3.49 \(\int x^3 \sqrt{1+x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{5} \left (x^2+1\right )^{5/2}-\frac{1}{3} \left (x^2+1\right )^{3/2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0271528, antiderivative size = 27, 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 \[ \frac{1}{5} \left (x^2+1\right )^{5/2}-\frac{1}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[1 + x^2],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.80429, size = 19, normalized size = 0.7 \[ \frac{\left (x^{2} + 1\right )^{\frac{5}{2}}}{5} - \frac{\left (x^{2} + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(x**2+1)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.00782998, size = 20, normalized size = 0.74 \[ \frac{1}{15} \left (x^2+1\right )^{3/2} \left (3 x^2-2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[1 + x^2],x]

[Out]

((1 + x^2)^(3/2)*(-2 + 3*x^2))/15

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 17, normalized size = 0.6 \[{\frac{3\,{x}^{2}-2}{15} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(x^2+1)^(1/2),x)

[Out]

1/15*(x^2+1)^(3/2)*(3*x^2-2)

_______________________________________________________________________________________

Maxima [A]  time = 1.51911, size = 30, normalized size = 1.11 \[ \frac{1}{5} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} x^{2} - \frac{2}{15} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 1)*x^3,x, algorithm="maxima")

[Out]

1/5*(x^2 + 1)^(3/2)*x^2 - 2/15*(x^2 + 1)^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.212709, size = 134, normalized size = 4.96 \[ -\frac{48 \, x^{10} + 100 \, x^{8} + 35 \, x^{6} - 40 \, x^{4} - 25 \, x^{2} -{\left (48 \, x^{9} + 76 \, x^{7} + 3 \, x^{5} - 35 \, x^{3} - 10 \, x\right )} \sqrt{x^{2} + 1} - 2}{15 \,{\left (16 \, x^{5} + 20 \, x^{3} -{\left (16 \, x^{4} + 12 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 5 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 1)*x^3,x, algorithm="fricas")

[Out]

-1/15*(48*x^10 + 100*x^8 + 35*x^6 - 40*x^4 - 25*x^2 - (48*x^9 + 76*x^7 + 3*x^5 -
35*x^3 - 10*x)*sqrt(x^2 + 1) - 2)/(16*x^5 + 20*x^3 - (16*x^4 + 12*x^2 + 1)*sqrt
(x^2 + 1) + 5*x)

_______________________________________________________________________________________

Sympy [A]  time = 0.891849, size = 37, normalized size = 1.37 \[ \frac{x^{4} \sqrt{x^{2} + 1}}{5} + \frac{x^{2} \sqrt{x^{2} + 1}}{15} - \frac{2 \sqrt{x^{2} + 1}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(x**2+1)**(1/2),x)

[Out]

x**4*sqrt(x**2 + 1)/5 + x**2*sqrt(x**2 + 1)/15 - 2*sqrt(x**2 + 1)/15

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.207732, size = 26, normalized size = 0.96 \[ \frac{1}{5} \,{\left (x^{2} + 1\right )}^{\frac{5}{2}} - \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^2 + 1)*x^3,x, algorithm="giac")

[Out]

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