3.2355 \(\int (1+x)^3 \sqrt{2+2 x+x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{1}{5} (x+1)^2 \left (x^2+2 x+2\right )^{3/2}-\frac{2}{15} \left (x^2+2 x+2\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0345012, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{5} (x+1)^2 \left (x^2+2 x+2\right )^{3/2}-\frac{2}{15} \left (x^2+2 x+2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^3*Sqrt[2 + 2*x + x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.84433, size = 32, normalized size = 0.84 \[ \frac{\left (x + 1\right )^{2} \left (x^{2} + 2 x + 2\right )^{\frac{3}{2}}}{5} - \frac{2 \left (x^{2} + 2 x + 2\right )^{\frac{3}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0193852, size = 26, normalized size = 0.68 \[ \frac{1}{15} \left (x^2+2 x+2\right )^{3/2} \left (3 x^2+6 x+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)^3*Sqrt[2 + 2*x + x^2],x]

[Out]

((2 + 2*x + x^2)^(3/2)*(1 + 6*x + 3*x^2))/15

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Maple [A]  time = 0.006, size = 23, normalized size = 0.6 \[{\frac{3\,{x}^{2}+6\,x+1}{15} \left ({x}^{2}+2\,x+2 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(x^2+2*x+2)^(3/2)*(3*x^2+6*x+1)

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Maxima [A]  time = 0.75329, size = 55, normalized size = 1.45 \[ \frac{1}{5} \,{\left (x^{2} + 2 \, x + 2\right )}^{\frac{3}{2}} x^{2} + \frac{2}{5} \,{\left (x^{2} + 2 \, x + 2\right )}^{\frac{3}{2}} x + \frac{1}{15} \,{\left (x^{2} + 2 \, x + 2\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222819, size = 227, normalized size = 5.97 \[ -\frac{48 \, x^{10} + 480 \, x^{9} + 2260 \, x^{8} + 6560 \, x^{7} + 12915 \, x^{6} + 17922 \, x^{5} + 17645 \, x^{4} + 12080 \, x^{3} + 5440 \, x^{2} -{\left (48 \, x^{9} + 432 \, x^{8} + 1804 \, x^{7} + 4564 \, x^{6} + 7647 \, x^{5} + 8739 \, x^{4} + 6751 \, x^{3} + 3357 \, x^{2} + 952 \, x + 111\right )} \sqrt{x^{2} + 2 \, x + 2} + 1425 \, x + 157}{15 \,{\left (16 \, x^{5} + 80 \, x^{4} + 180 \, x^{3} + 220 \, x^{2} -{\left (16 \, x^{4} + 64 \, x^{3} + 108 \, x^{2} + 88 \, x + 29\right )} \sqrt{x^{2} + 2 \, x + 2} + 145 \, x + 41\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*(48*x^10 + 480*x^9 + 2260*x^8 + 6560*x^7 + 12915*x^6 + 17922*x^5 + 17645*x
^4 + 12080*x^3 + 5440*x^2 - (48*x^9 + 432*x^8 + 1804*x^7 + 4564*x^6 + 7647*x^5 +
8739*x^4 + 6751*x^3 + 3357*x^2 + 952*x + 111)*sqrt(x^2 + 2*x + 2) + 1425*x + 15
7)/(16*x^5 + 80*x^4 + 180*x^3 + 220*x^2 - (16*x^4 + 64*x^3 + 108*x^2 + 88*x + 29
)*sqrt(x^2 + 2*x + 2) + 145*x + 41)

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Sympy [A]  time = 0.660184, size = 85, normalized size = 2.24 \[ \frac{x^{4} \sqrt{x^{2} + 2 x + 2}}{5} + \frac{4 x^{3} \sqrt{x^{2} + 2 x + 2}}{5} + \frac{19 x^{2} \sqrt{x^{2} + 2 x + 2}}{15} + \frac{14 x \sqrt{x^{2} + 2 x + 2}}{15} + \frac{2 \sqrt{x^{2} + 2 x + 2}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*sqrt(x**2 + 2*x + 2)/5 + 4*x**3*sqrt(x**2 + 2*x + 2)/5 + 19*x**2*sqrt(x**2
+ 2*x + 2)/15 + 14*x*sqrt(x**2 + 2*x + 2)/15 + 2*sqrt(x**2 + 2*x + 2)/15

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GIAC/XCAS [A]  time = 0.205617, size = 38, normalized size = 1. \[ \frac{1}{15} \,{\left ({\left ({\left (3 \,{\left (x + 4\right )} x + 19\right )} x + 14\right )} x + 2\right )} \sqrt{x^{2} + 2 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(((3*(x + 4)*x + 19)*x + 14)*x + 2)*sqrt(x^2 + 2*x + 2)