 1) + 7*x)

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Sympy [A]  time = 3.17738, size = 53, normalized size = 1.32 \[ \frac{x^{6} \sqrt{x^{2} + 1}}{7} + \frac{x^{4} \sqrt{x^{2} + 1}}{35} - \frac{4 x^{2} \sqrt{x^{2} + 1}}{105} + \frac{8 \sqrt{x^{2} + 1}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**6*sqrt(x**2 + 1)/7 + x**4*sqrt(x**2 + 1)/35 - 4*x**2*sqrt(x**2 + 1)/105 + 8*s
qrt(x**2 + 1)/105

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GIAC/XCAS [A]  time = 0.199367, size = 38, normalized size = 0.95 \[ \frac{1}{7} \,{\left (x^{2} + 1\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{2} + 1\right )}^{\frac{5}{2}} + \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 1)*x^5,x, algorithm="giac")

[Out]

1/7*(x^2 + 1)^(7/2) - 2/5*(x^2 + 1)^(5/2) + 1/3*(x^2 + 1)^(3/2)