3.10 \(\int \frac{\left (a x^2+b x^3+c x^4\right )^2}{x^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7} \]

[Out]

(a^2*x^3)/3 + (a*b*x^4)/2 + ((b^2 + 2*a*c)*x^5)/5 + (b*c*x^6)/3 + (c^2*x^7)/7

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Rubi [A]  time = 0.0606733, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^2 + b*x^3 + c*x^4)^2/x^2,x]

[Out]

(a^2*x^3)/3 + (a*b*x^4)/2 + ((b^2 + 2*a*c)*x^5)/5 + (b*c*x^6)/3 + (c^2*x^7)/7

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Rubi in Sympy [A]  time = 13.1869, size = 48, normalized size = 0.89 \[ \frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b c x^{6}}{3} + \frac{c^{2} x^{7}}{7} + x^{5} \left (\frac{2 a c}{5} + \frac{b^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**3+a*x**2)**2/x**2,x)

[Out]

a**2*x**3/3 + a*b*x**4/2 + b*c*x**6/3 + c**2*x**7/7 + x**5*(2*a*c/5 + b**2/5)

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Mathematica [A]  time = 0.0101182, size = 54, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{1}{5} x^5 \left (2 a c+b^2\right )+\frac{1}{2} a b x^4+\frac{1}{3} b c x^6+\frac{c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^2 + b*x^3 + c*x^4)^2/x^2,x]

[Out]

(a^2*x^3)/3 + (a*b*x^4)/2 + ((b^2 + 2*a*c)*x^5)/5 + (b*c*x^6)/3 + (c^2*x^7)/7

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Maple [A]  time = 0.001, size = 45, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{3}}{3}}+{\frac{ab{x}^{4}}{2}}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{5}}{5}}+{\frac{bc{x}^{6}}{3}}+{\frac{{c}^{2}{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^3+a*x^2)^2/x^2,x)

[Out]

1/3*a^2*x^3+1/2*a*b*x^4+1/5*(2*a*c+b^2)*x^5+1/3*b*c*x^6+1/7*c^2*x^7

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Maxima [A]  time = 0.752253, size = 59, normalized size = 1.09 \[ \frac{1}{7} \, c^{2} x^{7} + \frac{1}{3} \, b c x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{5} \,{\left (b^{2} + 2 \, a c\right )} x^{5} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^3 + a*x^2)^2/x^2,x, algorithm="maxima")

[Out]

1/7*c^2*x^7 + 1/3*b*c*x^6 + 1/2*a*b*x^4 + 1/5*(b^2 + 2*a*c)*x^5 + 1/3*a^2*x^3

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Fricas [A]  time = 0.267614, size = 59, normalized size = 1.09 \[ \frac{1}{7} \, c^{2} x^{7} + \frac{1}{3} \, b c x^{6} + \frac{1}{2} \, a b x^{4} + \frac{1}{5} \,{\left (b^{2} + 2 \, a c\right )} x^{5} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^3 + a*x^2)^2/x^2,x, algorithm="fricas")

[Out]

1/7*c^2*x^7 + 1/3*b*c*x^6 + 1/2*a*b*x^4 + 1/5*(b^2 + 2*a*c)*x^5 + 1/3*a^2*x^3

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Sympy [A]  time = 0.107811, size = 48, normalized size = 0.89 \[ \frac{a^{2} x^{3}}{3} + \frac{a b x^{4}}{2} + \frac{b c x^{6}}{3} + \frac{c^{2} x^{7}}{7} + x^{5} \left (\frac{2 a c}{5} + \frac{b^{2}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**3+a*x**2)**2/x**2,x)

[Out]

a**2*x**3/3 + a*b*x**4/2 + b*c*x**6/3 + c**2*x**7/7 + x**5*(2*a*c/5 + b**2/5)

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GIAC/XCAS [A]  time = 0.260175, size = 62, normalized size = 1.15 \[ \frac{1}{7} \, c^{2} x^{7} + \frac{1}{3} \, b c x^{6} + \frac{1}{5} \, b^{2} x^{5} + \frac{2}{5} \, a c x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, a^{2} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^3 + a*x^2)^2/x^2,x, algorithm="giac")

[Out]

1/7*c^2*x^7 + 1/3*b*c*x^6 + 1/5*b^2*x^5 + 2/5*a*c*x^5 + 1/2*a*b*x^4 + 1/3*a^2*x^
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