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

Optimal. Leaf size=13 \[ \frac{1}{3} \left (x^2+1\right )^{3/2} \]

[Out]

(1 + x^2)^(3/2)/3

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Rubi [A]  time = 0.00497958, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[x*Sqrt[1 + x^2],x]

[Out]

(1 + x^2)^(3/2)/3

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Rubi in Sympy [A]  time = 0.76266, size = 8, normalized size = 0.62 \[ \frac{\left (x^{2} + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**2 + 1)**(3/2)/3

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Mathematica [A]  time = 0.00328751, size = 13, normalized size = 1. \[ \frac{1}{3} \left (x^2+1\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[1 + x^2],x]

[Out]

(1 + x^2)^(3/2)/3

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Maple [A]  time = 0.003, size = 10, normalized size = 0.8 \[{\frac{1}{3} \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.34017, size = 12, normalized size = 0.92 \[ \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^2 + 1)^(3/2)

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Fricas [A]  time = 0.234433, size = 93, normalized size = 7.15 \[ -\frac{4 \, x^{6} + 9 \, x^{4} + 6 \, x^{2} -{\left (4 \, x^{5} + 7 \, x^{3} + 3 \, x\right )} \sqrt{x^{2} + 1} + 1}{3 \,{\left (4 \, x^{3} -{\left (4 \, x^{2} + 1\right )} \sqrt{x^{2} + 1} + 3 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(4*x^6 + 9*x^4 + 6*x^2 - (4*x^5 + 7*x^3 + 3*x)*sqrt(x^2 + 1) + 1)/(4*x^3 -
(4*x^2 + 1)*sqrt(x^2 + 1) + 3*x)

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Sympy [A]  time = 0.216007, size = 22, normalized size = 1.69 \[ \frac{x^{2} \sqrt{x^{2} + 1}}{3} + \frac{\sqrt{x^{2} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(x**2+1)**(1/2),x)

[Out]

x**2*sqrt(x**2 + 1)/3 + sqrt(x**2 + 1)/3

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GIAC/XCAS [A]  time = 0.203724, size = 12, normalized size = 0.92 \[ \frac{1}{3} \,{\left (x^{2} + 1\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(x^2 + 1)^(3/2)