3.235 \(\int \left (b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5} \]

[Out]

(b^2*x^3)/3 + (b*c*x^4)/2 + (c^2*x^5)/5

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Rubi [A]  time = 0.0383359, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^2,x]

[Out]

(b^2*x^3)/3 + (b*c*x^4)/2 + (c^2*x^5)/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b^{3} \int x\, dx}{5 c} - \frac{b^{2} x^{3}}{15} + \frac{\left (b + 2 c x\right ) \left (b x + c x^{2}\right )^{2}}{10 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**2,x)

[Out]

-b**3*Integral(x, x)/(5*c) - b**2*x**3/15 + (b + 2*c*x)*(b*x + c*x**2)**2/(10*c)

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Mathematica [A]  time = 0.00276049, size = 30, normalized size = 1. \[ \frac{b^2 x^3}{3}+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^2,x]

[Out]

(b^2*x^3)/3 + (b*c*x^4)/2 + (c^2*x^5)/5

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \[{\frac{{b}^{2}{x}^{3}}{3}}+{\frac{bc{x}^{4}}{2}}+{\frac{{c}^{2}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^2,x)

[Out]

1/3*b^2*x^3+1/2*b*c*x^4+1/5*c^2*x^5

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Maxima [A]  time = 0.691813, size = 32, normalized size = 1.07 \[ \frac{1}{5} \, c^{2} x^{5} + \frac{1}{2} \, b c x^{4} + \frac{1}{3} \, b^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*c^2*x^5 + 1/2*b*c*x^4 + 1/3*b^2*x^3

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Fricas [A]  time = 0.190344, size = 1, normalized size = 0.03 \[ \frac{1}{5} x^{5} c^{2} + \frac{1}{2} x^{4} c b + \frac{1}{3} x^{3} b^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*x^5*c^2 + 1/2*x^4*c*b + 1/3*x^3*b^2

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Sympy [A]  time = 0.085504, size = 24, normalized size = 0.8 \[ \frac{b^{2} x^{3}}{3} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**2,x)

[Out]

b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5

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GIAC/XCAS [A]  time = 0.206736, size = 32, normalized size = 1.07 \[ \frac{1}{5} \, c^{2} x^{5} + \frac{1}{2} \, b c x^{4} + \frac{1}{3} \, b^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*c^2*x^5 + 1/2*b*c*x^4 + 1/3*b^2*x^3